# Task-Oriented Evaluation of the Feasible Kinematic Directional Capabilities for Robot Machining
## 2 Materials and Methods
### Proposed Method
The ratio of these magnitudes denoted by \(h\in\mathbb{R}\) defines the relative importance of the synchronized translational and rotational motion independently of the velocity profile scaling:  
\[h=V/\Omega, \tag{26}\]  
where \(V=\|v\|_{2}\) is the magnitude of the linear velocity, and \(\Omega=\|\omega\|_{2}\) is the magnitude of the angular velocity. The parameter \(h\) is a task-dependent velocity ratio factor that reflects the synchronization of translational and rotational motion. The linear velocity vector and the angular velocity vector can be represented by:  
\[v=V\cdot\dot{u}_{T}, \tag{27}\]  
\[\omega=\Omega\cdot\dot{u}_{R}, \tag{28}\]  
where \(\dot{u}_{T}\in\mathbb{R}^{3}\) and \(\dot{u}_{R}\in\mathbb{R}^{3}\) are the unit vectors in the desired direction of the linear and the angular velocities in the task space, respectively. When the end-effector maintains a constant orientation, there is no rotational motion required (\(\omega=0\)), thus the ratio \(h\) approaches infinity and the motion is purely linear. The maximum directional kinematic capabilities can be obtained by the translational manipulability polytope \(P_{strong}^{T}\) in the strong sense. If the velocity ratio factor \(h\) is zero, the motion is pure rotation. The maximum directional kinematic capabilities can be determined by the rotational manipulability polytope \(P_{strong}^{R}\) in the strong sense. Combining the task requirements and the robot's velocity capabilities leads to the following:  
\[\left\|\widetilde{J}_{T}^{\mathrm{t}}V\dot{u}_{T}+\widetilde{J}_{R}^{\mathrm{t }}\Omega\dot{u}_{R}\right\|_{\infty}\leq 1. \tag{29}\]The maximum linear and angular velocity can be found by scaling under the constraints (8), considering (26). This yields the following equations:  
\[V_{\rm max}=\left\|\widehat{T}_{T}^{\dagger}\hat{a}_{T}+h^{-1}\cdot\widehat{J}_{ k}^{\dagger}\hat{a}_{R}\right\|_{\infty}^{-1}, \tag{30}\]  
Figure 3:0  
where we consider that the linear velocity is synchronized with the angular velocity, thus \(\Omega_{\rm max}=h^{-1}\cdot V_{\rm max}\) and \(V_{\rm max}=h\cdot\Omega_{\rm max}\). In contrast to the existing methods, the proposed DTF method gives us the physically meaningful and accurate information about how fast the robot can move when a simultaneous translational and rotational motion is needed in directions \(\hat{a}_{T}\) and \(\hat{a}_{R}\), respectively. The maximum linear (angular) velocity vector, considering the desired angular (linear) motion constraint, can be obtained as follows:  
Figure 3:1  
Figure 3:2  
which can be combined further into the maximum feasible twist vector \(t_{\rm max}^{T}=\left[v_{\rm max}^{T}\quad\omega_{\rm max}^{T}\right]\) for the given base task parameters (\(\hat{a}_{T}\), \(\hat{a}_{R}\), \(h\)) and under the constraints (8). To provide an intuitive understanding of the proposed DTF method, the maximum linear and angular velocity vectors (\(v_{\rm max}\) and \(\omega_{\rm max}\)) are shown graphically by the twist manipulability polytope in Figure 3:3 In the case of the 3 DOF planar robotic mechanism, the twist manipulability polytope lies in the 3-dimensional space, which consists of the 2-dimensional translational subspace (the \(v_{x}-v_{y}\) plane) and the 1-dimensional rotational subspace (\(\omega\)-axis). The maximum linear velocity capability is illustrated as a vector \(v_{\rm max}\) in red, with the direction \(\hat{u}_{T}\) and the magnitude \(V_{\rm max}\), which lie in the \(v_{x}-v_{y}\) plane, and it can be interpreted as the orthogonal projection of the twist vector onto the translational subspace. The maximum angular velocity capability is shown as a vector \(\omega_{\rm max}\) in blue in the \(\omega\)-axis, with the direction \(\hat{a}_{R}\) and the magnitude \(\Omega_{\rm max}\). The tip of the resultant twist vector \(t_{\rm max}\) in purple is touching the surface of the twist manipulability polytope.