# Constraints-driven control of robotic systems

Ecological studies on the evolution and the distribution of species show that the environment and its ecological constraints (such as favorable climate and availability of resources) are more effective than goal-driven behaviors in shaping characteristic features and behaviors of species.

In robotic applications, the fragility coming from a cost-optimization formulation may additionally lead to sub-optimality or even infeasibility of the designed controllers in presence of unmodeled phenomena.

For these reasons, a more desirable formulation is the following constraint-based energy-minimization:

\begin{aligned} \underset{u}{\mathrm{minimize~}} &\| u \|^2\\ \mathrm{subject~to~} &h(u) \le 0 \end{aligned}

where $u$ represents the control effort of the robots and $h$ is the function encoding the constraints.

A slight modification of this optimization program allows the robots to take into account the execution of desired goals:

\begin{aligned} \underset{u}{\mathrm{minimize~}} & \| u - u_\text{nom}\|^2\\ \mathrm{subject~to~} & h(u) \le 0 \end{aligned}

where $u_\text{nom}$ is the nominal input required to accomplish the desired goal.

The use of Lyapunov functions and barrier functions allows the formulation of an optimization-based control framework where set stability and set invariance can be leveraged to realize complex behaviors in a constraint-based fashion.

Robotic implementation of this idea can be found in the following applications:

A natural philosophical perspective

Embodiments of this concept can be found numerous in nature: the low-energy lifestyle of three-toed sloths is an iconic example. In philosophy, constraints-driven control may be recognized in Arthur Schopenhauer’s theory according to which “A man can do what he wants, but not want what he wants”, or the more practical “A man’s actions are determined by necessity, external and internal” by Albert Einstein.