Hyperbolic trajectories for pick-and-place operations to elude obstacles

Trajectories of hyperbolic type have been proposed for pick-and-place operations where the initial and the final positions are known in both Cartesian and Joint spaces. The method only requires defining the inverse kinematics in these positions. This process is performed with the help of a normalized hyperbolic trajectory, which may be a symmetric curve for the simpler case. This curve goes from zero to unity during a nondimensional time interval of 1 and is constructed with the composition of a hyperbolic tangent function and another function to rescale the time interval from [0, 1] to [-infinity, +infinity], The final function is such that the velocity, acceleration, jerk, and higher derivatives are all zero at the extremes points. The normalized trajectory is then scaled for every variable in the joint space. Immediately, direct kinematics is applied for all those points that compose the entire discrete trajectory in joint space to generate a smooth Cartesian path. The hyperbolic approach permits the definition of a third intermediate position to avoid obstacles, which may produce a change in the symmetry of the normalized trajectory in joint space. The intermediate point is tuned with the modulation of the parameters of the hyperbolic curve. The method was applied to a simulator of 6 degree-of-freedom robot arm developed within this research. The simulator was able to avoid a specific obstacle.