# Reconfiguration for the Maximum Dynamic Wrench Capability of a Parallel Robot
## 2 Dynamics of the Stewart-Platform Robot with Sliding Lockable Base Joints  
The geometric structure of the proposed Stewart-platform robot with sliding lockable base joints is illustrated in Figure 1, in which a base and a moving platform type of end-effector are connected to the base points through six extensible links \(i\) (\(i\) = 1,..., 6) with a spherical joint \(\mathrm{B}_{i}\) at their upper end and a universal joint \(\mathrm{A}_{i}\) at their lower end. Each link is driven by a motor through a transmission comprising a ball screw system and a connected rod. Both ends of the connected rod are respectively attached to a linear guide installed on the ball screw and to the moving platform. Accordingly, each link in the parallel robot structure can be seen with two parts, _i.e._, a lower extensible part (designatedhereafter as segment (1) comprising the ball screw system, and an upper fixed-length part (designated hereafter as segment (2) consisting of the linear guide and connected rod. When the ball screws are driven by the motors, the connected rods move in a direction along the corresponding link axes such that the positions and orientations of the platform type of end-effector can vary according to a desired task planning. Moreover, in order to optimally adjust the architecture geometry for the required performance, each universal joint A\({}_{i}\), _i.e._, the base point, can be displaced along the linear guideway of the base. In addition, the applied dynamic wrench force \(\mathbf{f}\) and wrench moment \(\mathbf{n}\) are also introduced to the center of mass of the moving platform for the maximum dynamic wrench capability analysis of the reconfigurable parallel robot. Therefore, by displacing the base locations of each link along a linear guideway, the proposed reconfigurable parallel robot is capable of varying its geometry to fulfill a specified task with the maximum dynamic wrench capability. In studying the optimal architecture of a reconfigurable parallel robot for maximum dynamic wrench capability, an appropriate inverse dynamics form is required. In comparing with other formatting algorithms for the dynamic equations of parallel robots, several methods such as the Newton-Euler formulation [12; 13; 14], virtual work principle [15; 16], [PERSON]'s method [17], kinematic influence coefficient theory [18], screw theory [19], and the Lagrangian formulation in generalized coordinates [20; 21; 22] are proposed to model and simulate the dynamics of parallel manipulators. However, these expressions are not structured enough, and are not appropriate for the dynamic wrench capability analysis. Therefore, following our previous development, the Boltzmann-Hamel-d'Alembert formula will be used to develop the required dynamics. As shown in Figure 2, the associated frames are first assigned in developing the dynamics of the parallel robot for the dynamic wrench capability analysis. Here, an inertial coordinate system \(N\)-frame, \(\chi_{N}y_{N}z_{N}\), is fixed to the base and a second coordinate system \(B\)-frame, \(\chi_{y}y_{k}z_{b}\), is attached to the moving platform with its origin located at the platform's center of mass, B, whose position vector with respect to the \(N\)-frame is denoted as \(\mathbf{p}\). Moreover, two parallel local coordinate systems, _i.e._, \(i_{1}\)-frame and \(i_{2}\)-frame with the corresponding origins A\({}_{i1}\) and A\({}_{i2}\), are attached to the segment 1 and segment 2 of the link \(i\), respectively, and the position vector of A\({}_{i1}\) with respect to the \(N\)-frame is represented as \(\mathbf{a}_{i}\). The position vector of the ball joint B\({}_{i}\) with respect to the \(B\)-frame is defined as \(\mathbf{b}_{i}\), and the length vector of link \(i\) from the universal joint A\({}_{i1}\) to the ball joint B\({}_{i}\) with respect to the \(i_{1}\)-frame is expressed as \(\mathbf{I}_{i}\). Because the length vector of each link is an addition of the variable length  
Figure 1: Geometric structure of a reconfigurable motor-driven parallel robot.