Kinematic and stiffness analysis of the Orthoglide, a PKM with simple, regular workspace and homogeneous performances

The question whether a parallel-kinematic machine (PKM) is globally more suitable for rapid machining than a serial machine or not, is difficult to answer [1,2] and still open. PKMs and serial machines have their own merits and drawbacks. Today, most industrial 3-axis machine-tools have a serial kinematic architecture with orthogonal linear joint axes along the x, y and z directions. Thus, the motion of the tool in any of these directions is linearly related to the motion of one of the three actuated axes. Also, the performances are constant throughout the Cartesian workspace, which is a parallelepiped. The main drawback is inherent to the serial arrangement of the links, namely, poor dynamic performances. The Orthoglide is a translational 3axis Delta-type [3,4] PKM that was designed to meet the advantages of serial machine tools but without their drawbacks [3]. Starting from a Delta-type architecture with three fixed linear joints and three articulated parallelograms, an optimization procedure was conducted on the basis of two kinematic criteria (i) the conditioning of the Jacobian matrix of the PKM and (ii) the transmission factors. The first criterion was used to impose an isotropic configuration where the tool forces and velocities are equal in all directions. The second criterion was used to define the actuated joint limits and the link lengths with respect to a desired Cartesian workspace size and prescribed limits on the transmission factors. The Orthoglide has a Cartesian workspace shape that is close to a cube whose sides are parallel to the planes xy, yz and xz respectively.

In this paper, a new method is proposed to analyze the stiffness of overconstrained Delta-type manipulators, such as the Orthoglide. The Orthoglide is then benchmarked according to geometric, kinematic and stiffness criteria: workspace to footprint ratio, velocity and force transmission factors, sensitivity to geometric errors, torsional stiffness and translational stiffness. Next section recalls the geometric and kinematic parameters of the Orthoglide prototype. The main contribution compared to our previous results [1], [3], [5] - [8] is related to the Orthoglide stiffness analysis, which is based on the developed analytical pseudo-rigid model. This allows comparing the Orthoglide to the similar performances of other manipulators [9,10].

Figure . 1 shows a photography (left) and a CAD-model (right) of the Orthoglide prototype. This machine has three parallel PRPaR identical chains (where P, R and Pa stand respectively for prismatic, revolute, and parallelogram joints). The actuated joints are the three orthogonal prismatic ones. The output body is connected to the prismatic joints through a set of three parallelograms, so that it can move only in translation. Note that because only two parallelograms would be sufficient to restrict the motion in translation, the Orthoglide is overconstrained. The Orthoglide has been designed so that it has an isotropic configuration in its workspace, that is, a configuration where the Jacobian matrix is isotropic. This configuration is reached when all parallelograms are orthogonal to each other (Fig. 2). To have a kinematic behavior close to the one of a serial 3-axis machine tool, we have also imposed that, in the isotropic configuration, the velocity transmission factors must be equal to 1. This condition implies that for each leg, the axis of the linear joint and the axis of the parallelogram are collinear. Since at the isotropic configuration, the parallelograms are orthogonal, this implies that the linear joints are orthogonal (Fig. 2). An optimization scheme was developed to calculate automatically the geometric parameters as function of the size L workspace of a prescribed Cartesian workspace, which is a cube [3]. The three main steps of this scheme are briefly recalled. First, two points Q 1 and Q 2 are determined in the prescribed cubic Cartesian workspace (Fig. 2) such that if the transmission factor bounds are satisfied at these points, they are satisfied in all the prescribed Cartesian workspace. These points are then used to define the leg length as function of L workspace . Finally, the positions of the base points and the linear actuator range are calculated such that the prescribed cubic Cartesian workspace is fully included in the Cartesian workspace of the Orthoglide. The prototype built at IRCCyN was designed using this optimization scheme on the basis of the following data: the size of the prescribed workspace is L workspace = 200 mm and the minimal and maximal velocity transmission factors λ are ½ and 2, respectively as shown in Fig. 3 [7]. These factors are defined as the eigenvalues of the product of the Jacobian matrix J by its transpose The resulting length of the three parallelograms is 310 mm and the resulting range of the linear joints is 257 mm. The values of the transmission factors inside the prescribed cubic workspace were confirmed using interval analysis

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