Sensitivity of the Optimum Bucket Trajectory in Controlled Excavation

Abstract

In recent years there has been increasing interest in the development and deployment of excavation automation. Several works has been devoted to the controlled motion of an excavator bucket along assumed trajectory. Among them are works by Bernold [1], Bradley and Seward [2], Hiller and Schneider [6,8]. However, some difficulties in realization of such a motion with prescribed accuracy has been observed. This can be caused, among others, by discontinuous motion of hydraulic actuators. The latter are moving according to the openings and closings of hydraulic valves distributing the oil. Electro-modulated hydraulic systems applied in modern excavators embraces both, electro-hydraulic controls, and servo-systems operating as a close-loop with feedback. In any open-loop electromodulated system input is provided by a reference voltage fed to an electrical regulator, passing on information to an electro-servo valve. The latter in turn causes the motion of hydraulic actuator. Good response and correlation is usually assured for systems with constant parameters. This is because the electrohydraulic systems cannot react properly to any discrepancies between input signal and resulting output. In the cases when constant values cannot be assured. then differences between input signal and the assumed actuator motion can occur. The discrepancy can be minimized in close-loop electro-modulated systems. This is due a transducer added, allowing the output to provide a feedback signal directed to the input side. However, also in this case mechanical inertia of the system may give not the expected output. A significant discrepancy between inputs and resulting outputs can be also caused by unpredicted, sharp variations of soil properties, such as stones or other obstacles. A rapid variation of a force on the bucket tip may cause change of the oil pressure in actuators and then change the assumed output of hydraulic system. In this paper, the sensitivities of the bucket motion with respect to small variation of hydraulic actuators' lengths are discussed. The problem is investigated for kinematically induced digging process performed by a backhoe excavator. It is assumed that all three actuators of the machine can work independently and simultaneously. This gives possibility of moving the bucket, along three degrees of freedom independently. Other words, controlling the flow of hydraulic oil into actuators it is possible to control motion of the bucket in a unique way. The discussed processes are of dynamic nature. (V21ha and Skibniewski [9,101) and sensitivity analysis should incorporate dynamic analysis of the system. However, not all phenomena occurring here are known, and their mathematical model can not be incorporated into our discussion. This is the reason for which only kinematic analysis of the sensitivities is considered. The sensitivity analysis is based on the variation of actuator lengths. The latter, of course, depends on dynamic behaviour of the system, which means that presented results are having evaluative significance. The first section of the paper is devoted to the derivation of implicit functions joining components of the bucket motion with actuator lengths. The former is two displacements of the bucket edge, and its rigid rotation. The second section deals with discussion of small changes of the bucket coordinates, in terms of variations of actuator lengths. Applying the rule of chain differentiation, we arrive to the discussed sensitivities. In the next section, evaluation of small actuators' motions is discussed, in view of openings and closing of proportional valves. The latter are opened and closed in finite distance of time, depending upon the finite lengths to which the prescribed trajectory is divided. The last part of the paper is presenting numerical examples of sensitivities of discussed coordinates. Calculations deal with sensitivities of coordinates defining an optimum trajectory.