## Section: Scientific Foundations

### Dynamic models of motion

- Animation:
Models and algorithms that produce motion accordingly to the animator specification.

- Physically Based Animation:
Animation models which take into account the physical laws in order to produce motion

- Hybrid System:
dynamic system resulting of the composition of a part which is differential and continuous and a part which is a discrete event system.

- State Vector:
data vector representing the system at time t , for example: position and velocity.

The use of 3D objects and virtual humans inside a virtual environment imply to implement dedicated dynamic models. However, the desired interactivity induces the ability to compute the model in real-time. The mathematical model of the motion equations and its corresponding algorithmic implementation are based on the theory of dynamic systems and uses tools from mechanics. The general formulation of these equations is a non-linear second order (in time) differential system coupled with algebraic equations (DAE: Differential Algebraic Equation) defined by:

where is the mass matrix, are the actions and non-linear effect (Coriolis) and are the output parameters describing the system. In case are known and is unknown, inverse dynamic approaches are mandatory to solve the problem. If we concentrate on deformable objects, the equation becomes a second order (in time) and first order (in space) differential system defined point wise on the domain D occupied by the object:

where stands for the current position, is the stress tensor in the material and is related to the deformation tensor by the relation ( is the constitutive material law tensor), is the specific mass and is a given force by volume unit (say gravity). These equations have to be solved by approximation methods (Finite Element Method: FEM) which may be difficult in real time. When contact or collisions occur, they lead to discontinuities in the motion. To solve the above DAE system, we prefer to use a discontinuous formulation expressed in terms of measure that is issued from Non-Smooth Contact Dynamics (NCSD) where is the density of the contact impulsion. As a collision involves a local deformation of the contacting objects, another choice is to consider the deformation of the object. This resolution is expected to be more precise but also to violate the real time constraint.

For motion control, the structure of the dynamic model of the motion becomes a hybrid one, where two parts interact. The first one is the above-mentioned differential part while the second one is a discrete event system:

In this equation, the state vector is related to the command vector .