# Task-Oriented Evaluation of the Feasible Kinematic Directional Capabilities for Robot Machining
## 2 Materials and Methods
### Translational and Rotational Manipulability
Since translational subspace is considered separately from rotational subspace, multiple solutions may exist in the joint space, which satisfies constraint (8). The minimum infinity norm solution, illustrated by \(Q_{\infty}^{T}\) in Figure 1a, can be found numerically based on the structured linear algebra algorithm [46], which is rather computationally extensive. Alternatively, this problem can be expressed as a linear programming problem, where the optimal solution exists at the intersection of the explicitly given linear inequality constraints. Besides polytope analysis, performance indices based on the vector expansion method [49] offer a computationally faster solution compared to the polytopes approach, since only the robot's capabilities are investigated in the direction of interest in the task space. The maximum achievable linear (angular) end-effector velocity in the \(\hat{u}_{T}\) (\(\hat{u}_{R}\)) direction, based on the vector expansion method, can be estimated by the following:  
\[v_{2}=\left\|{{{I}_{T}^{\dagger}}{{\hat{u}_{T}}}}\right\|_{\infty}^{-1} \tag{16}\]  
\[\omega_{2}=\left\|{{{I}_{R}^{\dagger}}{{\hat{u}_{R}}}}\right\|_{\infty}^{-1} \tag{17}\]  
The value \(v_{2}(\omega_{2})\) matches the vector length of a manipulability polytope \(P_{2}^{T}\) (\(P_{2}^{R}\)) in the \(L_{2}\) weak sense, with the least Euclidian norm solution of joint velocities in the direction defined by \(\hat{u}_{T}\) (\(\hat{u}_{R}\)). The polytope \(P_{2}^{T}\) (\(P_{2}^{R}\)) is inscribed in the translational (rotational) manipulability polytope \(P_{\infty}^{T}\) (\(P_{\infty}^{R}\)) in the \(L_{2}\) weak sense, defined as follows:  
\[P_{2}^{T}=\left\{{v\in\mathbb{R}^{3}}\right|v={{I}_{T}}{{\hat{\theta}}},\ \left\|{{{\hat{ \theta}}}}\right\|_{\infty}\leq 1,\ {{\hat{\theta}}}={{I}_{T}^{\dagger}}{{v}}\right\} \tag{18}\]  
\[P_{2}^{R}=\left\{{\omega\in\mathbb{R}^{3}}\right|\omega={{I}_{R}}{{\hat{ \theta}}},\ \left\|{{{\hat{\theta}}}}\right\|_{\infty}\leq 1,\ {{\hat{\theta}}}={{I}_{R}^{\dagger}}{{\omega}}\right\} \tag{19}\]  
which determines a set of joint velocities defined by the intersection \(Q\cap Row({{I}_{T}})(Q\cap Row({{I}_{R}}))\) in the joint space, where \(Row({{I}_{T}})\) (\(Row({{I}_{R}})\)) is the row space of the translational (rotational) Jacobian matrix. The polytope \(P_{2}^{T}\) is depicted in Figure 2. In Figure 2b,c, the polytope \(P_{2}^{T}\) in relation to the polytopes \(P^{twist}\) and \(P_{\infty}^{T}\) is shown in two different views in the task space, where it can be seen that \(P_{2}^{T}\) is inscribed in the \(P_{\infty}^{T}\). Unlike the polytope \(P_{\infty}^{T}\), which defines the  
Figure 1: Example of a manipulability polytope representation based on the 3 DOF planar manipulator: (a) Joint velocity polytope \(Q\) and the minimum \(L_{\infty}\) norm solution of joint velocities represented by \(Q_{\infty}^{T}\); the task space polytopes \(P^{twist}\) and \(P_{\infty}^{T}\) in the isometric view (b), and in the top view (c). The polytope \(P_{2}^{T}\) is depicted by Figure 2. In Figure 2b,c, the polytope \(P_{2}^{T}\) in relation to the polytopes \(P^{twist}\) and \(P_{\infty}^{T}\) is shown in two different views in the task space, where it can be seen that \(P_{2}^{T}\) is inscribed in the \(P_{\infty}^{T}\). Unlike the polytope \(P_{\infty}^{T}\), which defines the minimum \(L_{\infty}\) norm solution of the joint velocities, the polytope \(P_{2}^{T}\) represents the maximal linear velocities with the least \(L_{2}\) norm solution of the joint velocities under the constraint (8), i.e., \(\left\|\hat{\theta}\right\|_{\infty}\leq 1\), illustrated by \(Q_{2}^{T}\) in Figure 2a. In terms of trajectory planning optimization, an analysis of the translational manipulability polytope in the \(L_{2}\) or \(L_{\infty}\) weak sense can offer a trajectory planning that reaches the goal pose faster [28] or with lower kinematic effort, assuming that the orientation of the end-effector is not important.