## Learning objectives

The course is conceived to provide students with the necessary tools that are required for the development of linear and nonlinear dynamic models of the most common physical systems. Part of the course is devoted to the study of the industrial manipulators control systems and, more in general, of the mechatronic control systems.

In particular, the course will cover the following topics:

- Analisys of simple mechanical systems, electrical networks, hydraulic circuits, heat transmission and development of the respective models;

- Analysis of complex mechanical systems and development of dynamic models by taking into account the inertia and the friction effects;

- Nonlinear control techniques that are used for the management of the industrial manipulators;

At the end of the course, students will be able to:

- Autonomously implement dynamical models of the most common physical systems;

- Autonomously implement the model of an industrial manipulator or of a mechatronic system;

- Study the behavior of an industrial manipulator or of a mechatronics system;

- Tune the control system of an industrial manipulator;

## Prerequisites

The course requires a preliminary knowledge of some basic notions concerning the manipulator kinematics. Students must know and be able to use operators like rotation matrices, homogeneous transformation matrices, etc.. Short recalls, concerning some basic concepts, will be provided at the beginning of the course.

## Course unit content

Measure units and dimensional analysis (4 hours)

- The international system of units.

- The Pi theorem of Buckingham.

Generalized components (6 hours)

- Flux accumulator (generalized inductor): definition stored energy, co-energy and related properties.

- Intensity accumulator (generalized capacitor): definition stored energy, co-energy and related properties.

- Dumper (generalized resistor): definition, content, co-content, and related properties.

- Flux and intensity generators.

- Paynter diagrams.

- Quadrupole: definitions. The transformer and the gyrator cases.

Mechanical components for the linear motion (4 hours)

- Definition of the graph associated to a set of translational mechanical components.

- Inertia: definition and associated kinetic energy.

- Spring: definition, the Hooke law, elastic limit, and yield strength.

- Dumper and friction: the Coulomb friction, the viscous friction, and the Stribeck number.

- Classic and mobility analogies: choice of the flux of the intensity variables in the two cases and equivalent electrical components.

Mechanical components for the rotational motion (2 hours)

- Definition of the graph associated to a set of translational revolute components.

- Moment of inertia, shock absorber, and rotational damper.

Simple machines (2 hours)

- Lever, wheel axis, transmission belt, gear, pinion/rack transmission.

DC electrical motor (1 hour)

- The Lorentz law.

- Torque/current relationship.

- The equivalent electrical component.

Fluid dynamic components (3 hours)

- Definition of the graph associated with a hydraulic circuit

- Hydraulic capacity: derivation of the characteristic equation and capacity of the fluid.

- Hydraulic inertia: derivation of the characteristic equation and inertia of the fluid.

- Hydraulic dissipator: definition of dynamic viscosity, derivation of the Hagen-Poiseuille law, derivation of the Darcy

- Weisbach formula by dimensional analysis.

- Reynolds number, difference between laminar and turbulent regimes.

Hydraulic machines (2 hours)

- Hydraulic lever, pressure intensifier, piston, pump, turbine: the kinematic relationships and the equivalent electrical components.

Heat transmission (2 hours)

- First principle of thermodynamics.

- Definition of the graphs associated to the networks of thermal components.

- Law of the Gibbs phases.

- Thermal capacity at constant volume and constant pressure, specific thermal capacities.

- Heat transmission:

- Conduction: Fourier law, coefficient of thermal conductivity.

- Convection: convection coefficient.

- Irradiation: Stefan-Boltzmann's law.

Short review of the manipulator kinematics (4 hours)

- The ellissoid of manipulability

Short review of the manipulator static equations (4 hours)

- The virtual works principle

The manipulators dynamics (20 hours)

- The center of gravity of the rigid systems

- The inertia tensor of the rigid systems

- The Stainer theorem

- The inertia tensor of composite systems

- Review of the Newton-Euler recursive algorithm

- The Euler-Lagrange approach

- The passivity property

- Solution of the direct dynamics problem

The manipulators control (16 hours)

- Reviews and extensions of the independent joints control techniques

- The Proportional-Derivative centralized control

- The inverse dynamics control

- The inverse dynamics control in the operational space

- The impedence control

- The force-position control

## Full programme

## Bibliography

R. Kurth, "Dimensional analysis and group theory in astrophysics", Pergamon Press (1972)

András Recski, "Matroid Theory and its Applications in Electric Network Theory and in Statics", Springer-Verlag, Berlin, Germany, 1989

A. G. J. MacFarlane, "Dynamical System Models", GEORGE G. HARRAP & Co. LTD, London, 1970

Brian C. Fabien, "Analytical System Dynamics, Modeling and Simulation", Springer, USA, 2009

Clarence W. de Silva, "Modeling and Control of Engineering Systems", CRC Press, USA 2009

C. Guarino Lo Bianco, "Modellistica e controllo dei manipolatori industriali“, www.lulu.com, 2020.

L.Sciavicco e B.Siciliano, "Robotica industriale: modellistica, pianificazione e controllo'', terza edizione, McGraw-Hill Italia, 2008.

## Teaching methods

The course is taught by means of oral lessons, which contemplate both theoretical arguments and exercises.

A cycle of Lab lessons is planned in order to experimentally verify the acquired notions (14 hours).

Depending on the Covid emergency, lessons could alternatively be taught through streaming sessions.

## Assessment methods and criteria

The final test is divided into two written parts:

in the first one (Part A), which lasts 2h:30m, the student must solve some problems of dynamics (maximum score 32/30);

in the second one (Part B), which lasts 1h:30m, he must answer to questions concerning the course theoretical topics (maximum score 32/30).

The exam is passed if a score higher than 18/30 is gained in both the two parts.

"Part A" can be passed by means of an intermediate test session carried out during the lesson period.

The final score is the mean value between the scores of the two parts.

The "Laude" grade is assigned if the final score is higher than 30/30.

Depending on the Covid emergency, the exam tests will be executed in a classroom (by fulfilling the prescribed safety requirements) or online.

## Other information