Analysis and control of nonlinear systems: the Pendubot (thesis)
An electronic engineering thesis on controlling an underactuated robot, the Pendubot: why it is hard, the tools of nonlinear theory and the swing-up plus balancing scheme.
The Pendubot is a double pendulum with a single motor: the first link is driven by an electric motor, the second is free to rotate about its joint. It is one of the classic test beds of nonlinear control theory, and the electronic engineering thesis presented on this page, “Analysis and control of nonlinear systems with application to an underactuated robot” (written in Italian), uses it as a case study after introducing the main results of nonlinear dynamical systems theory.
Why the Pendubot is hard
The system has two degrees of freedom, the angles of the two links, but only one control input, the motor torque. It is called underactuated: the second link cannot be driven directly, only through the forces transmitted by the first one. The equations of motion, derived with the Lagrangian method, are strongly nonlinear: they contain sines, cosines, and centrifugal and Coriolis terms that couple the two links.
The usual goal is to bring both links to the upright vertical position and keep them there: an unstable equilibrium, like balancing a stick on your palm, except that your hand is attached to the first link.
The tools of nonlinear theory
The first part of the thesis summarises the theory needed. The main tools for this kind of problem are:
- equilibrium points and linearisation: around an equilibrium the system is approximated by a linear one, and its stability is studied through the eigenvalues of the Jacobian matrix;
- Lyapunov stability: you look for an “energy” function that decreases along the trajectories, the most general way to prove stability of a nonlinear system;
- controllability of the linearised system, which tells whether a single input can really govern both links near the equilibrium;
- feedback linearisation, which uses a change of variables and a suitable control law to make part of the dynamics linear.
How it is controlled: swing up, then balance
In the Pendubot literature control is usually split into two phases:
- Swing-up: starting from rest, with the links hanging down, the motor makes the system swing, adding energy at every pass, until the links come close to the vertical. Nonlinear techniques are used here, often energy-based ones.
- Balancing: near the vertical the system is almost linear, and a linear state-feedback regulator (for example designed with the LQR method) stabilises it. The switch between the two phases happens when the state enters the region where the linear regulator works.
The same pattern, energy to get there and linear control to stay there, appears in the inverted pendulum on a cart, the Furuta pendulum and many walking or jumping robots.
Who it is for
The thesis is a good companion to systems theory and automatic control courses: it shows on a concrete case why linear methods are not enough and how they are combined with nonlinear ones. It is also useful to anyone choosing a thesis topic in control engineering. The document is in Italian.
