## 2 Theoretical Background  
We consider the planar motion of a two-wheeled robot. Its position is described by the coordinates \((x_{c},y_{c})\) of a point fixed at the center of the axis of the wheels in the inertial frame, and its orientation relative to this frame is given by \(\theta\). Assuming a non-skid condition for both wheels, the kinematics of the robot can be represented by the following unicycle model:  
\[\hat{x}=X_{1}(x)u_{1}+X_{2}(x)u_{2}, \tag{1}\]  
where \(x=\left[x_{1}\ x_{2}\ x_{3}\right]^{T}=\left[x_{c}\ y_{c}\ \theta_{c}\right]^{T}\in\mathcal{X}\subseteq \mathbb{R}^{3}\) is the configuration (state) and \(u=\left[u_{1}\ u_{2}\right]^{T}\in\mathcal{U}\subseteq\mathbb{R}^{2}\) is the control input, representing linear and angular velocity controls, and  
\[X_{1}(x)=\left[\cos x_{3}\ \sin x_{3}\ 0\right]^{T}\in\mathbb{R}^{3},\quad X_{2} (x)=\left[0\ 0\ 1\right]^{T}\in\mathbb{R}^{3} \tag{2}\]  
are control vector fields. The kinematic structure (1) allows it to be defined on a Lie group isomorphic to the Special Euclidean Group \(SE(2)\)[27]. This property can be used efficiently for control design. To facilitate the notation used in this paper, let us briefly recall the basic designations associated with representation on the Lie group. * \(G\)--a Lie group;
* \(g\)--a group element;
* \(g_{1}g_{2}\)--a group operation of the elements \(g_{1}\) and \(g_{2}\), i.e., \(g_{3}=g_{1}g_{2}\) creates a new group element \(g_{3}\);
* \(L_{g_{1}}\)--the left translation on \(G\), where \(g_{1}\in G\), i.e., \(L_{g_{1}}g_{2}=g_{1}g_{2}\);
* \(X(g)\)--a vector field (_vf_) \(X\in\mathcal{X}\) evaluated at the point \(g\in G\), where \(\mathcal{X}\) is a set of all smooth _vfs_;
* \(X_{i,j}\)--a matrix consisting of \(j-i+1\)_vfs_, i.e., \(X_{i,j}=\left[X_{i}\ X_{i+1}\ \ldots\ X_{j-1}\ X_{j}\right]\), where \(0<i<j\);
* \(e\)--the neutral element, i.e., \(ge=eg=g\), where \(e\in G\);
* \(g^{-1}\)--the inverse element of \(g\), i.e., \(gg^{-1}=g^{-1}g=e\);
* Ad--the group adjoint representation, where \(\mathrm{Ad}_{g}:T_{e}G\to T_{e}G\), i.e., \(\mathrm{Ad}_{g}=\mathrm{d}c_{g}(e)\), \(\mathrm{d}\) is a symbol of a differential, \(c_{g}(e)=geg^{-1}\); additionally \(\mathrm{Ad}\) can be defined with use of the derivative of left translation as \(\mathrm{Ad}_{g}=\mathrm{d}L_{g}(g^{-1})\mathrm{d}L_{e}(g^{-1})\), where \(g\in G\);
* \(\mathrm{Ad}_{g}^{X}\)--the adjoint representation projected on \(X(e)\), i.e., \(\mathrm{Ad}_{g}X(e)=X(e)\mathrm{Ad}_{g}^{X}\); by extension of this definition, we allow for the usage of \(\mathrm{Ad}_{g}^{X}\) when considering the adjoint representation projected on a matrix \(X_{a,b}(e)\), i.e., \(\mathrm{Ad}_{g}X_{a,b}(e)=\mathrm{Ad}_{g}[X_{a}(e)\ \ldots\ X_{b}(e)]=\left[\mathrm{Ad}_{g}X_{a}(e)\ \ldots\ \mathrm{Ad}_{g}X_{b}(e)\right]=\left[X_{a} \mathrm{Ad}_{g}^{X}\ \ldots\ X_{b}\mathrm{Ad}_{g}^{X}\right]\) \(=X_{a,b}(e)\mathrm{Ad}_{g}^{X}\). For a broader introduction to representation on the Lie group, the reader is referred to [28]. Following the methodology used in this paper, we consider control _vfs_ in (1) as generators of a Lie algebra \(\mathfrak{g}\) and assume that the following set of _vfs_\(\mathcal{X}_{1,3}=\{X_{1},X_{2},X_{3}\}\subseteq\mathfrak{g}\), with \(X_{3}=[X_{1},X_{2}]\) being the Lie bracket, constitutes the algebra basis. This choice of algebra basis is rather standard, and a broader discussion of this topic can be found in [27]. Taking advantage of matrix vector notation, we define the matrix \(X_{1,3}(x)=\left[X_{1}(x)\ X_{2}(x)\ X_{3}(x)\right]\) that uses _vfs_ from \(\mathcal{X}_{1,3}\). Using this basis, it is possible to represent system (1) as  
\[\hat{x}=X_{1,3}(x)\hat{C}\hat{u}, \tag{3}\]where \(C:=[C_{1}\ 0_{3\times 1}]\in\mathbb{R}^{3\times 3}\), \(C_{1}:=[\mathbb{I}_{2\times 2}\ 0_{1\times 2}]^{T}\in\mathbb{R}^{3\times 2}\), \(\mathbb{I}_{2\times 2}\in\mathbb{R}^{2\times 2}\) is an identity matrix, and \(\mathbb{0}_{3\times 1}\in\mathbb{R}^{3\times 1}\) and \(\mathbb{0}_{1\times 2}\in\mathbb{R}^{1\times 2}\) are zero matrices.