## 2 Constraint Analysis at the Velocity Level  
The kinematics of parallel manipulators, in general, and lower mobility manipulators, in particular are usually described with a well-established procedure at the velocity level known as screw-based rate kinematics. Consequently, the control scheme of most of these manipulators is designed based on the rate kinematics algorithms. This section's main objective is to create a foundation for analyzing the moving platform's motion by examining the legs' structural constraints. Therefore, two approaches to studying the PMs will be employed, i.e., (a) several serial limbs restrict the moving plate's motion, and (b) analyzing the PMs by considering the moving plate as a rigid body that restricts the legs. Although these two approaches of analyzing the motion and constraints of the PMs are connected, this paper will discuss the two in separate sections. To analyze each serial limb's kinematic feature that connects the fixed and the moving plates, the entire PM is breakdown into subsystems by isolating the limbs and then integrating back to describe the PM's motion. A general manipulator has instantaneous motion \(\dot{\mathbf{x}}\in\mathbb{R}^{6}\). But when subjected to certain constraints, some regions can or cannot be reached by the manipulator, and consequently, the kinematic analysis needs to consider those regions. Hence, two sub-spaces need to be considered for a complete description of the robot manipulators. These regions, in general, consist of the sub-space that the manipulator is free to move in, and the sub-space that the manipulator is restricted from entering. These two regions are described with two complementary sub-spaces named Instantaneous Motion Space (IMS) and Instantaneous Restriction Space (IRS) [27]. As discussed earlier, since PM has resulted from multiple serial limb combinations, the entire manipulator's kinematic behavior can be determined by separately analyzing the limbs IMS and IRS. Instantaneous Motion Space (IMS): It is the tangent space of SE(3) spanned by the linearly independent motion vectors at the moving-plate (\(\text{IMS}\subset\mathbb{R}^{6}\)). From the leg point of view, this space is spanned by the intersection of twists in all legs, and it can be physically interpreted as the motion of the moving platform in the region where all legs commonly operate. Instantaneous restriction Space (IRS): It is an orthogonal complementary subspace of the instantaneous motion space in \(\mathbb{R}^{6}\) - IMS, which belongs to the unreachable region due to the structural constraints and formed by the union of all restriction screws induced from each leg. PMs' constraints are a union of those legs restriction, and thus the entire motion is described by the legs IMS and IRS. In a serial manipulator, IMS (\(\mathcal{C}(\mathbf{J}_{j})\)) is obtained from the forward rate kinematics whereas, IRS (\(\mathcal{C}(\mathbf{J}_{c})\)) is obtained by taking the orthogonal complement (\(\mathcal{N}(\mathbf{J}_{j}^{T})\)). Hence, the IMS fully describes the serial manipulator's motion, and  
Figure 3: Kinematic architecture of the 3-PvRS manipulator. the corresponding IRS is \(\{\text{IR}^{6}\text{IMS}\}\) which represents the space that the manipulator cannot reach. Thus, the generic extended forward-rate kinematics (FRK) of an isolated limb can be written with the \((6\times 6)\) extended Jacobian matrix \(\mathbf{J}_{e}\) as follows:  
\[\dot{\mathbf{x}}=\mathbf{J}_{e}\dot{\mathbf{q}}_{e}= \begin{bmatrix}\mathbf{J}_{j}&;&\mathbf{I}_{c} \end{bmatrix}\begin{bmatrix}\dot{\mathbf{q}}\\ \mathbf{0}\end{bmatrix} , \tag{8}\]  
where \(\dot{\mathbf{x}}\) and \(\dot{\mathbf{q}}_{e}\) denote the velocity of the end-effector and extended joint velocities, respectively. \[\dot{\mathbf{x}}\in\mathcal{C}(\mathbf{J}_{j})\text{ and }\dot{\mathbf{q}}\in\mathcal{C}(\mathbf{ J}_{j}^{T}). \tag{9}\]  
The dimension of \(\mathbf{J}_{j}\) is a \((6\times f)\) Jacobian matrix and that of \(\mathbf{I}_{c}\) is a \((6\times(6-f))\) constraint matrix, where \(f\) is the total DoF.