Reducing Deviations from Trajectory Components with Input Shaping

Input shaping, a technique for reducing residual vibration, was evaluated for use with trajectory following applications. The performance of input shaping for trajectory following was investigated by simulating the response of a two-dimensional system with orthogonal modes. Due to the difficulty of predicting the response of a flexible system to an arbitrary trajectory command, the investigation is limited to the two most common trajectory components: corners and arcs. For a wide range of parameter values the shaped inputs result in better trajectory following than unshaped inputs. The improvements in trajectory following are shown to be robust to parameter uncertainty. Introduction Input shaping reduces residual vibration of computer controlled machines. Input shaping is implemented by convolving a sequence of impulses, known as the input shaper, with a desired system command to produce a shaped input that is then used to command the system. The input shaper is derived from a set of equations that constrains the residual vibration of the system. Additional constraint equations are used to ensure robust performance in the presence of modeling errors. Several methods for obtaining a desired level of robustness have been proposed[8-11, 13]. Additional information regarding input shaping can be found in several sources [2-11, 13, 15]. The ability of input shaping to suppress residual vibration is well established; however, one question that naturally arises is how input shaping affects the intermediate motion of the system. Can input shaping be used to suppress motional vibration (as opposed to residual vibration) and improve tracking of general curvilinear trajectories? An arbitrary trajectory command is composed of multiple command signals. Each signal is convolved with the input shaper to produce a shaped trajectory command which differs from the original trajectory. Because the process of input shaping alters the desired command signals, there is no reason to believe that the system would follow the desired trajectory using a shaped input. There is some evidence that input shaping actually improves tracking performance for trajectories which are only functions of the spatial coordinates (not time). Spatial trajectories compose a large percentage of trajectory following applications. In water-jet cutting, painting, and certain types of scanning operations, the shape of the path traced by the machine tool is considerably more important than the position of the tool as a function of time. Experiments show a five-bar-linkage manipulator follows a clover pattern more accurately with shaping than without [1]. Other experiments show that a flexible structure mounted onto an XY stage follows circular and square trajectories better with input shaping over a wide range of system parameters and command speeds[12]. Because it is difficult to predict how a flexible system will respond to an arbitrary trajectory command, this investigation will examine the differences between the unshaped and shaped response to some rudimentary trajectory commands. For an arbitrary trajectory, each component of the trajectory (e.g. corners, arcs, straight lines) evokes a dynamic response from the system. The concatenation and/or superposition of these responses forms the total system response to that trajectory command. To keep the problem tractable, the investigation will be limited to the two most common trajectory components: corners and arcs. Our purpose in this paper is not to predict the exact response of a flexible system to an input shaped trajectory command; rather, it is to explore the difference between unshaped and shaped responses. The investigation is based upon simulations of a simple two-dimensional flexible system. The model, shown in Figure 1, represents a system with two orthogonal modes under PD control. The flexibility and damping of both the controller and the mechanical system are lumped together into a single spring and damper for each mode. The inputs to the system are xand y-position commands. This model is representative of gantry robots, coordinate measuring machines, and any mechanism which is mounted onto an XY-stage. The results obtained with this model should not be extrapolated to systems with significantly different dynamics without further study. The trajectory following performance of the system is dependent upon various system parameters, the desired command trajectory, and the type of input shaper that is used. If two systems are identical, for instance, with the exception of differing spring constants, the system with the stiffer spring will generally exhibit better performance. For this model, the system parameters consist of the natural frequencies (ωx and ωy) and the damping ratios (ζx and ζy). The command speed of the desired motion also has a large influence on trajectory following. Reducing the time allowed for a given motion will usually degrade the tracking.