# Application of Elliptic Jerk Motion Profile to Cartesian Space Position Control of a Serial Robot
## 2 Architecture of the RRFbR Robot  
The considered architecture (Figure 1) is derived from the industrially widespread SCARA manipulator, RRPR (Revolute, Revolute, Prismatic, Revolute [15]), by introducing an articulated parallelogram (four-bar, Fb), which replaces the vertical prismatic joint. This solution avoids the friction issues related to the prismatic joint and makes the static balancing of the manipulator easier [11]. This 4-DOF manipulator has the same mobility as the SCARA since the end-effector performs three translations and one rotation around a vertical axis. This mobility, named Schoenflies' motion [16], is sufficient for a very wide range of handling and assembly tasks, and this is the reason for the popularity of the SCARA architecture.  
The seven links of the RRFbR manipulator, L0... L6, are connected by seven revolute joints. According to the Chebychev-Grubler-Kutzbach criterion [1], six moving rigid bodies (L1... L6) and seven revolute joints correspond to 1-DOF mobility (\(6\times 6-7\times 5=1\)), but the planar four-bar mechanism has three redundant constraints since its joint axes are parallel. By eliminating these redundant constraints, the 4-DOF mobility is obtained. The four actuated revolute joints (J1... J4) are represented in red in Figure 1. The corresponding rotation angles, shown in the kinematic scheme of Figure 2, represent the internal coordinates of the robot, collected in the vector \(\theta=[\theta_{1}\), \(\theta_{2}\), \(\theta_{3}\), \(\theta_{4}]^{T}\). The four external coordinates, composing the vector \(\mathbf{x}=[x\), \(y\), \(z\), \(\theta]^{T}\), are the three Cartesian coordinates of the end-effector central point E with respect to the frame O(\(x\),\(y\),\(z\)) and the end-effector rotation \(\theta\) about the \(z\)-axis of the same frame (Figure 2).  
The kinematic and dynamic models of the manipulator are discussed in [11]. The dynamic model defines the relationship between the robot motion, the vector of the actuation torques \(\boldsymbol{\tau}=[\tau_{1}\), \(\tau_{2}\), \(\tau_{3}\), \(\tau_{4}]^{T}\), and the vector of the generalized forces applied by the end-effector to the environment in the directions of the four external coordinates, \(\mathbf{F}=[F_{\mathrm{x}}\), \(F_{\mathrm{y}}\), \(F_{\mathrm{z}}\), \(M_{\mathrm{z}}]^{T}\)[11].  
Due to the robot architecture, gravity acts only on joint 3, since the rotations of joints 1, 2 and 4 do not influence the vertical position of the Center Of Mass (C.O.M.) of any link. Therefore, static balancing can be applied by adding a torsional spring acting on joint 3 in parallel with the actuator. The balancing spring, characterized by stiffness \(k_{3}\) and neutral position \(\theta_{3p}\), can be mounted on any joint of the four-bar mechanism without changing its effect. As discussed in [17], even if the balancing spring should be tuned for any task to minimize energy consumption, a suitable compromise is to impose that the arm is exactly balanced when \(\theta_{3}=0\) and in the absence of payload. This condition, which will be applied in this work, leads to a negative value of \(\theta_{3p}\). The equations that define the spring parameters \(\theta_{3p}\) and \(k_{3}\) fulfilling this condition as a function of the robot's geometrical and mass properties are discussed in [17].  
Figure 1: The RRFbR robot (red: actuated revolute joints; green: passive revolute joints).