Oscillation-free point-to-point motions of planar differentially flat under-actuated robots: a Laplace transform method

Abstract

Differentially flat under-actuated robots are characterized by more degrees of freedom (DOF) than actuators: this makes possible the design of lightweight cheap robots with high dexterity. The main issue of such robots is the control of the passive joint, which requires accurate dynamic modeling of the robot. Friction is usually discarded to simplify the models, especially in the case of low-speed trajectories. However, this simplification leads to oscillations of the end-effector about the final position, which are incompatible with fast and accurate motions. This paper focuses on planar $n$ -DOF serial robotic arms with $n-1$ actuated rotational joints plus one final passive rotational joint with stiffness and friction properties. These robots, if properly balanced, are differentially flat. When the non-actuated joint can be considered frictionless, differentially flat robots can be controlled in open loop, calculating the motor torques demanded by point-to-point motions. This paper extends the open-loop control to robots with a passive joint with viscous friction adopting a Laplace transform method. This method can be adopted by exploiting the particular structure of the equations of motion of differentially flat under-actuated robots in which the last equations are linear. Analytical expressions of the motor torques are obtained. The work is enriched by an experimental validation of a $2$ -DOF under-actuated robot and by numerical simulations of the $2$ - and $4$ -DOF robots showing the suppression of unwanted oscillations.

Motion Planning of Differentially Flat Planar Underactuated Robots

“…A great deal of research has been carried out on the planning of point-to-point motions exploiting differential flatness properties [12][13][14][15]. In order to achieve differential flatness, the last links of the robot must have a particular mass distribution with the center of mass (COM) of some links lying on joint axes [12,16].…”

“…The equations of motion of the planar robot can be obtained using Lagrange's method. The system of equations of motion can be expressed in matrix form as follows [15]:…”

“…The theory of differential flatness is then used to define a reduced set of variables y (called flat variables), equal to the number of actuators [23]. For a planar robot equipped with only one passive joint, the flat variables are defined as follows [15]:…”

“…This equation is linear since the inertia term I * n is constant. As a result, the Laplace transform can be calculated [15]:…”

“…This test is the most realistic case. Because of the presence of damping, the polynomial orders increase [15].…”