## 3 Literature Classification and Analysis
### Mathematical Models  
The representatives using dual quaternions are points, lines, planes, spheres, infinite cylinders, coordinate systems, twists, and wrenches. Given the imaginary units \(\hat{i}\), \(\hat{j}\), \(\hat{k}\) that satisfy \(\hat{i}^{2}=\hat{j}^{2}=\hat{k}^{2}=\hat{i}\hat{j}\hat{k}=-1\) and the dual unit \(\varepsilon\), which satisfies \(\varepsilon\
eq 0\) and \(\varepsilon^{2}=0\), the quaternion set is defined as [4]:  
\[H\triangleq\Big{\{}h_{1}+\hat{h}h_{2}+\hat{j}h_{3}+\hat{k}h_{4}:h_{1},h_{2},h_{ 3},h_{4}\;\epsilon R\Big{\}}, \tag{16}\]  
and the dual quaternion set is defined as  
\[\mathcal{H}\triangleq\{h_{1}+\varepsilon h_{2}:h_{1},h_{2}\;\epsilon H\}. \tag{17}\]  
where \(h_{1},h_{2},h_{3},h_{4}\) are scalar components of the dual quaternion, and they belong to the set of real numbers. \(\hat{i}\), \(\hat{j}\), \(\hat{k}\) are imaginary unit vectors in quaternion algebra, which extend the concept of complex numbers to four dimensions. \(\hat{i}\) represents the unit vector along the x-axis, \(\hat{j}\) represents the unit vector along the y-axis, and \(\hat{k}\) represents the unit vector along the z-axis. This set is used to represent various geometric and kinematic properties of robots. The set of dual quaternions is particularly useful for representing rotations, translations, rigid motions, twists, wrenches, and other geometric primitives. For instance, the set \(S^{3}\triangleq\big{\{}h\in H:\|h\|=1\big{\}}\) is used to represent rotations, with \(\big{\|}h\big{\|}\triangleq\sqrt{hh^{*}}=\sqrt{h_{1}^{2}+h_{2}^{2}+h_{3}^{2}+h_ {4}^{2}}\) being the quaternion norm and \(h^{*}=h_{1}-\big{(}h_{2}\hat{i}+h_{3}\hat{j}+h_{4}\hat{k}\big{)}\) being the conjugate of \(h=h_{1}+h_{2}\hat{i}+h_{3}\hat{j}+h_{4}\hat{k}\). Translations and rotations can be grouped into dual quaternions. In dual-quaternion algebra, geometric primitives like points, planes, and lines have representatives, which is particularly useful for incorporating geometric constraints into robot motion controllers. For example, a plane expressed in a frame \(\mathcal{F}_{a}\) is given by [4]:  
\[\underline{\pi}^{a}=n^{a}+\epsilon d^{a} \tag{18}\]  
where \(n^{a}=n_{x}\hat{i}+n_{y}\hat{j}+n_{z}\hat{k}\) is the normal to the plane, \(d^{a}=\langle p^{a},n^{a}\rangle\) is the signed distance between the plane and the origin of \(\mathcal{F}_{a},p^{a}\) is an arbitrary point on the plane and \(\varepsilon\) is a dual unit used in dual-quaternion algebra. A line in a frame \(\mathcal{F}_{a}\), with direction given by \(l^{a}=l_{x}\hat{i}+l_{y}\hat{j}+l_{z}\hat{k}\) and passing through point \(s^{a}=p_{x}\hat{i}+p_{y}\hat{j}+p_{z}\hat{k}\), is represented in dual quaternion algebra as:  
\[l^{a}=l^{a}+\epsilon s^{a}\times l^{a}. \tag{19}\]  
One test, for example, showed a mobile manipulator tracing a circle while avoiding a wall as well as cylindrical obstacles, effectively evading obstacles despite slight impairments in the tracking of a trajectory. Furthermore, dual quaternion algebra robotics is being adopted because of the ease of use presented by the design of DQ Robotics, so it can be applied to education as well as self-learning. The versatility and efficiency in practical use are also demonstrated through its capacity for independently creating robot models and managing multiple robotic systems [4]. Dynamic movement primitives aim to learn complex behaviors in such a way, representing tasks as stable, well-understood dynamical systems. By modeling movements over the SE(3) group, modeled primitives can be generalized for any robotic manipulator capable of full end-effector 3D movement. [PERSON], [PERSON] et al. presented a robot-agnostic formulation of discrete DMP based on the dual quaternion algebra, oriented towards modeling throwing movements. DMP considers adapted initial and final poses and velocities, all computed from a projectile kinematic model and from the goal at which the projectile is aimed.