# Physics of Realistic Modelling and Animation (FMAR)

Credits Dept.
7.5 (6.0 ECTS) FIS

## Instructors

 Person in charge: (-) Others: (-)

## General goals

The aim of this subject is give students an understanding of physics-and in particular mechanics-in order to enable them to build physically realistic mathematical models of articulated systems (robots, vehicles, animated bodies with skeletons, etc.). The models they study will enable them to describe the kinematics and dynamics of the physical systems they study, and they will also be introduced to the numerical integration methods used for obtaining the resulting movement, which thus yields a physically realistic form of animation.

## Specific goals

### Knowledges

1. Transformation relationships between reference systems. Mathematical modelling of rigid, articulated systems. Denavit-Hartenberg formalism.
2. Mathematical modelling of the physical properties of large bodies (a rock, a rigid element), articulated rigid systems (robots, industrial handling devices). Mass distribution, inertia tensor.
3. Kinematics and dynamics of multi-particle systems. Conservation theorems. Types of forces acting: gravity, aerodynamic resistance, elastic forces. Collisions.
4. Dynamics under constraints. Lagrange formalism.
5. Animation of physically realistic systems. Integration methods. Visualisation of objects and systems in movement.

### Abilities

1. Construction of a geometric mathematical model suitable for describing a physical system (robot, vehicle, etc.).
2. Identification of the variable space corresponding to possible system configurations. Ability to determine the values of articulation variables in order to attain a static configuration.
3. Learn the fundamental physical laws underlying the modelling of a system. Determining the equations that describe a system"s dynamics.
4. Identification of generalised variables applicable to systems operating under constrained dynamic conditions. Determination of constrained dynamic equations.
5. Construction of an animation based on the numerical solution of the system"s dynamic equations.

### Competences

1. Use of mathematical formalism to construct physical models of the real world.
2. Ability to summarise and present results.

## Contents

Estimated time (hours):

 T P L Alt Ext. L Stu A. time Theory Problems Laboratory Other activities External Laboratory Study Additional time

1. Geometric transformations in space.
T      P      L      Alt    Ext. L Stu    A. time Total
2,0 1,0 2,0 0 0 5,0 0 10,0
Reference systems. Position and orientation. Transformation relationships in reference systems.

• Laboratory
Visualisation of objects. Position and orientation. Relationships between geometrical transformations.

2. Rigid, articulated systems.

T      P      L      Alt    Ext. L Stu    A. time Total
6,0 3,0 6,0 0 0 15,0 0 30,0
• Laboratory
Movements of rigid elements. Denavit-Hartenberg description. Articulated robots. Pseudo-inverse Jacobean determinant.

3. Systems of N interacting bodies.
T      P      L      Alt    Ext. L Stu    A. time Total
4,0 2,0 4,0 0 0 10,0 0 20,0
Kinematics and dynamics of multi-particle systems. Conservation theorems. Types of forces acting: gravity, aerodynamic resistance, elastic forces. Collisions.

• Laboratory
Conservation theorems. Elastic forces. Collisions.

4. Kinematics and the dynamics of a rigid solid.
T      P      L      Alt    Ext. L Stu    A. time Total
4,0 2,0 4,0 0 0 10,0 0 20,0
• Laboratory
Movement of a free solid rigid body. Movements under forces and impacts.

T      P      L      Alt    Ext. L Stu    A. time Total
6,0 3,0 2,0 0 0 15,0 0 26,0
• Laboratory
Dynamics and movement in constrained systems.

6. Animation of physically realistic systems.
T      P      L      Alt    Ext. L Stu    A. time Total
6,0 3,0 10,0 0 8,0 15,0 0 42,0

Advanced methods for integrating equations expressing movement.

Trajectory. Collisions - detection and treatment.

Dynamic visualisation of moving objects and systems employing kinematic constraints.

• Laboratory
Creation of a practical computer, drawing on the knowledge acquired throughout the course and applying it to a practical case based on basic numerical computer calculation (See section on teaching methodology).
Preparation of a report on the practice session.

 Total per kind T P L Alt Ext. L Stu A. time Total 28,0 14,0 28,0 0 8,0 70,0 0 148,0 Avaluation additional hours 2,0 Total work hours for student 150,0

## Docent Methodolgy

The teaching methodology will be based on theory classes, classes of problems, practical exercises, and a practical session covering computer animation and drawing upon the knowledge acquired during the course and on basic numerical computer calculation. The practical session forms an integral part of the course. Students will carry out the work in pairs.

This will consist of a physically realistic animation of an autonomous robot (or of a chosen physical system of similar complexity). The physical characteristics of the system, the environment in which it operates, and desired movement are used to determine the movements of each robot articulation. These movements are then considered in the light of the system"s overall evolution and the physical laws governing movement.

In carrying out this practical work, students must complete all of the following stages:

1) Mathematical modelling of the robot: determination of

Denavit-Hartenberg table parameters, the inertia tensors of the various elements, and the maximum forces of each articulation.

2) Specification of the set of variables and the linking conditions of the movement one wishes to create.

3) Automatic, explicit generation of transformation matrices and of the relevant Jacobean matrix. Numerical resolution of the inverse kinetic problem and obtaining the theoretical forces for each of the articulations.

4) Obtaining the real forces acting on the robot, and introducing these in the robot model equations to generate real movement.

5) Exporting movement to a rendering system, and generating the animation. Submission of the results of the practical work accompanied by a report on the same.

The practical work will be carried out during lab tutorials, during which students will have banks of tests at their disposal so that they can check their progress towards the objectives set for each stage. Only the physical part of the problem will be solved during the practical sessions. To help students generate the graphic animation, they will receive help in the form of a complete description of the physical system in the animation"s first frame. Students" work will therefore be limited to calculating the translation and rotation increments of the various elements in the following frames, using numerical calculation to solve the dynamic equations (which will be read by the rendering system to generate the animation images).

## Evaluation Methodgy

Assessment will be based on two exams (a part exam and a final exam) and the lab grade (Nota_lab).

The weights of the non-eliminatory part exam and the final exam will be 25% and 75% respectively, (0% and 100%, respectively, where the final exam grade is higher than the grade for the part-exam). The assessment will take into account the extent to which students achieve the objectives set for each stage of the practical work.

Nota_curs = (Nota_ex + Nota_lab) / 2

## Basic Bibliography

• Domingo García Senz, Elvira Guàrdia Manuel Elements de mecànica aplicada a la robótica, Edicions UPC, 1996.
• William F. Riley, Leroy D. Sturges. Ingeniería mecánica, Reverté, 1995-1996.
• J.M. Selig. Introductory robotics, Prentice Hall, 1992.

## Complementary Bibliography

• Miquel Grau Sánchez, Miquel Noguera Batlle Càlcul numèric, Edicions UPC, 1993.
• Herbert Goldstein, Charle Poole, John Safko Classical mechanics, Addison-Wesley, 2002.
• Ferdinand P. Beer ... [et al.] Mecánica vectorial para ingenieros, McGraw-Hill, 2005.

(no available informacion)

## Previous capacities

Knowledge of mathematical analysis. Vectorial and matricial formalism. Notions of differential calculus.

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