Initial Investigations into the Effects of Input Shaping on Trajectory Following

The effects of input shaping on trajectory following were investigated by simulating the response of a fourth-order system with orthogonal modes and conducting experiments on an XY positioning stage. For nearly all values of damping and frequency ratio, the shaped inputs result in significantly better trajectory following than unshaped inputs. When the trajectory improvement techniques in this paper are used, the shaped inputs perform considerably better for all parameter values. Introduction Input shaping improves settling time and positioning accuracy by reducing residual vibrations in computer controlled machines. The method requires only estimates of the natural frequencies and damping ratios. Input shaping is implemented by convolving a sequence of impulses, an input shaper, with a desired system command to produce a shaped input that is then used to drive the system. The input shaper is determined from a set of constraint equations that limit the residual vibration of the system. The constraint on vibration amplitude can be expressed as the ratio of residual vibration amplitude with shaping to that without shaping. This percentage vibration is given by: %Vibration = e−ζωtn{(ΣAieζωti cos(ω 1−ζ 2 ti ))2 + (ΣAieζωti sin(ω 1−ζ 2 ti ))2} 1 2 (1) where Ai and ti are the amplitudes and time locations of the impulses, tn is the time of the last impulse, ω is the vibration frequency, and ζ is the damping ratio. Specifications for an input shaper usually require some amount of insensitivity to errors in the system model. The insensitivity constraint proposed by Singer and Seering requires the derivative of the percentage vibration equation (Eq. 1) to be zero at the modeling frequency [9,10]. These constraints yield a zero vibration and zero derivative (ZVD) input shaper. An alternate constraint achieves significantly more insensitivity by relaxing the constraint of zero vibration at the damped modeling frequency [11,12]. By limiting the residual vibration at the modeling frequency to some small value, V, instead of zero, the zero vibration constraint can be enforced at two frequencies close to the modeling frequency. This set of constraints leads to extra-insensitive (EI) input shapers that are essentially the same length in time as the ZVD shapers presented in [10], but have significantly more insensitivity. Figure 1 compares the sensitivity curves, plots of residual vibration vs. system frequency for ZVD and EI shapers. The EI shaper results in low levels of vibration over a wider range of frequencies than the ZVD shaper, hence the name, extra-insensitive. A great variety of restrictions can be added to the constraint equations to customize input shaping to a specific system. For multiple mode systems, a set of equations consisting of one 0 5 10 15 20 0.7 0.8 0.9 1.0 1.1 1.2 1.3 ZVD Shaper