WO2000048130A1 - Method and apparatus for animating two-dimensional and three-dimensional images - Google Patents

Abstract

The invention concerns a method of animating an image comprising a chain of N segments i (i = 0, 1, 2, ..., N-1) between a pseudo-root segment 0 and a pseudo-leaf end segment N-1, each segment being defined by an orientation in space (α, β, γ, the pseudo-leaf segment having to reach a target angular orientation (αtarget, βtarget, γtarget), the method comprising the following steps: defining an image field function comprising: an attractive component associated with the target angular orientation and of zero value for that orientation, and if angular constraints are to be complied with, a repulsive component associated with each constraint, determining the motion of each segment from an equation of motion as a function of the field for the initial orientation of that segment, and animating the chain of segments in accordance with the motion thus determined. The method can be implemented using geometrical transformations or quaternions.

Description

METHOD AND APPARATUS FOR ANIMATING TWO-DIMENSIONAL AND THREE-DIMENSIONAL IMAGES

Technical field: prior art The invention is generally concerned with generating two-dimensional or three-dimensional animated shapes. Although shapes, in particular three-dimensional shapes, can now be generated, animating them in real time still represents an unsolved technical problem.

Various fields of activity, such as video games and cinema film production, require tools for generating shapes, in particular three-dimensional shapes, and simultaneously animating and/or modifying them in real time. Similarly, in the field of computer-aided design of technical components, it is important not only to be able to represent the components but also to be able to observe their behavior in the event of simulated deformation.

At present there is a great disparity between what is needed in the above fields and the solutions available.

"Real time" methods exist, but require the use of large computer systems .

Prior art methods in this field first generate three-dimensional shapes in a conventional way and then "add" motion, in the same way as when creating an animated drawing. The prior art methods have little capacity for animation. In a field like video games, animation capability and motion are essential, much more so than is realism of the non-animated shape itself. Software for approximating a real image by means of a skeleton is available. Such software has a "tree" structure. It is not applicable to real time transformation or to interactive fields.

Another aspect of the problem currently faced is that of three-dimensional models whose data can be sent via the Internet, including data for animating such models, and possibly audio data. The Internet uses telephone lines with a low bit rate. This information carrier is therefore unable to transmit data defining shapes, in particular moving shapes, which are conventionally defined by a large number of points. The problem therefore arises of generating three-dimensional objects using code that is sufficiently optimized for transfer and control of information via the Internet, comprising information concerning the motion of the object concerned. Another technical problem that arises, independently of the problem of transmission via the Internet, is that of storing objects in the form of efficient and very compact data, in real time, with a low demand for memory. The invention also concerns inverse kinematics. This technique is used to animate 3D models.

For example, controlling a 3D universe and animating 3D characters in real time, amounts to more than geometrical computations and fast display of textures. Enabling virtual beings to "live" and to be fully animated in real time requires complex algorithmic computations such as inverse kinematics.

Inverse kinematics is used to compute the motion of an object from the "target" of the motion, for example. To be more precise, inverse kinematics is a method of controlling hierarchical structures in which the user determines the positions of the objects at the extremity of the hierarchy tree. The position and orientation of all the other objects of the tree are then determined. Hierarchizing a skeleton begins with placing the highest objects in the hierarchy (the "parents") and then descending the hierarchy to position each "child". Inverse kinematics is a control method based on the concept of a target: a "child" is placed and then the positions and orientations of the "parents" are computed. Inverse kinematic type computation techniques are used already (for example, the Jacobian method) , but in practice they cannot be used in real time unless very powerful computers are used. One aim of the invention is therefore to make a contribution to these inverse kinematic methods to the extent that this is compatible with "real time" use. Inverse kinematics in accordance with the invention are also used to model collisions. Turning to another aspect, in "conventional" animation, it is necessary to create animation keys for all the key points of the objects (for example, for the arm of a character: the shoulder, the forearm, the arm, the hand, etc.) . The other intermediate positions of the character are then generated by interpolation. This obliges the animator to create all these keys manually.

Moreover, each animation key is then stored in the form of a complete 3D model of the object or character, and all these models have to be stored, compressed and computed. Accordingly, 10 animations each with 10 keys necessitate the creation and storage of 100 different 3D models of the same object, which represents a vast amount of data, especially if the models are to be transmitted via the Internet. Even using the NURBS technology, this amounts to 500 kilobytes of data.

The problem therefore arises of finding an "inverse kinematic" type technique compatible with real time. The problem also arises of finding an "inverse kinematic" type technique that is economical in terms of memory occupancy.

Summary of the invention First of all, the invention provides a method of animating an image comprising a chain of N segments i_ (i = 0, 1, 2, ..., N-l) between a segment 0 called a pseudo- root segment and an end segment N-l called a pseudo-leaf segment, each segment being defined by an orientation in space (α, β, ψ) , the pseudo-leaf segment having to reach a target angular orientation (α_targetr βtargetr ψtarget) , the method comprising the following steps:

- defining an image field function comprising: • an attractive component associated with the target angular orientation and of zero value for that orientation, and

• if angular constraints are to be complied with, a repulsive component associated with each constraint,

- determining the motion of each segment from an equation of motion as a function of the field for the initial orientation of that segment, and

- animating the chain of segments in accordance with the motion thus determined.

The invention also concerns a method of sending animation data of an image comprising a chain of segments, between a pseudo-root segment and a pseudo-leaf end segment as defined above, each segment being defined by an orientation in space (α, β, ψ) , the pseudo-leaf segment having to reach a target angular orientation (o-targetr βtargetr ψtarget) / the method comprising the following steps:

- defining an image field function comprising: • an attractive component associated with the target angular orientation and of zero value for that orientation, and

• if angular constraints have to be complied with, a repulsive component associated with each constraint,

- determining the motion of each segment from an equation of motion as a function of the field for the initial orientation of the segment, and

- sending over a communications network data relating to the segments (for example: angular orientations) and the motion thus determined.

In accordance with the invention, an image field value can be associated with any segment of a shape or skeleton to be animated, represented on the image, for any discrete time n. The field is a vector function E defined for any segment of the shape or part of the shape and any orientation of the segment. Any segment that is part of the shape can be identified in a system of angular coordinates by three angles (α, β, ψ) , for example. The invention adds to these angular coordinates "influence" coordinates or parameters which in fact correspond to the value of the image field at the segment concerned having a defined orientation at the time n concerned. Consequently, with these additional coordinates, it is possible to define the influences to which each segment of the shape is subject and to determine the appropriate motion for the pseudo-leaf to reach its target.

A second embodiment adds to the field or potential coordinates dynamic coordinates ki and k_a which are called respectively an "inertia" parameter and a "flexibility" parameter. The flexibility (and therefore the coefficient k_a) represents the fact that the object at the point concerned is damped to a greater or lesser degree according to the degree to which it is influenced by other points or other parts of the object. The inertia corresponds to the concept of inertia in classical dynamics (mathematical parameter) .

According to the invention, the description of the motion is therefore contained in the definition of the additional dynamic coordinates. The description of the motion is therefore economical in terms of memory occupancy in a computing device implementing the method of the invention.

The method of the invention provides for immediate modification of the dynamic coordinates constituted by E_n (the value of the field for a particular segment) or its potential and/or k_a and/or ki (damping and inertia coefficients) . It therefore provides for a high level of interaction .

The invention also concerns a method of animating an image or for sending animation data of an image via a communications network, the image comprising a chain of N segments i (i = 0, 1, 2, ..., N-l) between a segment 0 called a pseudo-root segment and an end segment N-l called a pseudo-leaf segment, each segment i_ being identified relative to the father segment i-1 of the segment i_ by a geometrical transformation dTRF₍₁₎ = dTRF ₎*δTRF₍₁₎ made up of a rest transformation dTRF ₎ and a transformation δTRF₍₁₎ representing the motion, each of these transformations comprising an associated rotational component dROT₍₁₎, dROT0₍₁₎ and δROT₍₁₎; the method comprises the following steps: 1. determining a target to be reached for the pseudo-leaf,

2. selecting the pseudo-leaf as the current segment,

3. computing for the current segment the rotation δROTB₍₁₎ to reach the target fixed for that segment, 4. determining the target to be reached for the father segment of the current segment,

5. selecting the father segment of the current segment as the current segment and reiterating step 3,

6. moving the chain of segments as a function of the rotations thus determined, or for sending the data concerning the chain of segments and the rotations thus determined via a communications network.

Each target to be reached can be determined by three angles C⁽.t_arget r βt≤irget / ψt_arget • According to this method, each segment, or bone, operates as if it were isolated from the others but makes its contribution to the pseudo-leaf reaching its target. The action effected by a bone depends on the action effected by the other bones: if one bone tends towards an attractor, this modifies the attractor of the parent bones, and therefore modifies their action, although they have exactly the same mechanism.

According to this logic, which starts from the "pseudo-leaf" and works back to the pseudo-root, the target of the father segment depends on the target of the son segment.

The invention also provides a method of moving and/or animating an articulated system displayed on an image, the system comprising N segments i (i = 0, 1, 2, ..., N-l) between segment i = 0 called a root segment and an end-segment i = N-l called a pseudo-leaf segment and the segments being joined in pairs by articulations, a target orientation of the pseudo-leaf being determined, each segment i_ being identified relative to the father segment i-1 by a quaternion dQ__. and relative to an absolute frame of reference by a quaternion Q_x, the target orientation to be reached being represented in the absolute frame of reference by a quaternion Q_B, the method comprising:

- for each section i , defining an image field quaternion comprising: • an attractive component associated with the target orientation, and • if constraints are imposed, a repulsive component associated with each constraint,

- computing the speed quaternion for each section i, and

- moving each section of the articulated system in the image as a function of the speed thus obtained, or sending the data concerning the articulated system and the speeds thus obtained via a communications network. Using quaternions solves the problems of discontinuities in the angles, i.e. sudden jumps in the motion, which can occur when using the previous embodiment .

Brief description of the drawings In any event, the features and advantages of the invention will become more apparent in the light of the following description. The description refers to embodiments given by way of explanatory and non-limiting example and refers to the accompanying drawings, in which:

- Figure 1 represents an object point and a target point in a plane, - Figure 2 represents an object point, a target point and an obstacle in a plane,

- Figure 3 shows a skeleton to be animated on a screen, - Figure 4 shows the evolution of the component of a field in accordance with an angular variable,

- Figure 5 shows the relative motion of two segments, or bones, of an articulated chain to be animated, - Figures 6A and 6B are diagrammatic representations of two embodiments of a method of the invention,

- Figure 7 illustrates the problem of angular discontinuities,

- Figures 8 and 9 are diagrammatic representations of the relative position of two segments, or bones, of a skeleton to be animated,

- Figures 10 to 16 show inverse kinematic algorithms using quaternions, and

- Figures 17A and 17B are diagrammatic representations of the structure of apparatus for implementing a method in accordance with the invention. Detailed description of embodiments of the invention The invention will doubtless be better understood after a description of a model developed by the inventors for animating points of an image represented in two dimensions (2D) or three dimensions.

A first example concerns a two-dimensional (2D) representation .

Figure 1 shows a reference plane (0, X, Y) . An object or character (or point or pixel) P, also referred to as the "animate" or object point, with coordinates (x, y) is initially at the origin 0 and is to move towards a target P_B with coordinates (x_B, y_B) . In accordance with the invention, the target generates an attractive field and thereby propagates its influence throughout space in accordance with a particular law. A field E (vector) of this kind can derive from a potential V (scalar) using the equation E(P) = -W(P) where V is the "Gradient" operator.

This means that the components E_x, E_y, etc. of the vector E(P) are given by the equations:

E_x = - (δV/dx) , E_y = - (dV/δy) , etc.

The potential produced by the target is of the form V_B(P) = k.|PP_B)², for example, where k has a non-zero value. A potential of this kind, which is an increasing function of the distance I PP_B I , represents the fact that the influence of the target increases with the distance of the object point from it.

In field terms (in 2D) :

E_x = -(δV_B/δ_x) = -2.k. (x-x_B) , and E_y = -(δV_B/5_y) = -2.k.(y-y_B)

The field lines generated by the target P_B then constitute a bundle of straight-line segments passing through P_B . In this example the field is radial and oriented toward P_B. Its intensity is zero at P_B and increases with distance.

The field E is proportional to the speed (dP/dt) of the point or of the object, for example. Under these conditions, the speed of the point is obtained, in the case of the above potential, from the equations: (dx/dt) = -λ. (x-x_B) and (dy/dt) = -λ. (y-y_B) (1)

If the field is zero, the speed is zero also, and the character therefore stops (has no inertia) . For this, for a target P_B to be reached a field function is used that is zero at P_B. This is achieved if the potential function is a minimum or a maximum at P_B, for example .

Integrating the above equations yields : x (t ) = x_B+B . exp (-λ . t ) and y (t ) = y_B+B' . exp ( -λ . t ) in which B and B' are integration constants. Accordingly, the point reaches its attractive target. Its speed is initially high (it is equal to the vector of the components [λx_B, λy_B] on leaving the origin) and then decreases progressively toward 0.

In a variant the above equations could yield the direction of the point-, rather than the speed, the modulus of the speed being constant (or in any event controllable.) The modulus δ of the speed expresses the required incremental motion of the object point. For the speed norm to be made equal to δ, the following equations of motion apply, and can readily be generalized for three dimensions : (dx/dt) = (δ/||E||) .E_x and (dy/dt) = (δ/||E||).E_y (2)

In one particular embodiment, in some cases, the motion of a point P may be deemed to be possible only in "quantified" directions, i.e. "leftward", "rightward", "upward" or "downward". The motion appears continuous if seen "from afar", as is the case if the point P represents a pixel on a screen. Under these conditions, a motion algorithm needs only to look at the signs of the quantities (dx/dt) and (dy/dt), i.e. the signs of the quantities E_x and E_y. For example, if E_x>0 and E_y>0, the pixel must be moved "rightwards" and "downwards". Until the target is reached, the signs of the quantities E_x and E_y are examined and the pixel is moved accordingly.

The situation will now be described in which there is an obstacle centered at P₀ (with coordinates (x₀, yo) ) ι the point P having to reach the target by circumventing or avoiding the obstacle (Figure 2) .

An obstacle P₀ generates a repulsive field whose influence decreases with distance and is higher locally, in the vicinity of the obstacle, for example. There can in particular be an area or a distance of influence around the obstacle beyond which the repulsive field of the obstacle has no effect. It is a potential field, for example : where || PPo| < di ( influence barrier at distance di from P₀) :

V₀ ⁽P⁾ = k' - ll PPoll ^"1 (3 ) elsewhere : Vo ( P) = k' . di^"1 (a constant ) .

A function of the above kind is not discontinuous at the influence barrier (which would produce an infinite field at that point) . However, the gradient (the field) gives rise to a discontinuity and this can suddenly repulse the animate, or the point, as it crosses the influence barrier.

A function V₀(P) of this kind takes an infinite value for PP₀ = 0. However, in the case of an obstacle, it is in principle avoided and therefore never reached. In the area of influence, the gradient of the function V₀(P) = k' . HPPol^"1 gives a field: Eo(P) = + k' . PPo- ||PPo| ²

(the + sign indicates a repulsive field: k' must be greater than zero for the field to be a repulsive field) . To summarize, the field produced by the obstacle P₀ is:

- for 0 < ||PPo|| < d_{l r} E₀ = k' .PP₀.||PPo|r²

The total field E to which the character or the point is subject is the sum of the various fields (principle of additive fields) :

E = E_B + Eo . Whence :

(dx/dt) = (δ/|(E_B + Eo)||).(E_B + E₀)_x (dy/dt) = (δ/|(E_B + E₀)||).(E_B + E₀) _y (4)

(dz/dt) = (δ/||(E_B + Eo)||).(E_B + E₀)z Consider therefore the most general case of the point P attempting to reach a target P_B (generator of a potential V_B(P) = k. ||PP_B||²) in a universe (a 2D universe, for example) containing an obstacle P₀. The influence of the obstacle (generator of a potential V₀(P) = k' . ||PP_B||^_1) is represented by equations (5) :

(dx/dt) = (δ/|| (E_B+Eo) ||) . [-k. (x-x_B)+k' . (x-xo) . ||PP₀||^"2] (dy/dt) = (δ/ || (E_B+E₀) ||) . [-k. (y-y_B)+k' . (y-y₀) . i|PPo||^"2] (5) For a qualitative analysis, only the signs of these quantities are considered:

Sgn (dx/dt) =S_gn[-k. (x-x_B)+k' . (x-x₀) .||PPo|^"2] Sgn (dy/dt) =S_gn[-k. (y-y_B)+k' . (y-y₀) .||PPoH^"2] (6) Because of the discontinuity of the field near the influence barrier of the obstacle, the moving point P can suffer a sudden change of trajectory there. There will then be the "surprise" of encountering an unexpected obstacle. This is understandable because the concept of action at a distance, inherent to the field concept, is eliminated beyond the influence barrier, given the definition of that barrier. The discontinuity of the field can be expressed as a vector ΔE with components:

ΔE_X = (x_d - x₀) .d ² and ΔE_y = (y - y₀) > d_λ ^~2 (7) where x_d and y_d are the components of the point P at the distance d_x from P₀ (at the influence barrier) . If the amplitude of the components ΔE_X and ΔE_y is such that Sgn (dx/dt) and Sgn (dy/dt) change, the situation is that of a kind of "impact" (although there is no contact), with a change of direction. Entering the influence area causes the trajectory to be deviated. In the case of a plurality of obstacles (N obstacles) P₀ι, P02, -, Pot generalization yields the following equations (in which the value k' - is zero outside the area of influence of the obstacle P_θD) :

(dx/dt)=N. {-k. (x-x_B)+Σ_(D-ι,_N, [k'₃. (x-x₀₃) . ||PPo_||^"2] > ⁽dy/dt)=N.{-k. (y-y_B)+Σ_(D=1,_N) [k'_:. (y-y_0:) .||PPo₃| ²] } (8)

(dz/dt)=N.{-k. (z-z_B)+Σ_(D=1,_N) [k'_D. (z-z₀-) .||PP₀₃| ²] } with N = (δ/|| (E_B+∑(_J-I,N) (E_0D) ) ||) where E_θD designates the field associated with the obstacle P_0] . The model explained above ignores the influence of the inertia of an animate or a point. The corresponding trajectory is computed by deeming the total field E to which the point is subject to be directly proportional to the direction of its speed v. In mathematical terms, the speed vector v is given in this case by the equation: v = (δ/||E||)E in which the coefficient (δ/||E||) is intended to produce an incremental movement equal to δ.

The algorithm for computing P_n+ι as a function of P_n (also called the Actuator of the animate) is therefore as follows (P_n denotes the current position and P_n+ι the next position) :

P_n+ι = Pn + (δ/||En||)E„ (9) where E_n is the total field at P_n. In another embodiment the object point has a variable inertia, for example inertia that is constant "far" from a target to be reached and that decreases as the target is approached. This "smoothes" the trajectory when avoiding obstacles (the animate is entrained by its inertia) and prevents any possibility of undefined oscillatory motion about a target since its inertia decreases to the point of becoming very low or even zero once the target has been reached (this has the same effect as progressive "friction") .

To this end, a target P_B has a characteristic distance di from which the inertia decreases. Also, the Actuator of the animate then has the following structure:

1 ) V_n+1 = k_a . v_n + E_n

2 ) P_n+ι = Pn + (δ/ || v„₊₁ || ) . v_n+1 ( 10 )

The elasticity or damping coefficient k_a depends on the distance ||P_nP_B||. If IP_RP_BH > di the value k_a=l can be taken, for example. On approaching the target, k_a falls to 0 so that the influence of the inertia is felt progressively less and less. The coefficient k_a therefore progressively reduces the inertia of the animate as it approaches the target to be reached. If the value of k_a is very low the field is practically proportional to the speed and if k_a is very close to 1 the field is practically proportional to the acceleration. It can be shown that if k_a = 0 the situation is as previously where the inertia is zero.

Generally, a value of k_a is taken such that 0 < k_a < 1. The cases in which k_a = 0 or k_a = 1 have already been commented on above. For k_a > 1 the system diverges .

The law by which k_a decreases as a function of ||P_n B|| is a parabolic law, for example: k_a = (||p_np_B||/dι) . [2-(||P„P_B||/dι)] (11)

In a variant, an additional coefficient ki in the range from 0 to 1, for example, is introduced that is independent of the target and such that the speed of the animate is computed from the new equation (12) : V_n+1 = _ϊ . kaVn + E_n ( 12 )

This provides greater flexibility in adjusting the inertia of the animate.

In the method of the invention, the universe in which a point or an object moves is represented in field terms. If the universe is made up of obstacles to be avoided, for example, the fields are repulsive fields.

The algorithm for moving an animate at the object point can be defined by the following rules.

At time n the animate is in state ψ_n. This state comprises a set of kinematic data, for example, namely the position P_n of the animate, its speed v_n and its incremental movement δ.

A chain of action (or "actuator") of the animate capable of producing its new state ψ_n+ι at time n+1 as a function of the obstacles and the target to be reached has the following structure, for example:

1) Compute the total field E_n (attractive and repulsive component) at point P_n.

2) Compute the new speed using, for example, the equation:

Vn+1 = k_aV_n + E_n where k_a = ( || P„P_B|| /dι ) . [ 2 - ( || P„P_B || /dι ) ] for II PRPB || < i and, for example , k_a = 1 otherwise .

3 ) Compute the new position P_n+1 = P_n + (δ/ ||v_n+1 || ) . v_n+1 in a more general embodiment, the speed of the object point can be computed from the equation:

Vn+1 = k_aV_n + kiE_n in which :

• V_n+ι and V_n are respectively the speed at times n+1 and n,

• E_n is the value of the field (with its attractive component (s) and/or repulsive components) at the location of the point at time n, and

• k_a and ki are respectively a damping coefficient and an inertia coefficient.

The function of k_a has already been explained. A value is generally taken such that 0 < k_a < 1, the values k_a = 0 or 1 also being possible. Once again, k_a can be made dependent on the distance, for example using equation (11) above, with the same effects as already explained. The coefficient k (> 0 and generally in the range from 0 to 1) corresponds to inertia in classical physics (in the mathematical sense of the term) .

If ki is low, inertia is high (and vice versa) and if k_a is low, damping is high (or "flexibility" is low) and vice versa.

Note that if k_a = 0 the effect of the inertia is zero although the mathematical coefficient ki has a nonzero value.

Another coefficient k_e can also be introduced, by means of the following equation: v_n+ι = k_ek_aV_n + kiE_n in which: k_a and ki have the same meaning as above, k_e is a multiplier coefficient for k_a . The model described above can be transposed to the problem of transforming or animating skeletons, in particular three-dimensional (3D) skeletons, which approximate a real image .

To this end an image, or rather a skeleton schematically representing objects or shapes of an image (such as the representation of a character from Figure 3, for example) can be considered or described as a set of articulated "bones" 2, 4, 6, 8, 10, 12 connected together by articulations.

Each bone B has a father B^i . If it has no father, the bone B_± is the root and i = 0. The chain of bones is described as a series of relative geometrical transformations dTRF₍₁₎. The prefix d stands for "difference", and dTRF ₎ is the transformation of the bone B_x relative to the frame of reference of the father bone (i-1) . In the case of B₀, dTRF_(O) is the transformation of the bone B₀ in any

"absolute" frame of reference. In a frame of reference of this kind, each bone B-_. has a transformation TRF₍₁₎ = TRF(₁-ι₎*dTRF(₁) with TRF₍-i) = I (Identity) .

Consider first a kinematic chain, that is to say a series of "bones" connected by articulations. The end "bone" is called the pseudo-leaf.

An orientation in space can be defined by three angles α, β and ψ. α and β are the spherical coordinates of the rotation axis and ψ is the rotation angle about that axis. The pseudo-leaf bone seeks to reach a

"target" orientation also defined by three angles α_target βtcirget and ψt_arget •

A set of three angles constitutes a vector that defines an angular point P_a = (α, β, ψ) . For example, the pseudo-leaf is an angular point

P_a = (α, β, ψ) which must reach another angular point

P_atarget = (0-target , βtarget, ψtarget ) • The target ang lar point generates an attractor field E in the space of the angular points. The field is a vector function E (α, β, ψ) defined at any angular point P_a = (α, β, ψ) or for any direction defined by three angles α, β, ψ. P_a moves in this space of angular points due to the effect of this field.

What is more, a bone cannot always take any orientation: there may be constraints on it. These constraints can be materialized by one or more angular points Paback creating a repulsive field E' at any point in the space .

Transposing the equations given above yields the following algorithm:

V_n = K_aV_n-i + K^E_n + E'_n) (13-1)

{

Pan = P.(n-l) + _n (13-2)

in which: n is an integer which represents discrete time, P_an is the angular position at a particular time, P_a(n-i) is the angular position at the "preceding time", V_n is the "speed" vector of the angular point, E is the attractor vector field with components:

E [ K ( α_target ~ Ct (n-l ) ) ' K ( β_target ^~~ β (n-l ) ) / K ( ψtarget ~ ψ (n-l ) ) ] / K being the attractor coefficient (for example < 10^"1, which is already a high value) ,

. E' is the repulsive field, which is related, for example, to the fact that each parameter α, β and ψ can vary only within certain limits: αe[α_mm, o_maχ] βe [βmii βmaxl and ψe [ψmin, ψmax]- One or other of these parameters can reach a limit value.

For example, if ψ < ψ_mιn + r. (ψ_max - ψ_mιn) , ψ is said to be "close" to

£ being a proximity coefficient (which can be equal to 1/10, for example, or is typically in the range from 0 to 1) . There is then, for the component ψ of the particle, a repulsive field E'_ψ in the direction of ψ whose (positive) value is:

E'_ψ = -K' . [ψ - (ψ_mιn + r. (ψ_max - ψ _n) ) ] where K' is a "repulsor" coefficient (a value of a few units, is already a high value) .

Conversely, if ψ > ψ_max - r. (ψ _max - ψ_mιn) , ψ is "close" to ψ„a . This tendency is compensated by applying a repulsive field E'_ψ in the direction of ψ whose (negative) value is: E'_ψ = -K' . [ψ - (ψ_max - r. (ψ_max - ψ _n) ) ]

. Ki is the angular inertia coefficient of the bone influenced by the fields E and E' . A low value of K_x gives a high inertia (a strong field has less influence on a high inertia object) and vice versa.

If k_x is equal to 0, there can be no movement.

If ki is close to 0, a great deal of time is required for a bone to move from one position to the other (this is the problem of moving a body with high inertia) . k = 1 gives a low inertia, but this depends equally on the intensity of the vector E + E' , which is a variable parameter.

A value is generally chosen such that 0 < ki < 1

K_a is the angular damping coefficient of the bone influenced by the fields E and E' .

If K_a = 0 there is maximum damping, which means that v_n = 0 if (E + E' ) = 0; or, in other words, the particle stops if the total field applied to it is nil.

A value of K_a equal to around 0.5 rapidly damps the process .

If K_a = 1, the sum (E + E' ) is equal to the acceleration v_n - v_(n-i) and the particle does not stop if the total field applied to it is nil; in this case the damping is minimum. The value of K_a = 1 causes the point to oscillate about its target, without ever reaching it.

If a value is taken such that K_a > 1, the point never reaches its target but moves away from it irrevocably. (In fact this is true only in the sense of this being a very high "probability") .

The coefficient K_a is an intrinsic characteristic of the bone subject to the influence of the field (likewise Ki) . For preference a value is taken such that 0 < K_a < 1.

For some degree of "reliability" of operation, for example in the case of large oscillations, whilst continuing to compute the speed v_n from equation (13-1) it is possible to modify equation (13-2) slightly by introducing a limit speed v_lim that the particle at "angular" position P_an must never exceed, or an angular speed that the bone must not exceed.

For this, the "norm" of the speed is computed:

|| Vn || = I V_αn I + I V_βn I + I V_ψn I

If ||v_n|| < v_lιm, then equation (13-2) is unchanged. Otherwise, equation 13-2 is modified (to become equation 13-2') as follows:

Pan = Pa .n-l ) + V_n. V_lιm/ 1| V_n || ( 13 -2 ' )

The inertial properties are retained, because the speed v_n is computed in the same manner; however, the particle "absorbs" the high speed fluctuations unless the system is damped too slowly.

This limitation of the speed also has a role if the system attempts to exceed limit values . The limit angular values are in fact not "thresholds" in the strict sense of the term. These limit values can be exceeded, in particular if the system is set up with a high inertia. This does not occur in practice if the precaution is taken of artificially limiting the "speed" of the particle in the same manner as described above in the case of large oscillations.

To illustrate the influence of the total field on the particle, equation 13-1 can be represented graphically (Figure 4) on one component only (for example ψ) . Projecting equation 13-1 onto the ψ axis yields: V_ψn = K_a . Vψ (_n-i ) + K_. . K ( ψtarget " ψ (n-l ) ) + K_X . E , wi th :

• E ' ψ = E ' ψi = -K' . [ψ- (ψ _n + r . (ψ_max - ψπ n ) ) ] _/ if ψ < ψ_mιn + r . (ψ_max - ψ_mιn) , and :

• E ' ψ = E ' _ψ2 = -K' . [ψ- (ψ_max - r . (ψ_max - ψ _n) ) ] if ψ > ψ_max - r. (ψ_max - ψ_mιn) . If K_a = 0, the field E_ψ + E'_ψ represents the variation v_ψn that must be imparted to ψ. The graph shows that if ψ > ψ_target the variation is negative, which tends to reduce the difference (ψ - ψ_targ_et) • Conversely, if ψ < ψtarget the variation is positive, which tends to reduce the difference (ψt_arge_t - ψ) • The target is therefore a point of convergence.

From the geometrical transformation point of view it may be said that, in an initial state, a chain of bones is a series of relative transformations that are fixed by construction. Let dTRF0₍₁₎ represent these "rest" transformations. When the chain moves, each transformation dTRF₍₁₎ is subject to a variation δTRF₍₁₎ about dTRF0₍₁₎ in accordance with the following law: dTRF ₎ = dTRF0₍₁₎* δTRF ₎

Accordingly, it is the series of variations δTRF that produces the motion of the pseudo-leaf towards the required orientation.

For a complete description of the kinematic chain the constraints on δR0T₍₁₎ can be specified for each bone

(i) :

- First of all the initial rotation axis of δROT₍₁₎ is indicated. Each δROT_(i) is associated with two angles α₀ and βo which indicate the position of the initial axis.

In fact the initial rotation angle ψo about this axis is always nil because the initial variation of δROT0₍₁₎ gives the identity matrix. - Then the constraints are indicated by "angular

V ctors _amln ( Ctnun Piαin / ψmin ) and C_amax ( Ct_maχ , Pme f ψmax ) which give the authorized limit values of the three angles α, β and ψ.

For example, if α_mιn = α₀ - α_max and β _n = β₀ - β_maχ the maximum constraint is that on the position of the rotation axis, since it is prohibited from moving. For example, if ψ_mιn = ψ₀ and ψ_max = ψo + 90°, this means that the bone can turn about its fixed axis only through an angle ψ between its initial value ψ₀ and the value ψo + 90°.

The convergence of an isolated bone toward an attractor can now be described in the following manner.

It is assumed that a bone i of the chain is isolated (and therefore that the bones are processed independently of each other) , this bone having to achieve an orientation ROTB₍₁₎ .

The algorithm enabling it to arrive there is as follows (ROTi designates the rotation associated with the bone i. in an absolute frame of reference) : 1 - Compute dROTB₍₁₎ to be reached from the equation: dROTB₍₁₎ = ROT^"1,_!-_!,* ROTB_{1); or dROTB_U) = R0TB , if i = 0. 2 - Compute δROTB₍₁₎ to be reached from the equation: δROTB₍₁₎ = dROT0^_1 ₍₁)* dROTB , .

3 - From δROTB₍₁₎, compute the three angles α_B, β_B and ψ_B to be reached (equations 13-1 and 13-2 above) .

4 - From the field theory described above, compute the three angles α_n, β_n and ψ_n actually reached (using equations 13-1 and 13-2; these operations are "projected" onto the "axes" α, β, ψ) .

5 - From the three angles α_n, β_n and ψ_n, compute the resulting matrix δROTB₍₁₎ . 6 - Compute dROT₍₁₎ to be applied to the bone from the equation: dROT , = dROT0 ₎* δROT_U). 7 - Repeat the process as many times as necessary.

The process described in the previous paragraph can be applied in exactly the same manner to all the bones of the chain. Each bone operates as if it were isolated from the others but makes its contribution to the pseudo- leaf reaching its target. The action of a bone depends on the action of the other bones: if a bone tends towards an attractor, this modifies the attractor of the parent bones and therefore modifies their action, although they have exactly the same mechanism.

The following algorithm applies for all the bones (assigning the index i = N to "the pseudo-leaf" of the chain) :

1- Compute all absolute rotations ROT_(i) for ie [0, N] .

2- Create an index i = N defining the "active" bone.

3- Store the target ROTB_(N) of "the pseudo-leaf" in a buffer ROTB. 4- Require bone number i to accomplish the algorithm described above by taking ROTB_(x) = ROTB. 5- Set i=(i-l) to set the father bone. 5-1- If there is no father, return to step 1.

5-2- If there is a father, compute its target from the equation ROTB = ROTB*dROT^"1 ₎ and return to step 4.

In this logic, which starts from "the pseudo-leaf and works back to the pseudo-root, the target of the father "agent" depends on the target of the son "agent" (this is the equation ROTB = ROTB*dROT^"1 ₍₁₎ ) .

The orientation to be reached is set by the three angles α_target, βtarget and ψtarget- given bone has its current orientation fixed by the three angles α, β and ψ. In fact, the process of convergence minimizes the angular distance D_a given by the equation

D_a = | α-α_target l + I β-βtarget I + I ψ " ψtarget I • α and β are the spherical coordinates of the rotation axis. For this axis to be able to travel through all of the space it is sufficient to take α € [-π, +π] and β e [-π/2, + π/2] .

The axis being given, it is not necessary for ψ to be able to vary by a complete rotation (ψ e [-π, +π] ) to express all possible rotations: to scan all rotations it is sufficient for ψ to be able to vary by a half- rotation, for example ψ € [0, +π] . Rotation through angle ψ about axis u is equivalent to rotation through angle -ψ about axis -u (the direction of the axis is changed) .

Thus there is always a choice of two possible angles for the same rotation. In particular, the three angles ^target? βtarget and ψtarget are equivalent to the three angles α' target / β ' target nd ψ ' target SUCh that : β ' target ⁼ ~βt_arget and ψ ' t_arget ⁼ "ψtβrget

O- t_arget ⁼ Ctt_arget + 7t i f Ctterget < 0 Or O.' target ^{= α}target ~ π i f Cttarget > 0

This latter relation conserves α' _targ_et e [~π, +π] . To decide between the two possible combinations, the one is chosen which, relative to the current orientation α, β and ψ, gives the minimum angular distance D_a.

The pseudo-leaf of a chain of bones can reach a given point in space independently of its orientation. A point in space can be reached by rotation of the bones of the chain and therefore with an angular attractor field. However, as the source of this field is here a point in space and not an orientation, this field is called a translation angular attractor field.

As shown in Figure 5, E₍₁-n) is the end of bone number i and B_x the point in space to be reached. A rotation through angle ψ_x about an axis perpendicular to the plane (E^_., E_(1+D, B has the effect of moving E_(1+D towards B^_..

Figure 5 shows the end E_(N+i) of the pseudo-leaf in the process of attempting to reach the point B_N if the number of the bone is i = N. To simplify the notation, let E_(N+i) = E and B_N = B . The following equation is an expression for the vector EB in the absolute frame of reference :

EB = OB - {TRL_(N) + ROT(_N) *[0, 0, -L_N]^T} (14) in which TRL_(N) and ROT_(N, are the translation and rotation components of the transformation TRF_(N) in the absolute frame of reference.

Each bone contributes to the pseudo-leaf reaching the target B: each bone has the same work to do, namely effecting a rotation through angle ψ_x . The formula giving the expression for the vector EB in the frame of reference of bone number i_ is obtained in the following manner. In this frame of reference, let E^_IJB_J. be the expression of the vector EB; then:

E_d÷nB, = [ROT₍₁₎]-¹*EB (15) where ROT ₎ is the component of the rotation of the bone i. in the absolute frame of reference.

At a given "time n", bone i is deduced from its father by the rotation dROT ₎ . For this bone i to contribute to the pseudo-leaf moving towards B, its end E(_1+D must effect a rotation d²ROTB_U), the matrix expressed in the frame of reference of the bone i.

Consequently, and relative to its father, the bone i is subject to a rotation dROTB ₎ = dROT₍₁₎ *d'ROTB₍₁₎ .

This rotation to be achieved is expressed, for bone i., in its construction frame of reference and as a function of reference (construction) transformations dROT0₍₁), in accordance with the following equation: δROTB₍₁₎ = dROT0^"1 ₍₁)*dROT_(l)*d²ROTB₍₁₎

In conclusion, each bone i_ sees its rotation dROT₍₁₎ relative to its father tend towards the rotation δROTB ₎ . This corresponds to three angles α_B, β_B and ψ_B which define the translation angular target.

It is therefore sufficient to compute d²ROTB₍₁₎. The rotation through angle ψ is about an axis u whose direction is given by the vector product:

In the frame of reference of the bone i_ the components of u are (ignoring a multiplier coefficient) :

U_x = (E +uB y, U_y = -(E,_x+ι,B-.)x and U_z = 0

The Z axis component being nil, the matrix d²ROTB₍₁₎ is given by:

u_x". (1-cosY) +COSΪ u_x.u_y. (1-cosY) u_γ. sinY d'ROTB_(l) u_x.u_γ. (1-cosY) u_γ ². (1-cosY) +cosY -u_x.sinY

^•Uv. sinY u_x.sinY cosY

Scalar product E_IE₍₁₊D .E₍₁₊i)Bι is used to find cosψ: cosψ = K. [L₁-(EB)_lZ] where: K = {[(E₍₁₊₁₎B _X]² + [(E₍₁₊₁₎B₁)y]² + [ .-tEu._DB z]²}-¹'²

Because sinψ = K. { [ (E₍₁₊ι,B₁) _x] ² + [ (E _+1]BX ,]²}^l/2 , there is no need to compute u_x and u_y, which are normalized at 1, explicitly, but the following coefficients of the matrix d²ROTB _! are found directly:

u_Y.sinY = - K. (E_U+1)B _X Also: u_x ². (l-cosY)+cosY = [K. (E ₊₁₎B₁)_Y]²/(l + cosY) + cosY u_γ ². (l-cosY)+cosY = [K. (E ₊₁₎B _x]²/(l+cosY) + cosY u_x.u_γ. (1-cosY) = [-K. (E₍₁₊₁₎B₁)_X] . [K. (E₍₁₊₁₎B₁)_Y]_/(l+cosY)

Thus all coefficients of the matrix d²R0TB_U) have been computed.

The calculation of K can yield an infinite value if the quantity { [ (E,_1+ι,B₁) _x] ² + [ (E_u+ι,B _γ]² + [L,- (E ,_l)B,) _z] ²} is too low.

If [(E₍₁₊ι,B₁)_x]²+[(E₍₁₊₁₎B₁)_γ]² « [L_X-(E₍₁₊₁₎B₁)_Z]²}, however, the angle ψ is either very close to 0 or very close to π. The comparison can therefore be done by taking a factor in the order of 10^"5, which is largely sufficient in terms of accuracy and guarantees that K will always be a number. If [L_x- (E₍₁₊i)Bι) z] > 0, the identity matrix is produced.

If [Li- (E_d+ijBi) z] < 0, there is more than one possible matrix because there is symmetry about an indeterminate axis. However, it is sufficient to take one of the possible matrices.

The algorithm for the influence of a point in space can therefore be as follows:

1 - compute all absolute rotations ROT₍₁₎ for ie[0, N] ,

2 - create an index I = N defining the number of the "agent" bone,

3 - compute EB from equation (14) above,

4 - compute Eu₊uB. from equation (15) above,

5 - compute the matrix d²ROTB₍₁₎, as previously described,

6 - deduce from it the matrix: δROTB₍₁₎ = dROT0^~1 >*dROT ₎*d²ROTB,-.>

7 - deduce the three angles α_B, β_B and ψ_B from the matrix δROTB₍₁₎,

8 - apply the field theory, regarding these three angles as an angular attractor field, and deduce δROT ₎ and then the rotation dROT₍₁₎ to applied to the bone (from the equation dROT ₎ = dROT0_U)* δROT ₎),

9 - set (i-1) → i to select the father bone, 9-1 - if there is no father, return to step 1,

9-2 - if there is a father, recompute TRF_(N) (which has just been modified) and then return to step 3.

How the pseudo-leaf and all the bones of the chain can be attracted in a common direction can now be described. This attraction direction constitutes a gravity field. This attractor is called the gravity angular attractor. The rotations of the bones are used to produce this effect; the root is as if fixed to a pivot (which does not react) and everything "falls" in the direction of the gravity field. If the articulations are unconstrained, the system "unwinds" and tends towards that direction.

The principle is the same as before except that the vector EB has three fixed components that do not depend on the position of the pseudo-leaf. In other words, the equation EB = OB - {TRL_(N) + ROT_(N)*[0, 0, -L_N]^T} is no longer to be used, because EB is a constant.

All the computations are exactly the same as before, except for finding cosψ, for which the scalar product EEji+_D.E_x is used (rather than EiE₍i₊ι_, .E(_i+i)Bι) . This is equivalent to making the single modification Li = 0 in the above equations. In the scalar product EiE_d+u .Ei, Ei is the expression of the gravity field E in the frame of reference of bone number i_, i.e. E = [ROT_(i) ] ^_1*E .

The algorithm for the gravity angular attractor field is a simple one: the axis of each bone of the chain (the axis of the bone, not the rotation axis) tends to be oriented toward the gravity field "of its own accord" (independently of the others) .

The rotation (or orientation) components of the motion are assumed to be subject to the influence of the angular fields. It follows from what has been explained already that there are four different types of angular field:

1 - Rotation angular attractor fields. These fields correspond to an orientation that the pseudo-leaf must achieve .

2 - Translation angular attractor fields. These fields correspond to a point in space that the pseudo- leaf must reach. 3 - Gravity angular attractor fields. These fields correspond to a direction in space which all the bones of the chain must reach.

4 - Constraint angular repulsive fields. These fields correspond to the orientations that the various bones of the chain must avoid.

The combination of the three attractor fields constitutes a global objective to be achieved.

Let E_aR, E_aT and E_aG be the three attractor fields and E_r the repulsive field. The total field E is E = E_aR + E_aT + E_aG + E_r.

^• Each attractor field E_a has the following components :

E_a [ K . ( α_target " Ot) , K . ( βtarget " β ) , K . (ψtarget " ψ ) ]

Accordingly, the components of the total field E_a are:

E_a=K_R . ( a_targe_tR-a ) + K_τ . ( a_tar_ge_tτ^_a ) + KQ . ( a_targe_tG-a ) + E_ra E_b ⁼K_R . (b_targe_tR-b ) + Kp . (b_targ_etT~b ) + KQ • (b_targe_tG~b ) + E_r Eγ=K_R . ( Yt_argetR^_Y ) + K_τ . ( Yt_argetT^~Y ) + KG • ( Yt_argetG^_Y ) ⁺ E_rγ

The above equations show that the three objectives are combined in linear fashion with the possibility of weighting the three influence coefficients K_R, K_τ and K_G.

Each bone of the chain corresponds to the pseudo- leaf reaching the given orientation. In one embodiment (Figure 6A) , the bones operate in a certain order .

Bone N-l operates first. Then the target of bone N- 2 is computed and it operates. Then the target of bone N-3 is computed and it operates, and so on, up to bone 0. This procedure is "linear upward" and gives good results. A different logic takes account of the operation of bone N-2 disturbing the previous operation of bone N, etc. It is then desirable to have bone N-l "react" to the operation of bone N-2, and thereafter to have only bone N-3 operate. Figure 6B shows this alternate procedure . The consequence of each operation of bone _i is then a "downward" reaction on the successive bones i+1, i+2, ..., N-l.

Regardless of the order in which the bones are processed, each of them is influenced by the motion of the others. Each bone adapts its operation not only to the target to be reached by the pseudo-leaf, of course, but also to the preceding operation of the other bones.

Overall, results are better with this second procedure. The algorithm effects "micro-resolutions" which can be reviewed at each new iteration. This means the process is not limited to one solution at time n if another, more beneficial solution, emerges at time n+1.

This new approach indicates that the target of bone i-1 can be computed from the objective of bone i because the "linear upward" order is no longer valid. The target of any bone is therefore computed absolutely. The

Computation Of the angles (Ct_targetR, β_targetR/ ψtargetR) / ( CttargetT / βtargetT / ψtargetϊ¹ ) and ( Ct_targetG/ βtβrgetG/- ψt rgetG ) for each of the three angular attractor fields is explained below.

1 - Rotation angular attractor field: (ROTB_N is the orientation to be achieved by the pseudo-leaf) . 1-1 - Compute ROTBx = ROTB_N* [ROT_N] ^"1*ROT₁.

1-2 - Compute dROTB_x = [ROTB₍₁- ] ^"^ROTB_.. or dROTBi = ROTB;,_. if i = 0.

1-3 - Compute δROTB_x = [dROTO ^-1*dROTBι .

1-4 - Deduce from δROTB_x the angles (α_targetR, βtargetR-- ψtargetR) •

2 - Translation angular attractor field: (TRLB_N is the point in space to be reached by the pseudo-leaf) .

2-1 - Compute EB = TRLB_N - TRL_N + ROT_N *[0, 0, L_N]^T.

2-2 - Compute EB_. = [ROT ^'^EB and deduce d²ROTB_x from it. 2 -3 - Compute δROTBi = [ dROTO ^"1*dROT₁*d²ROTB₁ .

2-4 - Deduce the angles (α_targetτ βtargetT ψtargetr) from δROTB_x .

3 - Gravity angular attractor field: (E_G is the direction to be reached by the pseudo-leaf) .

3-1 - Compute E_Gl = [ROT₁]^"1*E_G and deduce d²ROTB_x from it. 3-2 - Compute δROTB_x

. 3-3 - Deduce the angles (ottargetc- βargetc^ ψtargetc) from δROTB_x . When all the attractor angles have been computed, the total field E has also been computed. The remainder of the process that produces the action of bone i_ is written as follows:

4 - Compute from the field theory the angles (α, β, ψ) to be imparted to bone i_.

5 - Deduce the matrix δRO i from it.

6 - Deduce the matrix dROTi = dROT0₁*δROT₁, which is the new rotation matrix of bone i , from the matrix δROTi .

7 - If the process has not terminated (another bone is to be subjected to an action) , return to step 1, but after updating the transformations TRF _N and TRF_X which have just been modified.

The process described above uses descriptions of the motion and positions in terms of angles. The equations given have the advantage of being linear. In fact, this can give rise to a number of problems, especially in the case of combining the two angles (α, β) which represent the rotation axis and in the case of interpolation between two pairs (cti, βi) and (α₂, β₂) in which the components α and β are interpolated linearly but separately. This can lead to discontinuities in the angles, i.e. to sudden jumps in the motion. As shown in Figure 7, a rotation can then be effected along a path 14 (clockwise in Figure 7) rather than along the other path 16 (counter-clockwise) , in a manner that is difficult to predict .

One solution to this problem is to use a description in terms of quaternions. A rotation is then represented not by angles but a unit quaternion Q. The disadvantage of this form of description is that the field equations are non-linear. As already explained, to each bone B_. there corresponds a transformation dTRFj_.. The rotation part of this transformation is expressed in the form of a quaternion dQ__.. The theory of quaternions is explained in S.G. HOGGAR, "Mathematics for Computer Graphics", Cambridge University Press.

In physical terms, a quaternion dQ is a rotation through angle dψ (spin or "inherent rotation") about an axis dA which is a vector entirely defined by two angles dα (right ascension or "bearing") and dβ (declination or "azimuth") . Let dP_a be the set of these three angles: dP_a = (dα, dβ, dψ) (P_a signifying "angular point") . The value of dP_a determines that of dQ but the value of dQ gives two possible values for dP_a because a change of sign of dψ and a change of direction of the axis A give the same rotation.

In theory, to define the rotation between a bone and its father it sufficient to store dP_a in a database.

It is equally possible to introduce a breakdown of the rotation dQ_x into a plurality of terms. In an initial state the chain of bones is a series of relative transformations set by construction. Let dTRF₀(_D represent these "rest" transformations. When the chain moves, each transformation dTRF₍₁₎ is subject to a variation δTRF₍₁₎ about dTRF_0d. iⁿ accordance with the law: dTRF ) = dTRF₀(ι)*δTRF_u, . It is therefore the series of variations δTRF _. that produces the motion. In terms of quaternions, this is written: dQ^_. = dQo*δQ₁.

In other words, the quaternion dQ__. is considered as the combination of an equilibrium quaternion dQ_l0 and a variation δQ written: dQ_x = dQi_O.δQi (the product is here a product, or rather a combination, in the sense of this term in the theory of quaternions) . The quaternion dQ_x represents the orientation of Bj_. relative to B₍₁-_D and the orientation of _x relative to an absolute frame of reference is represented by the absolute quaternion Q_x in accordance with the recurrent equation Q = Q(ι-i)dQ_x. An additional breakdown leads to consideration of the rotation δQ as the product of two rotations: δQi = δQi_θ*d³Q_!, because the processing of the angular constraints imposes symmetry on the limit values of dψ (the minimum angle must be equal and opposite the maximum angle) .

In total, the absolute quaternion Q_x is written:

Qi = dQ₁-ι*δQ_l0*d³Q₁ (16) Finally, d³Q_x that can be considered as the true variable, dQ__. and δQ_l0 being merely variable "parameters", or imposed by construction or because of the constraints. This additional breakdown can also be interpreted in the following manner: the initial construction rotation dQo has now become the rotation d'Q₀ = dQo*d²Qo which finally leads to the expression: dQ = d'Qo*d³Q. At a given time n, all the absolute quaternions of the articulated system can be calculated from the variations δQi using the above equations .

The variations δQ_x can be deduced from the absolute quaternions Q from the inverse equation: δQα. = [dQ_l0]^_1. [Qu-i.r'.Qi (20)

Assume that bone B_x is attracted by an absolute orientation expressed by the quaternion Q_B. The rotation of the bone B_x (expressed by the quaternion Q ) is subject to the influence of an attractive rotation (expressed by an attractor quaternion Q_E) . Q_E has the same role as the field E from equation 13-1 and expresses the "intention" of the bone B_x to reach the orientation Q_B.

For a point, the expression of an attractor field at time n is written E_n = P_B - P_n. However, for a quaternion, this difference is replaced by the following multiplication :

^Q _En = [Qm]^_1.QB (21) For a quaternion: Q_En = Qi (Identity) if Q_ιn = Q_B, which satisfies the equation Q_En = [Qm] ^-1-QB.

However, equation 13-1 shows that the field E_n is subject to scalar multiplication by an inertia angular coefficient k_x. Here this coefficient has the same meaning as explained above; preferably 0 < k_x < 1. In the language of quaternions, the term k^En is replaced by the term:

Q'_B is the quaternion spherically interpolated between Q_ιn and Q_B in accordance with the coefficient k_lf written Q'_B = Q_xn(° -.)QB

The symbol (\) represents the operator performing spherical interpolation of coefficient k_x . Multiplication by a scalar is equivalent to spherical interpolation between quaternions and addition or subtraction is equivalent to multiplication of quaternions. The following field theory equations are therefore valid for quaternions: V_n = Q_En. [Qι(°_ka)V_(n_₁₎] (23)

Based on equation (22) , the expanded expression for V_n is: V_n = [Q₁n]^"1[Q₁n(⁰k₁.)Q_B] • [Qι(\_a)V_(n-ι>] k_a is the angular damping coefficient (see above explanations on this subject) ; once again, a value is preferably taken such that 0 < k_a < 1

From an algorithmic point of view, the operations that can be effected for a bone B to tend towards an absolute attractor Q_B (like a system slaved in orientation) are:

1 - Compute the variation δQ_Bn that must be imposed on the bone B for the attractor Q_B to be reached exactly, using equation (20) and replacing Q with Q_B:

The variation δQ_Bn has the index n (which represents time) because it depends on dQ₍i-ι_)n, which is generally variable .

2 - Compute the resulting field, using equation (22) and replacing Q_Bn by δQ_Bn: δQ_En = [Qin]^"1- [Qm(°k)δQ_Bn]

3 - Compute the speed quaternion V_n using equation (23) and replacing Q_En by δQ_En.

4 - Compute the quaternion δQ_ιn using equation (24) and replacing Q_in by δQ_in and Q_i(n-i) by δQ_i(n-i) • 5- Compute the quaternion Q_in = Qα-nn-dQio .δQi_n using equation (24) .

This algorithm, repeated for all the bones of a chain from its end to be slaved (to be made to tend toward a fixed target) , working back to its other end, constitutes a new inverse kinematic algorithm.

The angular constraints are defined in the following

9 9 manner: the quaternion d Q is associated with an axis dA and a rotation through angle d²ψ about that axis and the quaternion d²Qo is associated with an axis d²A₀ and a rotation through angle d²ψ₀ about that axis. d²Q can vary about d²Q₀ so that:

- the axis d²A is entitled to be contained in a cone with axis d²A₀ and half-angle γ, and

- the angle d²ψ is entitled to vary about d²ψo by ±Δψ/2, where Δψ/2 is the half-angle of variation authorized for d²ψ.

The angular constraints (denoted d³Ca) are therefore the set of two values γ and Δψ: d³Ca = (Δψ, γ) . d²Q is varied about d²Q₀ by the quaternion d³Q in accordance with the equation d²Q = d²Q₀*d³Q. Physically, d³Q expresses the variations of a rotation about a rotation axis d²A₀ through angle d²ψ₀.

To finish, all that needs to be stored in a database is the angles dPa_o, d²Pa₀, d³Ca and dPa. d²A₀ and d²Qo are deduced from d²Pa₀, dQ₀ and d³Q are deduced from dPa₀ and d³Pa, respectively. Also dQ₀ is deduced from d' Qo and dQ₀. The axis d²A₀ can be computed because it is used for the angular constraints. If the choice is made not to store the variation in the database (the rotation is stored for d³Q ='I), the need to store d³Pa in it can be avoided.

From the graphical point of view, consider two bones Bi and Bi-i (as shown in Figure 8) . The aim is to define the angular constraints at the articulation between the two bones . Consider the case in which the rotation about the articulation is a rotation about the axis d²A₀, the constraint on which is maximum (i.e. this axis has no degree of freedom, in other words γ=0) .

The current rotation dQ of the bone Bi corresponds to a rotation through angle d²ψ about the axis d²A₀. The bone Bi is made to turn about the axis d²A₀ to define the extreme limits d²ψ_Mι_N and d²ψ_A of the angle d²ψ. These extreme limits define:

• The variation Δψ = ²ψMA - d²ψ_MiN/ which gives d³Ca = (Δψ, γ = 0) .

• The quaternion d²Q₀, deduced from the axis d²A₀ and the angle d²ψ₀ = (d²ψ_MIN + d²ψ_MAx)/2. As for the axis, the two angles α and β which define it can be found, and d²Pao is also deduced from it. The "rest" position is that which corresponds to a nil variation (d³Q = I) . The bone Bi is made to turn about the axis d²A₀ until this is obtained. The rotation dQ obtained in this way and from which dQo (or dPao) is deduced is dQ = d'Q₀ = dQ₀*dQ₀. If the rotation axis d²A₀ is free, this freedom is reflected in the possibility of the axis being within a cone with half-angle γ. The axis of the cone is the rotation axis d²A₀. Also, regardless of the position of the rotation axis within the cone, the constraint on the angle d²ψ is always the same.

This means that the constraint on the angle d²ψ can be adjusted for any position of the axis d²Ao as indicated above .

Thereafter, for adjusting the angle γ, the cone may be visualized as shown in Figure 9.

As already stated, the axis of the cone is the rotation axis d²Ao. If the rotation axis is d²A₀' (see

Figure 9) and the constraint on the axis is required not to be equally distributed within the cone in this way, the procedure required is as explained above, with the only difference that the "rest" rotation this time has the axis d²A₀' not the axis d²A₀.

A detailed algorithmic description of the inverse kinematic in accordance with the invention using quaternions will now be given with reference to Figures 10 to 16. Consider a chain of N "bones" B₀ to B_(N-i) . B₍₁-_D is the father of B_x . B₀ is called the "pseudo-root" and B_(N-_D the "pseudo-leaf". The bone B_(N-i) is influenced by a set of fields consisting of a set of attractor fields that represent the target that it must reach and/or a set of repulsive fields that represent one or more obstacles to be avoided.

The set of fields is any combination of fields that can be classed in one of the following three categories:

- spatial fields: an attractor (respectively repulsive) field of this type represents the fact that the pseudo-leaf must reach (respectively avoid) a given point in space; this type of field also concerns or describes translation type transformations;

- directional fields: an attractor (respectively repulsive) field of this type represents the fact that the pseudo-leaf must be oriented along (respectively avoid) a given direction; this is a spatial field for which the point to be reached (respectively avoided) is at infinity; - rotational fields: an attractor (respectively repulsive) field of this type represents the fact that the pseudo-leaf must reach (respectively avoid) a given orientation, i.e. a given rotation.

The algorithm shown in Figure 10 explains how, at each time n, the pseudo-leaf attempts to reach its fixed target, in collaboration with all the other bones of the chain.

To be more precise, the algorithm produces for all the bones Bi of the chain values of the quaternion dQi relating to the father B -u . If there is no father (i = 0), dQi is identical to the absolute quaternion. On each pass of the algorithm the pseudo-leaf moves toward the fixed target at the same time as the other bones of the chain contribute to this result.

The algorithm is repeated as many times as required, for example indefinitely. Once the pseudo-leaf has reached the target, it may be useful to allow the algorithm to continue running. For example, if the quaternion dQi of a bone which is not the pseudo-leaf is forced to change when the target has been reached, the absolute quaternion Q_(N-_D of the pseudo-leaf is modified, unless the algorithm is still running. In other words, if the pseudo-leaf reaches its target, the loop 22 maintains it in that state.

In a first step 20-1, the value N-l (pseudo-leaf index) is assigned to i_ (bone index) . Then (step 20-2) the number N_f of attractor or repulsive fields of the bones Bi is computed: if N_f = 0, there is no field. The computation of these fields is described in detail below, with reference to Figure 11.

A test is performed to determine if N_f is equal to 0 (step 20-3) .

If N_f = 0 (no attractor or repulsive field) the dynamics of Bi (step 20-5) can nevertheless be computed to execute its transient relaxation conditions.

Step 20-4 computes the global field of the bone B_lf which is a combination of the various fields. The computation is described in detail below with reference to Figure 15. Step 20-5 corresponds to computing the speed and the quaternion relating to the bone B_x. This computation is described in detail below with reference to Figure 16.

The algorithm then moves on to the father bone B_^i (step 20-6) . A test is conducted (20-7) to determine if it is the pseudo-root that has just been processed. If i < 0, the algorithm stops.

This algorithm uses real time control (loop 22) . The conjoint use of quaternions, the theory of influence and real time control produces a functional and dynamic inverse kinematic.

Figure 11 shows the computation of the various fields of the bone B_x (here for the attractor fields, the same diagram being valid for repulsive fields) . First of all, the index j_ of a field of influence associated with the pseudo-leaf is initialized (step 30-1) .

Then (step 30-2) , a pointer pQ₀ is associated with the start of a table to be created; this table groups the structures each consisting of a (quaternion, intensity) combination. Each field of influence will be transformed into a structure of this kind.

The number N_f of existing, or non-inhibited, fields stored in the table to which pQ₀ points is then initialized (step 30-3) .

The next test (step 30-4) determines if a field of influence of index j_ of the list of fields of influence associated with the pseudo-leaf exists or is not inhibited. If this field exists and is not inhibited, the type for field E of index j_ is then determined (step 30-5) : is it a rotational, directional or spatial field?

Depending on the type determined, the algorithm then chooses one of the following processes (step 30-6) . During step 30-6, the variation δQ_. (also denoted dQ_Eι) to be applied to the quaternion dQ_x = dQ_l0.δQ_! for the pseudo-leaf to reach its target exactly, for example by a shorter path, is computed. Step 30-6 is described in more detail below with reference to Figures 12 to 14.

As soon as a field has been detected, the variable _f is incremented (step 30-7) and the algorithm moves onto the next element from the table (30-8) . The next field of influence is associated with the pseudo-leaf (step 30- 9) . If j_ is equal to the number N_f of fields of influence associated with the pseudo-leaf the algorithm stops.

Figure 12 shows in detail the computation of the variation δQ which applies in the case of a rotational field.

The rotational field of influence is defined by the absolute quaternion Q_E that the pseudo-leaf must reach (or avoid) and its intensity I_E. Q_E [Q_N-_I] ^_1Qι initializes an absolute quaternion Q.

Whether B has a father is then determined (step 40-2) :

- If it has no father, the algorithm goes direct to step 40-4. - Otherwise, the quaternion Q is replaced by the quaternion Q_U-_D.Q (step 40-3).

Step 40-4 computes the variation of the absolute quaternion Q_E: δQ_E = [dQ₀]^_1Q δQE and I_E are then stored in the table to which pQ points (step 40-5) .

Figure 13 shows in detail the computation of the variation δQ that applies in the case of a directional field. The directional field of influence is defined by its intensity I_E and an absolute vector V_E toward whose axis the pseudo-leaf tends to become parallel. The absolute vector V_E can be characterized by two angles borrowed from astronomy: the right ascension A_E and the declination D_E. The equations used to transform V_E (step 50-1) are:

V_E = (X,Y,Z) = (cosD_E.sinA_E, sinD_E, cosD_E. cosA_E) .

V_E is then computed in the frame of reference of the bone B ( step 50-2 ) .

Next (step 50-3) the quaternion Q associated with the rotation through angle (Ai, V_E) about an axis perpendicular to the plane formed by A (axis of the bone Bi in its local frame of reference) and V_E is sought. In other words, this is a rotation that turns Ai until it is parallel to V_E (the diagram would be similar with repulsive fields) . In the local frame of reference of the bone B_l the coordinates of A_x are (0,0, -Li), L_x being the length of the bone B_x . In Figure 13, the symbol ® means: find the quaternion Q associated with the rotation defined above.

When Q has been obtained, δQ_E is obtained by the operations defined in 50-4 and 50-5. δQ_E and I_E are then stored in the table to which pQ points.

Figure 14 shows in detail the computation of the variation δQ that applies in the case of a spatial field.

The spatial field of influence is defined by an absolute vector V_E whose three coordinates are those a point in the 3D space and its intensity I_E. It is the end of the pseudo-leaf that must reach (or avoid) V_E.

Let A(_N_i) be the axis of the bone B_(N-ι_. (the pseudo- leaf) in its local frame of reference. Its coordinates are therefore: A_(N-i) = (0, 0, -L_(N-i) ) , L_(N-_D being the length of the bone B_(N_i) . E_E is the expression for A_(N_i) in the absolute frame of reference.

Let T(_N-i) (respectively TJ be the absolute translation component of the transformation of the bone B(_N-i) (respectively B . V_A defined in step 60-2 is therefore the absolute vector whose origin is the origin of the bone B and whose end is the end of the bone B_(N-_D (the pseudo-leaf) .

V_A is then completed in the local frame of reference of the bone B_x (step 60-3) . The absolute vector V_B whose origin is the origin of the bone B_x and whose extremity is the point V_E in space to be reached is then computed (step 60-4) . V_E is then computed in the frame of reference local to the bone Bi (step 60-5) .

The quaternion Q associated with rotation through an angle (V_A, V_B) about an axis perpendicular to the plane formed by V_A and V_B is then deduced from V_A and V_B.

Finally, δQ_E is obtained by the operations defined in 60-7 and 60-8 and δQ_E and I_E are stored in the table to which pQ points (step 60-9) .

Figure 15 shows in detail the computation of the global field of the bone Bi, which is a combination of various attractor or repulsive fields (step 20-4 from Figure 10) .

At this point in the algorithm the array of structures {quaternion, intensity} contains a number of structures equal to N_f. If m is the index of this array, the quaternion is written δQ_(E(ιri)) and I_E(m) is its intensity. The aim of this "field Bi" block is to compute the global quaternion δQ_E and the global intensity I_E as a combination of the N_f quaternions and elementary intensities δQ_E( _. and I_E(_m) .

The index m is first initialized to 0 (step 70-1) . Then δQ_E is also initialized, to δQ_E(0) (70-2) . A coefficient k is also defined (step 70-3) . The equation given in Figure 15 is one possible example, but other equations can be used.

Step 70-4 corresponds to spherical interpolation of coefficient k between the quaternions δQ_E and δQ_E(m+D (this interpolation is symbolized by the symbol (°_k)) . I_E is then computed using the equation given in Figure 15 (step 70-5) . Again this is merely one example, and other combinations are possible. The arithmetic or geometric mean of I_Ei can equally be used, for example.

Then, m is incremented (step 70-6) and compared to N_f (step 70-7) . If N_f is reached, the algorithm stops. Otherwise it loops to step 70-3.

Figure 16 shows in detail the computation of the speed and the quaternion relating to the father bone (step 20-5 from Figure 10) .

In a first step, a coefficient I can be computed which represents the intensity (of the global field) which allows for the inertia of the bone Bi. Step 80-2 is a spherical interpolation of coefficient I between the quaternions δQ_x and δQ_E. This is, as it were, a step of taking account of the field.

The field δQ_E which allows for the global intensity I_E and the intensity _u is then computed (step 80-3) . The "speed" quaternion Q_v of the quaternion δQ_x is then computed (step 80-4) . Spherical interpolation between the identity quaternion Q_τ and the previous speed (that at the previous time n-l) Q_v determines the new speed. The coefficient of this interpolation is k_aι, the damping coefficient of the bone B .

The speed is multiplied (quaternion multiplication) by the field δQ_E (step 80-5) . Step 80-6 corresponds to computing the variation δQ_x that results from the speed. The variation dQ_. is then computed (step 80-7) . Any embodiment of the invention animates a character easily and quickly. Taking the character shown in Figure 3 as an example, it is sufficient in accordance with the invention to determine the position, or the target to be reached, for a few pseudo-leaves such as the hands, a foot, an eye, etc. It is not necessary to define the motions of each point of the character.

The technique in accordance with the invention is used to compute the motion of an object from fields of influence (attractor fields or repulsive fields) which define the environment of the object. The motion depends on the various influences to which the various bones of the object are subject. This process can also be combined with the model of animating points of an image explained at the beginning of this detailed description, in particular with reference to equations (1) through (12) .

The method of processing or animating images in accordance with the invention can be implemented using apparatus like that shown in figures 17A and 17B.

Figure 17A represents a graphics workstation comprising a microcomputer 120 appropriately configured for generating and processing two-dimensional or three- dimensional graphics objects using a method in accordance with the invention, an output peripheral device 122 and control peripheral devices (keyboard 124 and mouse 125) . The microcomputer 120 has a computation section with all the electronic, software and other components needed for generating images and processing or animating the images so obtained.

To be more precise, the system uses a PC type microcomputer of moderate power. The microcomputer comprises a central processor unit 120 which comprises a microprocessor 126, a memory 128, an input peripheral device, for example of the hard disc type 132, connected to a bus 130. An output peripheral device, for example of the screen type 122, or display apparatus controlled by a video card is used to display information. The microcomputer also has control peripheral devices, in this instance the keyboard 124 and the mouse 125. The programming language used is the C++ language, together with the assembler language for Pentium microprocessors. Models are written to and read from a hard disk. The models are displayed on the screen 122 of the computer by means of the graphics subsystem, in other words :

- Either using a ray tracing algorithm which writes the points directly into the graphics memory of the computer so that they are displayed on the screen by the video card (hardware part of the graphics subsystem) .

- Or using algorithms controlling control software (API) of a 3D card (able to display three-dimensional information), i.e. a card dedicated to displaying polygons rather than points; the 3D card thereafter converts the polygons into points to be displayed on behalf of the video card.

The control peripheral devices, in particular the keyboard 124 or the mouse 125, are used to manipulate the objects, for example to select and move the various "bones" or segments of an articulated system, the morphic points in the 3D space represented on the screen, the attractive or repulsive fields of the articulated system, etc., or to show and clear a pull-down menu, i.e. a display area made up of parameters. These can also be varied using other keys of the keyboard 124.

From the software (application) , the object is loaded from the hard disk into the main memory of the computer and displayed on the screen. The manipulator then uses the controls (control peripheral devices) to carry out the required operations. The program comprises the following facilities, for example:

- initializing the application.

- loading a model described by the method.

- displaying a model: - determining, for example from splines, initial settings of their parameters and the algorithms of the method (for example, the parameters of the inverse kinematic) , the three-dimensional elements (polygon) .

- transforming the three-dimensional elements into two-dimensional elements which can thereafter be displayed on a screen (the two-dimensional space) .

- displaying the two-dimensional elements generated in this way, either using a ray tracing algorithm or using a library of functions and dedicated hardware cards commonly used in this field (3D cards) .

- manipulating the objects (bones or sections or morphic points or fields or textures or light) by modifying their parameters (texture coordinates, position, length, Ki, K_a, etc.). - quitting the application.

A program for implementing the method of the invention is resident or stored on a storage medium (for example: diskette or CD-ROM or removable hard disk or magnetic storage medium) which can be read by a data processing system or by the microcomputer 120. The program concerns, for example, a method of animating an image comprising a chain of N segments i. (i = 0, 1, 2, ..., N-l) between a root segment 0 and a pseudo-leaf end segment N-l, each of the segments being defined by an orientation in space (ct, β, ψ) , the pseudo-leaf segment having to reach a target angular orientation (α_target/ βt_arge_tr ψta_rget) • It comprises, for example, instructions commanding the microcomputer to:

- define an image field function comprising:

• an attractive component associated with the target angular orientation and of zero value for that orientation, and

• if angular constraints are to be complied with, a repulsive component associated with each constraint,

- determine the motion of each segment from an equation of motion as a function of the field for the initial orientation of the segment, and

- possibly, animate the chain of segments in accordance with the motion thus determined.

The shape or the points are displayed on the graphics screen 122 in real time. The method of the invention enables the motion of the object to be displayed in real time.

Or the program concerns, for example, a method of animating an image comprising a chain of N segments i_ (i = 0, 1, 2, ..., N-l) between a root segment 0 and a pseudo-leaf end segment N-l, each segment i_ being identified relative to the father segment _i-l of the segment i by a geometrical transformation dTRF₍₁₎ = dTRF0₍₁)*δTRF₍₁) made up of a rest transformation dTRF0 ₎ and a transformation δTRF₍₁₎ representing the motion, each of these transformations comprising an associated rotational component dROT₍₁₎, dROT0₍₁₎ and δROT₍₁₎ . The program comprises, for example, instructions commanding the microcomputer to:

1. Determine a target to be reached for the pseudo- leaf . 2. Select the pseudo-leaf as the current segment.

3. compute, for the current segment, the rotation δROTB(i) to reach the target fixed for that segment.

4. Determine, for the father segment of the current segment, the target to be reached. 5. Select the father segment of the current segment as the current segment and reiterate step 3.

6. Possibly, move the chain of segments as a function of the rotations thus determined.

Again, the shape or the points are displayed on the graphics screen 122 in real time. The method of the invention enables the motion of the object to be displayed in real time.

Or the program concerns, for example, a method of moving and/or animating an articulated system displayed in an image, the system comprising N segments _i (i = 0, 1, 2, ..., N-l) between a root segment (i = 0) and a pseudo-leaf end segment (i = N-l) , and the segments being joined two by two by articulations, a target orientation of the pseudo-leaf being determined, each segment i. being identified, relative to the father segment i-1, by a quaternion dQi and, relative to an absolute frame of reference, by a quaternion Qi, the target orientation to be achieved being itself represented in the absolute frame of reference by a quaternion Q_B. The program comprises, for example, instructions commanding the microcomputer to:

- define, for each section i , an image field quaternion comprising:

• an attractive component associated with the target orientation, and

• if constraints are imposed, a repulsive component associated with each constraint. - compute the speed quaternion for each section _i, and

- possibly, move each section of the articulated system in the image as a function of the speed thus obtained.

Again, the shape or the points are displayed on the graphics screen 122 in real time. The method of the invention enables the motion of the object to be displayed in real time. The microcomputer comprises means for computing and storing in memory potential or field values and/or values of k_a and/or ki of the segments concerned or selected and the speed and new coordinates of the points. These values can be transmitted to a display control unit. The microcomputer 120 can be programmed to generate

NURBS surfaces as described above. It can also be programmed for the generation or computer-aided design (CAD) of shapes. The data relating to the shape, or the points that define it, in two or three dimensions, is stored in memory with the corresponding dynamic parameters .

The microcomputer 120 is also connected to other peripheral devices, for example printing devices (not shown in the figure) . It can be connected to an electronic communications network, of the Internet type, for example, for sending or receiving data relating to the shapes generated. For this purpose the microcomputer is equipped with means for connecting to the network, or input/output interface means for the network, for example a modem. In accordance with the invention, the data transmitted comprises data on the relative coordinates of the points defining the object and also data relating to the dynamics of the points and of the object itself and obtained by one of the methods in accordance with the invention.

The invention therefore also concerns the reception of such data, relating to the object or the chain or the skeleton and to its dynamics, and obtained by one of the methods described above.

Generally, the data of images and/or the data relating to the points or to the skeletons of objects can be stored in a memory area of the device 120 with the corresponding dynamic parameters.

In accordance with the invention, the animation or the movement of the segments of the articulated system or of the chain can be displayed or the corresponding data (data on the segment and animation data) sent via a wide area communications network (WAN) or local area communications network (LAN) to other display apparatus or to other data processing or storage apparatus, also connected to the network. As already mentioned in the introduction, the invention can be applied to cinema film production, video games and methods of designing technical components where it is important not only to represent the components but also to be able to observe their behavior in response to simulated deformation.

Claims

1. A method of animating an image comprising a chain of N segments i. (i = 0, 1, 2, ..., N-l) between a pseudo-root segment 0 and a pseudo-leaf end segment N-l, each of the segments being defined by an orientation in space (α, β, ψ) , the pseudo-leaf segment having to reach a target angular orientation (α_target, β_ta_rget Ψta_rget) . ^tne method comprising the following steps:

- defining an image field function comprising: • an attractive component associated with the target angular orientation and of zero value for that orientation, and • if angular constraints are to be complied with, a repulsive component associated with each constraint,

- determining the motion of each segment from an equation of motion as a function of the field for the initial orientation of that segment, and

- animating the chain of segments in accordance with the motion thus determined.

2. A method according to claim 1, further comprising computing the speed of each segment proportionately to the image field value for that segment , the segment being moved as a function of the computed speed.

3. A method according to claim 1, further comprising computing the speed of each segment from the equation

V_n = K_aV^ + ^ (E_n + E'_n) in which:

. n is an integer which represents (discrete) time, . V_n is the "speed" vector of the segment,

. E_n and E'_n respectively represent the attractor field and repulsive vector fields at time n, and

. K and K_a are respectively an inertia coefficient and a damping coefficient of each segment influenced by the fields E and E' .

4. A method according to claim 3, wherein the field E_n is proportional to the vector with components :

(^αtarget ^{~ α} (n-l) ) ' ( βtarget ^~ P (n-l) ) - (ψtarget " Ψ (n-l) ) ' where α^.^, β(_n-_D, and ψ_(n.i) are the angular components of the orientation of the segment concerned at time n-l.

5. A method according to claim 3 or claim 4, wherein K_a is made less than 1.

6. A method according to claim 5, wherein K_a is equal to 0.

7. A method according to claim 3, wherein the repulsive field E'_n represents the fact that each parameter ct, β and ψ can vary only within certain limits: α e [ct_min, ct_max] , β ^e [β_mi_n - P_ma and ψ € [ψ_min, ψ_maχ] .

8. A method according to claim 1, where at least one of the segments i₀ has a limit angular speed v_lim that it may not exceed.

9. A method according to claim 8, wherein determining the motion of the segment i₀ comprises the following steps:

- computing the "norm" of the speed:

|v_n| = I ^ l + | v_βn | + | v_ψn | , and if |v_n| < v_lim, determining the position of the segment from the equation an — ^ ain-l) in which : n is an integer that represents (discrete) time, P_an is the angular position at a certain time, and ^p _a(_n-i) is ^tne angular position at "the preceding time", or, otherwise, determining the position of the segment from the equation Pan = Pa (n-l) + ^Vn N_lim/ 1| V_a || ( 13 - 2 ' )

10. A method of animating an image comprising a chain of N segments i (i = 0, 1, 2, ..., N-l) between a root segment 0 and a pseudo-leaf end segment N-l, each segment i. being identified relative to the father segment i-l of the segment i by a geometrical transformation dTRF_(i) = dTRFO _(i)* δTRF_(i) made up of a rest transformation dTRFO _{i) and a transformation δTRF_(i) representing the motion, each of these transformations comprising an associated rotational component dROT_(i)/ dROT0_(i) and δROT_(i) ; the method comprising the following steps:

1. determining a target to be reached for the pseudo-leaf ,

2. selecting the pseudo-leaf as the current segment, 3. computing for the current segment the rotation δROTB_(i) to reach the target fixed of that segment,

4. determining the target to be reached for the father segment of the current segment,

5. selecting the father segment of the current segment as the current segment and reiterating step 3,

6. moving the chain of segments as a function of the rotations thus determined.

11. A method according to claim 10, wherein each target to be reached is determined by three angles ot_target, β_target' Ψtarget •

12. A method according to claim 10 or claim 11, wherein the step of determining the rotation δROTB_(i) for the current segment for reaching the target fixed for that segment comprises : - defining an image field function comprising:

• an attractive component associated with the angular orientation of the fixed target and of zero value for that orientation, and

• if angular constraints are to be complied with, a repulsive component associated with each constraint, and - determining the rotation δROTB_(i) from an equation of motion as a function of the field for the initial orientation of the segment.

13. A method according to claim 12, further comprising computing the speed of each segment proportionately to the image field value for that segment.

14. A method according to claim 12, further comprising computing the speed of each segment from the equation

in which:

. n is an integer which represents (discrete) time

. V_n is the "speed" vector of the particle . E_n and E'_n respectively represent the attractor field and repulsive vector fields at time n, and

. Ki and K_a are respectively an inertia coefficient and a damping coefficient of each segment influenced by the fields E and E' .

15. A method according to claim 14, wherein the field E_n is proportional to the vector with components :

( ^αtarget ^{" α} (n-l ) ) ' ( βtarget ~ P (n-l) ) - ( ψtarget ^" Ψ (n-l ) ) ' in which α_(n.₁₎, β_fn-_D, and ψ_(n-_D are the angular components of the orientation of the segment concerned at time n-l.

16. A method according to claim 14, wherein K_a is made less than 1.

17. A method according to claim 16, wherein K_a is equal to 0.

18. A method according to claim 10, further comprising a step of computing for each segment the additional rotation to be effected for the pseudo-leaf to reach a particular point in space.

19. A method according to claim 10, further comprising a step of computing for each segment the additional rotation to be effected for all the segments to be attracted or moved in a common direction.

20. A method according to claim 10, wherein the target and the rotation of each segment are computed in a linear upward order, from segment N-l to segment 0, the target to be reached for segment _i depending only on the target to be reached for segment i+1.

21. A method according to claim 10, wherein the target and the rotation of each segment i_ are computed absolutely .

22. A method of moving and/or animating an articulated system displayed on an image, the system comprising N segments i_ (i = 0, 1, 2, ..., N-l) between a root segment i = 0 and a pseudo-leaf end segment i=N-l, and the segments being joined in pairs by articulations, a target orientation of the pseudo-leaf being determined, each segment _i being identified relative to the father segment i-l by a quaternion dQ_x and relative to an absolute frame of reference by a quaternion Q_x, the target orientation to be reached being represented in the absolute frame of reference by a quaternion Q_B, the method comprising:

- for each section _i, defining an image field quaternion comprising:

• an attractive component associated with the target orientation, and • if constraints are imposed, a repulsive component associated with each constraint,

- computing the speed quaternion for each section i, and - moving each section of the articulated system in the image as a function of the speed thus obtained.

23. A method according to claim 22, wherein the speed quaternion at a time n is computed by spherical interpolation between the identity quaternion and the speed at time n-l.

24. A method according to claim 22 or claim 23, further comprising a step of determining the quaternion dQi for each segment i .

25. A method according to claim 22, wherein the quaternion V_n is given by the equation:

V_n = Q_En- [Q_l(°_ka)V_(n-₁₎] in which:

- n is an integer which represents (discrete) time, Qi_n being the quaternion Qi at time n,

- V_n is the "speed" quaternion of the segment i_ at time n,

- Q_EΠ represents the field quaternion at time n comprising the attractive components and any repulsive components, and

- the symbol °_Ka represents spherical interpolation of the coefficient K_a, the damping coefficient of each segment influenced by the fields E and E' .

26. A method according to claim 22, wherein each attractor field of each section is computed in the form of a unit attractor quaternion and a unit intensity and all of the elementary attractor fields are then combined to obtain a global attractor field in the form of a global attractor quaternion and a global intensity.

27. A method according to claim 26, wherein the global attractor quaternion is obtained by spherical interpolation between the various unit quaternions.

28. A method according to claim 26, wherein the global intensity is a linear combination of the unit intensities .

29. A method according to claim 22, wherein the attractor fields are of the spatial, directional or rotational type or a combination of two or three of these various types .

30. A method according to claim 29, wherein a directional field is defined by a vector and an intensity.

31. A method according to claim 29, wherein a spatial field is defined by an intensity and a vector whose three coordinates are those of a point in the 3D space.

32. A method according to claim 29, wherein a rotational field is defined by a quaternion and an intensity.

33. A method according to claim 29, further comprising, for each unit field, a step of computing the variation δQi to be applied to the quaternion dQ for the pseudo-leaf to reach its target.

34. Apparatus for animating an image comprising a chain of N segments i (i = 0, 1, 2, ..., N-l) between a root segment 0 and a pseudo-leaf end segment N-l, each segment being defined by an orientation in space (α, β, ψ) , the pseudo-leaf segment having to reach a target angular orientation (ct_target. βtarget' Ψ_ta_rget) the device comprising: - means for defining an image field function comprising:

• an attractive component associated with the target angular orientation and of zero value for that orientation, and

• if angular constraints are to be complied with, a repulsive component associated with each constraint,

- means for determining the motion of each segment from an equation of motion as a function of the field for the initial orientation of that segment, and

- means for animating the chain of segments in accordance with the motion thus determined.

35. Apparatus for animating an image comprising a chain of N segments i. (i = 0, 1, 2, ..., N-l) between a root segment 0 and a pseudo-leaf end segment N-l, each segment i. being identified relative to the father segment i-l of the segment i by a geometrical transformation dTRF_(i) = dTRFO _(i)*δTRF_(i) made up of a rest transformation dTRFO _(i) and a transformation δTRF_(i) representing the motion, each transformation comprising an associated rotational component dROT_(i), dROT0_{i) and δROT_(i) , the apparatus comprising means for:

1. determining a target to be reached for the pseudo-leaf , 2. selecting the pseudo-leaf as the current segment,

3. computing for the current segment the rotation δROTB_(i) for reaching the target fixed for that segment,

4. determining the target to be reached for the father segment of the current segment, 5. selecting the father segment of the current segment as the current segment and reiterating step 3,

6. moving the chain of segments as a function of the rotations thus determined.

36. Apparatus for moving and/or animating an articulated system displayed on an image, the system comprising N segments i (i = 0, 1, 2, ..., N-l) between a root segment 0 and a pseudo-leaf end segment i = N-l, and the segments being joined in pairs by articulations, a target orientation of the pseudo-leaf being determined, each segment i being identified relative to the father segment i-l by a quaternion dQi and relative to an absolute frame of reference by a quaternion Q, the target orientation to be reached being represented in the absolute frame of reference by a quaternion Q_B, the apparatus comprising: - means for defining, for each section i., an image field quaternion comprising:

• an attractive component associated with the target orientation,

• if constraints are imposed, a repulsive component associated with each constraint,

- means for computing the speed quaternion for each section i., and

- means for moving each section of the articulated system in the image as a function of the speed thus obtained.

37. Apparatus according to claim 34, further comprising means for modifying the field function and/or the equation of motion.

38. Apparatus according to claim 35, further comprising means for modifying the computation of the rotation and/or the determination of the target to be reached.

39. Apparatus according to claim 36, further comprising means for modifying the field quaternion and/or the computation of the speed quaternion.

40. Apparatus according to claim 37, wherein the means for modifying the field function and/or the equation of motion modify them in real time.

41. Apparatus according to claim 38, wherein the means for modifying the computation of the rotation and/or the determination of the target to be reached modify them in real time.

42. Apparatus according to claim 39, wherein the means for modifying the field quaternion and/or the computation of the speed quaternion modify them in real time.

43. Apparatus according to any one of claims 34 to 36 connected to an electronic communications network.

44. A storage medium having instructions therein for causing a computer to perform a method according to any one of claims 1, 10 and 22.

45. A computer program, loadable into the internal memory of a computer, comprising software code portions performing a method according to any one of claims 1, 10 and 22.