An algebraic method to check the singularity-free paths for parallel robots

Trajectory planning is a critical step while programming the parallel manipulators in a robotic cell. The main problem arises when there exists a singular configuration between the two poses of the end-effectors while discretizing the path with a classical approach. This paper presents an algebraic method to check the feasibility of any given trajectories in the workspace. The solutions of the polynomial equations associated with the trajectories are projected in the joint space using Gröbner based elimination methods and the remaining equations are expressed in a parametric form where the articular variables are functions of time t unlike any numerical or discretization method. These formal computations allow to write the Jacobian of the manipulator as a function of time and to check if its determinant can vanish between two poses. Another benefit of this approach is to use a largest workspace with a more complex shape than a cube, cylinder or sphere. For the Orthoglide, a three degrees of freedom parallel robot, three different trajectories are used to illustrate this method. INTRODUCTION One of the crucial steps in the trajectory planning is to check the singularity-free paths in the workspace for the parallel manipulators. It becomes a necessary protocol to validate the trajectory when the parallel robot is used in practical applications such as precise manufacturing operations. A trajectory verification problem is presented in [1] based on some validity criteria like whether the trajectory lies fully inside the workspace of the robot and is singularity-free. Singularity-free path planning for the Gough-Stewart platform with a given starting pose and a given ending pose has been addressed in [2] using the clustering algorithm is presented in [3]. An exact method and an approximate method are described in [4] to restructure a path close to the singularity locus into a path that avoids it while remaining close to the original path. Due to the geometry of the mechanism, the workspace may not cover fully the space of poses [3], hence it is necessary to analyze the workspace of the manipulator. The workspace of a parallel robot mainly depends on the actuated joint variables, the range of motion of the joints and the mechanical interferences between the bodies of the mechanism. There are different techniques based on geometric tools [5, 6], discretization [7–9], and algebraic methods [10–13] which are used to compute the workspace of a parallel robot. An algebraic method to solve the forward kinematics problem specifically applied to spatial parallel manipulators is described in [14]. The main advantage of the geometric approach is that, it establishes the nature of the boundary of the workspace [15]. A procedure to automatically generate the kinematic model of parallel mechanisms which further used for singularity free path planning is reported in [16] . An algorithm for computing singularity-free paths on 1 ar X iv :1 50 5. 06 84 2v 1 [ cs .R O ] 2 6 M ay 2 01 5 closed-chain manipulators is presented in [17], also this method attempts to connect the given two non-singular configurations through a path that maintains a minimum clearance with respect to the singularity locus at all points. The main drawback of any numerical or discretization methods is that there might be a singular configuration between two poses of the end-effectors while discretizing the path. This paper illustrates a technique based on some algebraic methods to check the feasibility of any given trajectories in the workspace : it allows to write the Jacobian of the manipulator as a function of time and to check whether its determinant vanishes between two poses. Also, when the trajectory meets a singularity, its location can also be computed. The outline of this paper is as follows. We first introduce the modelization and the basic algebraic tools for computing trajectories in the workspace for parallel robots. We then extended the process to check the singularity-free paths within the workspace and ensures the existence of a singular configurations between the two poses of the end-effector by analysing the Jacobian as a function of time. In later sections we give some detailed examples using three different trajectories for the Orthoglide, a three degrees of freedom parallel robot. METHODOLOGY To ensure the non existence of a singular configurations between two poses of the endeffector, it is necessary to express the Jacobian of the manipulator as a function of the time. The general procedure to check the feasibility of the trajectories in the workspace as well as a function to compute the projection of the trajectories in the joint space can be decomposed as follows : • Defining the constraint equations, articular and pose variables associated with the parallel mechanism; • Computing the singularities and their projections in the workspace and in the joint space; • Computing the workspace and joint space boundaries; • Computing a parametric form of the trajectories in the Cartesian space; • Computing the projections of the trajectories in the joint space; • Computing the Jacobian of the manipulator as a function of the time. By computing, we mean, by default, getting a full characterization as solutions of an exact system of algebraic equations. By abuse of language, computing a projection might thus mean computing the algebraic closure of the projection, for example by using classical elimination strategies based on Gröbner bases (which thus could introduce a subset of null measure made of spurious points). Kinematics involves the study of the position, velocity, acceleration, and all higher order derivatives of the position variables (with respect to time or any other parameter/variables). Hence, the study of the kinematics of manipulators refers to all the geometrical and time-based properties of the motion. For a translational manipulator, the set of (algebraic) relations that connects the input values ρ to the output values X will be denoted by