# Task-Oriented Evaluation of the Feasible Kinematic Directional Capabilities for Robot Machining
## 2 Materials and Methods
### Proposed Method  
The estimation of the maximum feasible linear and angular velocity for simultaneous linear and angular motion using the proposed DTF method will be explained in this section. The problem can be formulated as follows: We seek a maximum linear (angular) velocity in the desired direction, in a rotational (translational) motion synchronized with the specified direction, under joint velocity constraints (8). In the following, we assume an \(n\) degree-of-freedom, non-redundant, robotic serial-link manipulator in a non-singular configuration. Unlike the majority of existing methods that take the translational (rotational) Jacobian submatrix \(J_{T}\) (\(J_{R}\)) into consideration, the derivation of the proposed DTF method is based on the Jacobian submatrices, which fulfils the additional restrictions with the null space projection [36; 38]. Let \(\widetilde{J}_{T}\) denote a \(3 xn\) translational Jacobian submatrix:  
\[\widetilde{J}_{T}=J_{T}(I-J_{R}^{\dagger}J_{R}), \tag{22}\]  
where \(I\) is the identity matrix, and \((I-J_{R}^{\dagger}J_{R})\) is the projector to the rotational null subspace, with the operator \((. )^{\dagger}\) that stands for the Moore-Penrose pseudoinverse of a matrix. Similarly, the rotational Jacobian submatrix \(\widetilde{J}_{R}\) can be obtained. If we assume a square Jacobian  
Figure 3: Example of a manipulability polytope representation based on the 3 DOF planar manipulator: (**a**) Joint velocity polytope \(Q\), the minimum \(L_{\infty}\) norm solution of the joint velocities represented by \(Q_{\infty}^{T}\), the least \(L_{2}\) norm solution of the joint velocities represented by \(Q_{2}^{T}\), and the intersection between the joint velocity polytope \(Q\) and the null space \(N(J_{R})\) represented by \(Q_{strong}^{T}\); the task space polytopes \(P_{\infty}^{triust}\), \(P_{\infty}^{T}\), \(P_{2}^{T}\), and \(P_{strong}^{T}\) in the isometric view (**b**) and in the top view (**c**). matrix, then it can be shown that the inverse of the Jacobian (15) can be represented by two parts, which can be derived from the pseudoinverse of \(\widetilde{J}_{T}\) and \(\widetilde{J}_{R}\)[35; 36]:  
\[J^{-1}=\left[\widetilde{J}_{T}^{\mathrm{t}}\quad\widetilde{J}_{R}^{\mathrm{t}} \right], \tag{23}\]  
such that the joint velocity vector can be decomposed correspondingly as:  
\[\dot{\theta}=J^{-1}\cdot t=\dot{\theta}_{T}+\dot{\theta}_{R}, \tag{24}\]  
where \(\dot{\theta}_{T}\in N(J_{R})\) and \(\dot{\theta}_{R}\in N(J_{T})\). Note that in the latter equation, we assume that the joint velocity vector can be described in a form consisting of two parts related to translation and rotation. This can be rewritten as follows:  
\[\dot{\theta}=\widetilde{J}_{T}^{\mathrm{t}}v+\widetilde{J}_{R}^{\mathrm{t}}\omega, \tag{25}\]  
where the first part of (25) are the joint velocities \(\dot{\theta}_{T}\in N(J_{R})\) required solely by the translational end-effector displacement (i.e., with zero contribution to the rotational motion), while the second part are joint velocities \(\dot{\theta}_{R}\in N(J_{T})\) responsible solely for rotational end-effector displacement (i.e., with zero contribution to the translational motion). To find the maximal velocity capabilities in the task space, we need to analyze how well the task's requirements fit the robot's joint velocity constraints (8). A robot surface machining path consists of waypoints located along a curved surface. During the process, the robot needs to follow the prescribed path with a constant linear velocity while maintaining the end-effector orientation normal to the curved surface. Synchronization between the desired linear and angular velocity vectors within two neighboring waypoints can be determined by the curved surface geometry. In contrast to the existing methods, where the desired motion synchronization cannot be controlled (the maximum kinematic capabilities can be evaluated only based on the desired linear/angular velocity direction), the proposed method also takes the desired motion synchronization into consideration.