On the Use of Energy Tanks for Robotic Systems

Passivizing learned policies and learning passive policies with virtual energy tanks in robotics

“…Evaluating the variation of the closed-loop storage function E = E +V c one achieves the PBC objective (5), which shows that the interconnection achieves indeed a passive closed-loop system with respect to the s-environmental port. The scalar V c (x c ) indicates the amount energy that is still at disposal of the control mechanism implementing the action w before loosing the described formal passivity.…”

“…In other words, if V c (x c (t))| t=0 > e * , the tank never depletes along the whole task horizon. It is important to notice that e * does not depend on the task only, but also on the specific tank dynamics which is chosen: the choice u t in (8) is not unique, and many variations have been proposed [5] (e.g., recirculation in the tank of dissipation) which would change the value of task energy for the same task.…”

“…In fact there is no need to design the controller in a way that it mimics a physical system (e.g., in the PD case the controller can be seen as a control spring and damper): as long as a task is achieved though a control input w and e * is finite, a passive implementation is possible by means of the energy tank algorithm, just setting V c (x c (t))| t=0 > e * ; ii) The aforementioned initial energy assignment is a degree of freedom in the algorithm, whose implications are often naively addressed. In fact a very high (yet bounded) energy initialisation in the tank would technically still fulfill the PBC objective ( 5), yet creating so called "practically unstable behaviours" [5], [17]. As a consequence, a naive tank design de facto makes robust stability a property which is and not connected to any safety guarantee.…”

“…Furthermore this choice serves best to our purpose of interpreting the task energy as a metabolic metric to assess an initial energy budget: the energy in the tank can only decrease along the task execution. In this discrete time implementation of energy tanks we aim to achieve the following finite difference version of the PBC objective (5). We indicate E k = E(kt) the sampled mechanical energy of the robot and E k = E k + e k the sampled closed-loop storage function.…”

“…Passive systems route physical energy rather than producing it, which is the ultimate reason why stability proofs (using energy functions as Lyapunov candidates) are conveniently assessed in this framework [3], [4]. A powerful technique allowing the passivisation of any control action is represented by virtual energy tanks [5]- [7]. These are able to store information about physical energy flows undergoing the system, and introduce at a control level the scalar information representing the energy budget for the controlled robot.…”