Large and fragile systems may require motion trajectories with the highest attainable degree of smoothness. This paper introduces a method to create infinitely continuous motion profiles based on convolving an infinitely continuous FIR kernel with an underlying rate-limited position profile. Conventional motion profiling is limited to blending polynomial between linear segments. Due to the issues of controlling unwanted inflections and the computing of the polynomial coefficient, blending polynomials are usually limited to third order with special cases up to seventh order. Beyond third order, closed form equations do not exists for the coefficients, requiring a set of equations to be solved for each blend. With third order polynomials the highest derivative that can be controlled is jerk. While maximum jerk can be controlled, change in jerk is instantaneous. This inherent instantaneous change in jerk can send shock waves though a system with the possibility of causing damage. The infinitely continuous profiles introduced in this paper overcomes all the shortcomings of polynomial blending. With FIR smoothing of an underlying rate-limited position profile, only a single kernel is needed to be computed off-line before motion starts. The entire process of motion generation consists of rate-limiting desired position updates and then filtering them with an infinitely continuous FIR kernel. The output is an infinitely continuous profile that allows real-time updates of velocity. Two methods are introduced to create the kernels. In the first method the duration of the kernel is minimized for a given maximum value of any derivative. In the second method the maximum value of any derivative is minimized for a given kernel duration. The use of the convolution instead of polynomial fitting to create profiles is simpler, eliminates special cases, and allows real-time updates of velocity while producing maximally smooth profiles.

The twin-pulse motion profile is still perhaps the simplest profile in existence for reducing residual vibration. Its primary drawback is that of narrow bandwidth, i.e., it is effective for attenuating only a very narrow range of frequencies about its design frequency. This has led to the inclusion of more pulses to improve robustness to frequency variation. However, one cost of adding pulses is extending the duration of the resulting motion. This paper presents several techniques whereby the basic twin-pulse motion profile can be expanded to reduce its frequency sensitivity while still preserving its advantages. These techniques include intentionally mis-aligning the null points of ideal twin pulses and a convolved boxcar function, then adding a trapezoidal profile component, finally culminating in what we term a misaligned boxcar/trapezoidal/twin pulse (M- BC:TZ:TP) motion profile. This expanded profile and a basic twin-pulse profile were both evaluated on a ten-pendula system with natural frequencies varying over a range of nearly 10%. Simulation results showed an overall reduction of 97% with the expanded twin-pulse profile, and notably all ten frequencies were attenuated; only one frequency was attenuated with the standard twin-pulse profile. Experiments using a linear rail transport system and the ten pendula device validated the simulation. These expanded profiles retain the simplicity and short duration of the twin-pulse with increased robustness to frequency variations.

Motion profiles such as swing free and input shaping are intended to eliminate residual vibrations. This paper demonstrates a simple generalized method of creating and understanding pulse-based profiles, and presents an approach to suppress a wider band of induced vibration than previously attainable. The current approach of input shaping ignores the potential contributions the base profile and pulse shape can have on the profile. When examined from a signal processing point of view these pulse-based profiles can be thought of as sampled low pass filters. The sampling of the filter kernel defines the magnitude of the pulses, but as in any digital system it also produces aliasing. An interpolating filter can then be used to both remove the aliasing and to smooth and shape the base profile. This dual purpose of the interpolating filter allows the use of the swing free step input in conjunction with a rate-limit as a base profile. A generalized methodology is proposed which can create smooth motion profiles with continuous derivatives, while keeping the computational efficient of traditional input shaping. Experimental validation of two multi-DOF systems is shown.