@@ -26,8 +26,49 @@ But the main goal of this SDK is to consider a robot model $robot\_model$, its c

and computes a control vector $\mathbf{u}$

to make the robot performs actions.

<br>

The central concept of our control architecture is the notion of task.

All our controllers are defined as optimization problems. We used either solvers of the shelves or we designed our own solvers depending of the targeted applications.

The free variables are the control vector $\mathbf{u}$ to apply to the system.

They are constraints which can be of several kinds: the mechanical constraints of the system with the environment, the behavior that we want the robot to follow.

The next two following sections introduce our software framework to take into account those two constraints.

<h2> Necessary Constraints </h2>

<h3id="Dynamics_system"> Dynamic modeling of a free-floating multi-body system</h3>

The dynamic model of a free-floating multi-body system can be written this way:

with $\mathbf{q}$ the robot state variable, $\mathbf{M}(\mathbf{q})$ the inertia matrix of the multi-body system, $N(\mathbf{q},\mathbf{\dot{q}})$ the matrix

of the non-linear dynamical effect (Centrifugal and Coriolis forces) of the system motion, $\mathbf{G}(\mathbf{q})$ the gravity, $\mathbf{u}$ a control vector,

and $\mathbf{T}(\mathbf{q})$ the matrix mapping the control vector on the system.

<br>

The current software library we are using to computate such quantities and theirs derivatives

is <ahref="http://stack-of-tasks.github.io/pinocchio/">Pinocchio</a><sup>

<ahref="#fn1"id="ref1">1</a></sup>. Pinocchio include dynamical quantities such as angular momentum and its derivatives.

<h3id="Contact models">Contact models</h3>

When considering a rigid contact model several constraints needs to be considered:

<ul>

<li>No penetration in the ground</li>

<li>No sliding</li>

</ul>

Both constraints generate inequalities at the contact points with respect to the state vector:

$$\mathbf{\phi}_n(\mathbf{q}) = 0$$

$$\mathbf{\phi}_t(\mathbf{q}) = 0$$

To avoid penetration velocity and acceleration of the points must be constrained too. Using differentiation we obtain: