## 4 Craniomy Trajectories by the Spherical Parallel Mechanism  
### Kinematic and Static Analysis of the SPM  
In order to perform the craniotomy, an SPM was studied. This type of architecture is known for generating RCM kinematics as its end effector rotates around the Center of Rotation (CoR). This mechanism has a total of three angular DoF: the end effector rotates with two angles and can spin around an axis passing through its center and the CoR. In the present study, a 3-DoF mechanism is proposed to perform a task that requires only two DoF. The idea is to use the last DoF (the spinning motion) as a reconfiguration variable to improve the behavior of the mechanism in the same position. Indeed, as the end effector is supposed to hold a surgical drill, the spinning motion of the end effector will generate a rotation of the drill around its longitudinal axis. This motion will have no impact on the operation of the drill itself. However, it will alter the configuration of the mechanism as illustrated in Figure 8a and, therefore, modify its behavior. The mechanical architecture proposed is a 3-RRR SPM. It is a parallel mechanism that connects its base with its end effector using three identical RRR legs. Each leg is mounted with three revolute joints and two spherical linkages. The linkages are dimensioned by the angle between the revolute joint axes. The first revolute joint of each leg is active, and their rotation serves as the mechanism's input coordinates. A kinematic representation of the mechanism is shown in Figure 8b. As the three RRR legs are identical, the dimension of  
Figure 8: (**a**) Illustration of the 3-RRR SPM reconfigurability. (**b**) Kinematic representation of the 3-RRR Spherical Parallel Mechanism. Figure 7: Tangential interaction forces \(f_{tang}\) measured by the cobot during the surgical tests performed on the cadaver. the proximal and distal spherical linkages is given by the angles \(\alpha\) and \(\beta\), respectively. The size of the end effector is given by the angle \(\gamma\), which is measured between its center axis and the last revolute joint of any leg. The joint input coordinates are measured at the first revolute joint of each leg as \(\theta_{k}\), with \(k\) referring to the leg considered (\(k\) = A, B or C). For each leg \(k\), all joints are represented by axes \(\mathbf{z_{ki}}\), with \(l\) referring to the joint position on the leg. The mechanism base is represented by the axes of the three active revolute joints \(\mathbf{z_{AI}}\), \(\mathbf{z_{B1}}\) and \(\mathbf{z_{C3}}\). They are disposed as a regular trihedron so that the axis \(\mathbf{z}\) of the origin reference frame is equidistant for all \(\mathbf{z_{KI}}\) axes as shown in Figure 9a. The angle \(\lambda\) measured between the axes \(\mathbf{z}\) and \(\mathbf{z_{KI}}\) is adjusted so that all \(\mathbf{z_{KI}}\) axes form another reference frame. This specific disposition will ensure that the mechanism end effector is located at the origin reference frame at the home position. The end effector is rotated around the mechanism CoR with three DoF. Its motion is defined as three successive rotation of angles \(\psi\), \(\theta\) and \(\varphi\) (in this order) around the axes \(\mathbf{z}\), \(\mathbf{x}\) and \(\mathbf{z}\), respectively, as illustrated in Figure 9b. The resolution of the Inverse Kinematic Model (IKM) of this type of SPM is well documented in the literature [16], and it can be written as follows:  
\[\mathbf{z_{k2}}\cdot\mathbf{z_{k3}}=\cos\beta, \tag{6}\]  
With \(k\) = A, B or C. This model must be solved for each leg of the SPM. Deriving Equation (6) consists of calculating the coordinates of axes \(\mathbf{z_{k2}}\) and \(\mathbf{z_{k3}}\) for each leg as follows:  
\[\left\{