Kinematic Analysis and Trajectory Planning of the Orthoglide 5-axis

The subject of this paper is about the kinematic analysis and the trajectory planning of the Orthoglide 5-axis. The Orthoglide 5-axis a five degrees of freedom parallel kinematic machine developed at IRCCyN and is made up of a hybrid architecture, namely, a three degrees of freedom translational parallel manipulator mounted in series with a two degrees of freedom parallel spherical wrist. The simpler the kinematic modeling of the Orthoglide 5-axis, the higher the maximum frequency of its control loop. Indeed, the control loop of a parallel kinematic machine should be computed with a high frequency, i.e., higher than 1.5 MHz, in order the manipulator to be able to reach high speed motions with a good accuracy. Accordingly, the direct and inverse kinematic models of the Orthoglide 5-axis, its inverse kinematic Jacobian matrix and the first derivative of the latter with respect to time are expressed in this paper. It appears that the kinematic model of the manipulator under study can be written in a quadratic form due to the hybrid architecture of the Orthoglide 5-axis. As illustrative examples, the profiles of the actuated joint angles (lengths), velocities and accelerations that are used in the control loop of the robot are traced for two test trajectories. INTRODUCTION Parallel kinematics machines become more and more popular in industrial applications.This growing attention is inspired by their essential advantages over serial manipulators that have already reached the dynamic performance limits. In contrast, parallel manipulators are claimed to offer better accuracy, lower mass/inertia properties, and higher structural stiffness (i.e. stiffness-to-mass ratio) [2]. These features are induced by their specific kinematic architecture, which resists to the error accumulation in kinematic chains and allows convenient actuators location close the manipulator base. Besides, the links work in parallel against the external force/torque, eliminating the cantilevertype loading and increasing the manipulator stiffness. Unlike the Variax proposed by Gidding & Lewis in Chicago in 1994, the delta robot invented by Clavel [1] has known a great success for pick and place applications. One reason for this success is the simplicity of the kinematic and dynamic models compared to the models of the Gough-Stewart platform [2]. Indeed, the performance of parallel robots may vary considerably within their workspace, which is often small compared to the volume occupied by the machine. It is noteworthy that the inverse kinematics of a parallel manipulator is usually easy to calculate when the actuated joints are prismatic joints as the corresponding equations to be solved are quadratic. However, the inverse Jacobian matrix of such manipulators is more difficult to express and its computing time higher. Several five degrees of freedom (dof) parallel manipulators have been synthesized in the literature the last few decades. However, their complexity make them difficult to build and use in general. Moreover, the use of fully parallel manipulators leads to robots with five limbs whose mutual collisions or geometric constraints reduce the workspace size. The 1 ar X iv :1 50 5. 06 84 5v 1 [ cs .R O ] 2 6 M ay 2 01 5 Tripteron is one of the simplest translational parallel robot with three degrees of freedom that can be found in the literature [6]. However, this architecture is not suitable for machining operations since its legs are subjected to buckling. As a consequence, a five dof hybrid machine, named Orthoglide 5-axis, has been developed at IRCCyN. This machine is composed of three dof translational parallel manipulator, named Orthoglide 3-axis, mounted in series with a two dof spherical parallel manipulator, named Agile Eye 2-axis. The Orthoglide 3-axis has the advantages of both serial and parallel kinematic architectures such as regular workspace, homogeneous performances, good dynamic performances and stiffness. The interesting features of the Orthoglide 3-axis are large regular dextrous workspace, uniform kinetostatic performances, good compactness [13] and high stiffness [14]. Besides, the translational and rotational motions of the end-effector (tool) are partially decoupled with the hybrid architecture of the Orthoglide 5-axis. This paper is organized as follows. The next section deals with the kinematic modeling of the Orthoglide 5-axis. Then, some trajectories are generated using a simplified computed torque control loop and tested experimentally. Finally, some conclusions and future work are presented. THE ORTHOGLIDE 5-AXIS A hybrid architecture Figure 1 depicts a CAD modeling of the Orthoglide 5-axis and figure 2 shows a semi industrial prototype of the Orthoglide 5-axis located at IRCCyN. The Orthoglide 5-axis is a hybrid parallel kinematics machine (PKM) composed of a 3-dof translational parallel manipulator, the Orthoglide 3-axis, mounted in series with two dof parallel spherical manipulator, the Agile Eye 2axis. The Agile Eye 2-axis is spherical wrist developed at Laval University [5]. The architecture of the Orthoglide 5-axis was presented in [4] as well as in [9]. FIGURE 1. Digital mock-up of the Orthoglide 5-Axis FIGURE 2. Semi industrial prototype of the Orthoglide 5-Axis performing a machining operation Orthoglide 3-Axis The Orthoglide 3-axis is composed of three identical legs. Each leg is made up of a prismatic joint, a revolute joint, a parallelogram joint and another revolute joint. The first joint, i.e., the prismatic joint of each leg, is actuated and the end-effector is attached to the other end of each leg. Hence, the Orthoglide 3-axis is a PKM with movable foot points and constant chain lengths. The kinematics of the Orthoglide 3-axis was defined in [3]. Let ρ = [ρ1 ρ2 ρ3] denote the vector of linear joint variables and p = [x y z]t denote the Cartesian coordinate vector of the position of the end-effector. The loop closure of the Orthoglide 3-axis leads to the following three constraint equations: (x−ρ1) + y2 + z2 = l2 1 (1) x2 +(y−ρ2) + z2 = l2 2 (2) x2 + y2 +(z−ρ3) = l2 3 (3) Therefore, the inverse Jacobian matrix J−1 O of the Orthoglide 3axis takes the form: J−1 O =  1 − y ρ1− x − z ρ1− x − x ρ2− y 1 − z ρ2− y − x ρ3− z − y ρ3− z 1  (4) It appears that the Orthoglide 3-axis can have up to two assembly modes, i.e., two solutions to its direct geometric model, and up to eight working modes, i.e., eight solutions to its inverse geometric model. Moreover, those solutions can be easily obtained by solving simple quadratic equations [12]. The lengths of the parallelogram joints and the joint limits were obtained by using the method presented in [8] in order the manipulator to get a cubeshaped translational workspace of 500 mm edge. Agile Eye 2-axis Figure 3 illustrates a spindle mounted on the two dof spherical parallel manipulator, the latter being mounted in series on the Orthoglide 3-axis in order to get the Orthoglide 5-axis. The