Determination of Force Fields for Ode-based and Skeleton Driven

Character Animation

L. H. You

, X. S. Yang

, X. Jin

, E. Chaudhry

and Jian J. Zhang

National Centre for Computer Animation, Bournemouth University, Poole, U.K.

State Key Lab of CAD & CG, Zhejiang University, Hangzhou, China

Keywords: Skeleton-driven Character Animation, Skin Deformations, Ordinary Differential Equations, Time-dependent

Varying Force Fields.

Abstract: In the existing work, character modelling and animation are two separate tasks. After a character model is

built, a lot of time and efforts are still required to animate the character model. Ordinary differential

equation-based surface modelling and animation using physically based deformable curves to define and

deform skin surfaces of 3D character models. With such a technique, character modelling and animation can

be integrated into a unified framework by introducing time-dependent varying force fields into ordinary

differential equations. This paper addresses the issue of determination of the force fields and proposes three

models, i. e. linear transformation model, interpolation model and extrapolation model, to obtain time-

dependent varying force fields. Some application examples are presented which demonstrates the force

fields obtained from the three models create believable skin deformations of character models.

1 INTRODUCTION

Skin deformation plays a very important role in

realistic character animation. Various techniques of

animating skin deformation have been developed.

These techniques can be divided into three groups:

skeleton-driven, physics-based and data-driven ones.

Among various skeleton-driven techniques,

skeleton subspace deformation (SSD) is the most

popular and widely used one. It was investigated by

Thalmann et al., (1988), Lander, (1998; 1999),

Weber, (2000), Wang and Phillips, (2002), Mohr

and Gleicher, (2003), Yang et al., (2006) and Kavan

et al., (2005).

Physics-based techniques were investigated by a

lot of researchers such as Chen and Zeltzer, (1992),

Wilhelms and Van Gelder, (1997), Scheepers et al.,

(1997), Jane and Allen, (1997), Nedel and Thalmann,

(1998), Nedel and Thalmann, (2000), Aubel and

Thalmann, (2001), Capell et al., (2002), Maryann et

al., (2002), James and Pai, (2002), Larboulette and

Cani, (2004), Guo and Wong, (2005), Venkataraman

et al., (2005), Teran et al., (2005) and Capell et al.,

(2007).

Data-driven techniques were proposed to

improve the skeleton subspace deformation by some

researchers such as Lewis et al. (2000), Allen et al.,

(2002), Mohr and Gleicher, (2003), Kurihara and

Miyata, (2004), Rhee et al., (2006), and Weber et al.,

(2007).

In addition to the above approaches, curve-based

surface modeling has also been introduced. The

examples include Shen and Thalmann, (1994), Singh

and Fiume, (1998), Pyun et al., (2004), Hyun et al.,

(2005), Yoon and Kim, (2006), Nealen et al., (2007),

and Gal et al., (2009).

Although curve based surface deformations

become more active in recent years, how to

introduce the underlying physics into curve based

surface manipulation for more realistic deformations

remains an open problem.

Ordinary differential equation-based surface

modelling uses the same methodology as that of

beam bending and can be regarded as physics-based.

It integrates modelling and animation into a same

framework by introducing time-dependent varying

force fields.

Skin deformations of character models using

ODE curve-based surface modelling and animation

combine the strengths of skeleton-driven, physics-

based, and data-driven techniques together. It also

takes advantage of the high efficiency of curve-

based surface modelling.

Due to the importance of force fields in ODE-

based surface modelling and animation, this paper

115

You L., Yang X., Jin X., Chaudhry E. and Zhang J..

Determination of Force Fields for Ode-based and Skeleton Driven Character Animation .

DOI: 10.5220/0004297301150118

In Proceedings of the International Conference on Computer Graphics Theory and Applications and International Conference on Information

Visualization Theory and Applications (GRAPP-2013), pages 115-118

ISBN: 978-989-8565-46-4

 2013 SCITEPRESS (Science and Technology Publications, Lda.)

will address the issue how to determine the force

fields.

2 MATHEMATICAL MODELS

The determination of time-dependent varying force

fields is to change the force field at the rest pose to

those at required poses. In this section, we propose

three models: Linear transformation model,

interpolation model, and extrapolation model.

2.1 Linear Transformation Model

The linear transformation model uses the same

methodology as that of the skeletal subspace

deformation to transform a force field at the rest

pose to the required poses.

A force field consists of sculpting forces at the

curve vertices. Therefore, we achieve the linear

transformation of a force field by considering a

sculpting force acting at a curve vertex

mni

. The

sculpting force acting at the vertex at the rest pose

can be represented by

rmni

in the global coordinate

system. Here the subscript

is used to indicate the

rest pose.

First, we transform this sculpting force into the

local coordinate system of the jth bone. The

geometric transformation matrix from local to global

coordinate system for the jth bone at the rest pose

can be written as

Next, the bone is translated and rotated. The

sculpting force defined in the local coordinate

system of the jth bone is transformed back to the

global coordinate system at the new pose. And a

weight

is applied to scale the sculpting force to

create realistic skin deformation. The geometric

operation transforming the sculpting force back to

the global coordinate system is described by the

matrix

where the subscripts

indicate the

geometric transformation from the jth transformed

bone to the global coordinate system.

If there are

bones whose movements have the

influences on the curve vertex

mni

, the resultant

sculpting force at the curve vertex

mni

should be

the sum of those caused by each of the bones which

can be written as

rmni j gj rj rmni











PMMP

(1)

2.2 Interpolation Model

The interpolation model requires knowing the force

fields at the rest pose and the final pose. Then the

difference between the two force fields is found and

the linear interpolation operation is used to find the

force fields at the poses between the rest and final

poses.

Assuming that

and

is the force fields

at the rest post and final pose, respectively, we first

transform these two force fields to the pose



and

obtain the transformed force fields





and





Then, we determine the force field at the pose



using the following linear interpolation

rfr

mn mn mn







PP P



(2)

where



is the rotation angle from the rest post to

the final pose.

2.3 Extrapolation Model

The determination of the force field using the

extrapolation model is similar to that using the

interpolation model. Both models use the skin shape

at the rest pose. However, the extrapolation model

used the skin shape at the pose adjacent to the rest

pose. That is to say, we use the pose adjacent to the

rest pose to replace the final pose in the interpolation

model. Accordingly, equation (2) can be used to

determine the force fields at any poses beyond the

rest and adjacent poses by replacing the subscript

and superscript

with the subscript and superscript

which indicates the pose adjacent to the rest pose.

3 APPLICATION EXAMPLE

The three models proposed in above section were

used to determine the force fields of human fingers,

a horse front leg, and a horse rear leg. The obtained

force fields

)(uF

were incorporated into the

following equation

()



(3)

where

12(1 )







(4)

GRAPP2013-InternationalConferenceonComputerGraphicsTheoryandApplications

116

Solving Eq. (3) subjected to the following

boundary constraints where

)(),( uvu

CS 

)(

),(

)(),( 1

)(

),(

)(),( 0

vvuu













(5)

we obtain the mathematical expression of a surface

which is used to create skin deformations of these

models.

3.1 Deforming Human Fingers with

Linear Transformation Model

The linear transformation model was used to

determine the force fields of human fingers whose

skin shape at the rest pose is shown in Figure 1a.

The obtained results were shown in b-d of Figure 1.

a b c d

Figure 1: Skin deformation of human fingers from the

linear transformation model.

3.2 Deforming a Horse Front Leg with

Interpolation Model

In this subsection, we used the interpolation model

to determine the force fields of a horse front leg at

any poses between the rest pose and final pose. The

skin shapes at the rest pose was indicated in Figure

2a and that at the final pose was presented in Figure

2e. The deformed skin shapes created were

demonstrated in Figures 2b, 2c, and 2d.

a b c d e

Figure 2: Skin deformation of a horse front leg from the

interpolation model.

3.3 Deforming a Horse Rear Leg with

Extrapolation Model

Finally, the extrapolation model was employed to

determine the force fields of a horse rear leg at the

poses beyond the two adjacent poses. The skin

shapes at the two adjacent poses were given in

Figures 3a and 3b. The deformed skin shapes created

were depicted in Figures 3c, 3d and 3e.

a b c d e

Figure 3: Skin deformation of a horse rear leg from the

extrapolation model.

4 CONCLUSIONS

How to determine time-dependent varying force

fields for ordinary differential equation-based

surface modelling and animation has been addressed

in this paper. We proposed three models to generate

time-dependent varying force fields. They are linear

transformation model, interpolation model, and

extrapolation model.

In order to demonstrate the effectiveness of the

proposed three models, we present some examples

of skin deformations of character models. These

application examples indicate that the force fields

determined by the proposed three models create

believable skin shapes.

Our future work is to introduce time-dependent

varying force fields into dynamic ordinary

differential equations which involves a time variable

and considers dynamic effects of skin deformations.

ACKNOWLEDGEMENTS

This research is supported by the grant of UK Royal

Society International Joint Projects / NSFC 2010.

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