Underactuated Robots

Prof. Leonardo Lanari and Prof. Giuseppe Oriolo

Dipartimento di Ingegneria Informatica, Automatica e Gestionale
Sapienza Universit di Roma


schedule 9 Nov - 18 Dec 2020; Tue 16-00-18:00, Thu 16:00-19:00, room A4
office hours Thu 14:00-16:00, room A211, DIAG, Via Ariosto 25
e-mail lanari [at] diag [dot] uniroma1 [dot] it; oriolo [at] diag [dot] uniroma1 [dot] it

Important notice

According to the Sapienza Phase 3 procedures for the COVID-19 sanitary emergency, UR lectures will be given in class but will also be available via streaming (Zoom).  Instructions for connecting to the Zoom stream are published on the UR Google Group. Sapienza students that wish to take the UR course should ask to join the Google Group as soon as possible. Apply from within Google Groups and be sure to enter you first and last name as "Display Name" and your master program (MARR/MCER) as "Reason for joining". WARNING: only applications using the Sapienza institutional mail (@studenti.uniroma1.it) will be accepted!


This 3-credits module is part of Elective in Robotics, a 4-module course offered to students of the Master in "Artificial Intelligence and Robotics" at Sapienza University of Rome. It can also be taken by students of the Master in "Control Engineering" as one of the two modules of Control Problems in Robotics.


The course focuses on underactuation as a pervasive principle in advanced robotic systems and presents a review of modeling and control methods for underactuated robots.


1. Introduction
Motivation. Definition of underactuated system (generalized coordinates vs degrees of freedom). Examples of underactuated robots.

2. Modeling and Properties
Eulero-Lagrange modeling (classic and alternate). State-space form. Control problems of interest. Controllabiity (STLA, STLC, natural controllability). Comparison with fully actuated robots. Integrability conditions for passive dynamics. Equilibrium points and linear controllability.

3. Case Studies: Acrobot and Pendubot
Modeling. Approximate linearization at equilibria. Linear controllability. Balancing. Partial feedback linearization. Swing-up (1) via analysis of the zero dynamics (2) via energy pumping.

4. Zero dynamics in underactuated systems
Normal form and zero dynamics. Importance of the zero dynamics in control. Zero-dynamics in linear and nonlinear underactuated systems. The homoclinic orbit.

5. Passivity
Definition and physical interpretation. Linear and nonlinear mechanical systems examples. Dissipativity in state space representations. Feedback equivalence to a passive system. Output stabilization of passive systems

6. Energy-based control of underactuated systems
The convey-crane and reaction-wheel cases.

7. Optimization methods for Planning and Control
Introduction to Dynamic Programming. Hamilton-Jacobi-Bellman equation. Derivation of the Linear Quadratic Regulator
Linear-Time-Varying LQR. Trajectory optimization with Iterative LQR. Constrained optimization. Model Predictive Control (Linear, LTV and Nonlinear). LQR-trees.

Study material (with reference to the numbered topics of the syllabus)

1. Slides (videos not included)
2. De Luca, Iannitti, Mattone, Oriolo: Underactuated manipulators: Control properties and techniques, MIROC, 2002 (pdf)
    Oriolo and Nakamura: Control of Mechanical Systems with Second-Order Nonholonomic Constraints: Underactuated Manipulators, CDC, 1991 (pdf)

3. Spong: The swing-up problem for the Acrobot, IEEE Control Systems, 1996 (pdf)
    Lanari and Oriolo: The Pendubot, notes
4. Isidori: Nonlinear control systems, Springer, 1995 (ch 4 up to 4.5) (pdf)
5. Byrnes, Isidori, Willems: Passivity, Feedback Equivalence, and the Global Stabilization of Minimum Phase Nonlinear Systems, T-AC, 1991, (up to Sect IV)
Xin, Liu, Control Design and Analysis for Underactuated Robotic Systems, Springer, 2014 (ch 2: fundamentals of energy-based control; ch 4: Acrobot; ch 6: Pendubot)   
    Collado, Lozano, Fantoni, Control of convey crane based on passivity, ACC, 2000
    Spong, Corke, Lozano, Nonlinear control of the Reaction Wheel Pendulum, Automatica, 2001 (pdf)
7. Slides: Part 1 and Slides: Part 2
    Tedrake, Underactuated Robotics: Algorithms for Walking, Running, Swimming, Flying, and Manipulation, course notes for MIT 6.832 (ch 7: dynamic programming; ch 8: LQR)
    Bertsekas, Dynamic programming and optimal control, vol 1, Athena scientific, 2017
    Bemporad, Model Predictive Control, course slides, 2020


Any student who has attended at least 2/3 of the lectures can pass this module by either giving a presentation on a certain topic (based on technical papers) or developing a small project (typically involving simulations). For more details, see the main pages of Elective in Robotics or Control Problems in Robotics.

Master Theses at the Robotics Laboratory

Master Theses on the topics studied in this course are available at the DIAG Robotics Lab. More information can be found here.
Questions/comments: oriolo [at] diag [dot] uniroma1 [dot] it