Mechanical work reduction during manipulation tasks of a planar 3-DOF manipulator

Abstract

The problem addressed in this paper concerns a planar 3-DOF manipulator operating in horizontal plane, that must move from an initial position characterized by angles theta₁(0), theta₂(0) and theta₃(0) to a required final position characterized by angles theta₁(T_F), theta₂(T_F) and theta₃(T_F). The manipulator can follow any trajectory from the initial to the final position, the goal being to reduce as much as possible the overall mechanical work of this manipulation task in horizontal plane. The approach here is not an optimal theory one, so that to find the global minimum of the overall mechanical work function. Our proposal is just a simple engineering method to reduce the overall mechanical work, trying several possibilities of simple methods of trajectory generation: (1) variations of theta_i parameters corresponding to constant accelerations a⁺_i from t₀ = 0 to t_1/2 = T_F /2, followed by constant decelerations a^-_i =-a⁺_i from t_1/2 = T_F /2 to T_F ; (2) variations/ evolutions of theta_i parameters having the form of cosine functions; (3) evolutions of theta_i parameters corresponding to straight line variations, but using smooth departures at t0 and smooth arrivals at T_F , with limitations concerning the departure accelerations and arrival decelerations; (4) considering the evolutions of angles theta₁(t), theta₂(t) and theta₃(t) between the initial time t₀ = 0 and the final time T_F , as linear segments between t₀ = 0 and t_1/2 = T_F /2 on one hand, and as other linear segments between t_1/2 = T_F/2 and T_F on the other hand. In this fourth case of trajectory, smooth departures, arrivals and transitions from one straight linear segment to another were also considered, thus limiting the maximum torques in the actuators. The results are interesting from an engineering point of view: a non-optimal minimization of the overall mechanical work is analyzed. From the point of view of protecting the actuators and thus increasing their lifetime, a second important criterion was also taken into account: the maximum torques were compared for the four possibilities of trajectories considered here. Further work will find the optimal solution of this manipulation problem, which will better evaluate the performance of the method proposed in this paper for mechanical work and maximum torque reduction.