Figure 1 shows a simplified model
of a spherical robot.
In this model, the spherical robot comprises a spherical shell 1 \((C,\boldsymbol{\hat{x}}_1,\boldsymbol{\hat{y}}_1,\boldsymbol{\hat{z}}_1)\),
a massless drive mechanism 2 \((C,\boldsymbol{\hat{x}}_2,\boldsymbol{\hat{y}}_2,\boldsymbol{\hat{z}}_1)\)
and a counterweight 3 \((G,\boldsymbol{\hat{x}}_3,\boldsymbol{\hat{y}}_2,\boldsymbol{\hat{z}}_3)\), both enclosed
in the shell. It can be autonomous, or driven by remote control.
The locomotion principle of this robot is based on the
disturbance of the system’s equilibrium by moving its
center of mass.
More specifically,
Shell 1 is of radius \(R\), mass \(m_1\), mass center \(C\) and inertia operator \({\cal I}_{C,1} = J_1 \,{\cal I}d\) (\({\cal I}d\) is the identity, \(J_1\) is a given scalar). Denote by \(x(t), y(t)\) the coordinates of \(C\) such that \(\boldsymbol{r}_{OC}=x \boldsymbol{\hat{x}}_0+ y\boldsymbol{\hat{y}}_0 + R \boldsymbol{\hat{z}}_0\) and by \(\boldsymbol{\Omega} = \boldsymbol{\omega}_{1/0}\) the angular velocity of body 1.
Drive mechanism 2 (massless) is connected to shell 1 by a frictionless pivot of axis \((C,\boldsymbol{\hat{z}}_1)\). Two motors \({\cal M}\) mounted between 1 and 2 subject the shell to a couple \({\cal C}\boldsymbol{\hat{z}}_1\) (\({\cal C}\) is a given scalar).
Counterweight 3 of mass center \(G\), mass \(m_3\) is connected to body 2 by a frictionless pivot of axis \((A,\boldsymbol{\hat{y}}_2)\). Denote by \(J_3\) the moment of inertia of body 3 about axis \((A,\boldsymbol{\hat{y}}_2)\). Denote by \(a\) and \(l\) the two constant lengths such that \(\boldsymbol{r}_{CA}= a \boldsymbol{\hat{x}}_2\) and \(\boldsymbol{r}_{AG}= l \boldsymbol{\hat{x}}_3\).
The spherical robot is in motion in a referential 0, in point contact with a horizontal plane \(\Pi (O,\boldsymbol{\hat{x}}_0,\boldsymbol{\hat{y}}_0)\) at a point \(I\). The action of the contact forces due to \(\Pi\) on body 1 is modeled by the action screw \(\{{\cal A}_{\Pi\to 1}^c \}= \begin{Bmatrix} \boldsymbol{F} + N\boldsymbol{\hat{z}}_0 \\ \boldsymbol{0}\end{Bmatrix}_I\). The shell 1 is assumed to roll/pivot without slipping.
Question 1. What role does the counterweight play in the functioning of the robot? How would its path be precisely controlled?
Question 2. Identify all equations which would determine the evolution of \(\dot{x}\), \(\dot{y}\), \(\boldsymbol{\Omega}\), \(\dot{\theta}_2\) and \(\dot{\theta}_3\) during the motion of the system. Do not detail these equations. (These equations should not involve unknown forces or moments). Carefully justify the choice of each equation.
