Comparison of Command Shaping Methods for Reducing Residual Vibration

This paper compares command shaping techniques for controlling residual vibration of high-performance machines. Input shaping generates vibration-reducing shaped commands through convolution of an impulse sequence with the desired command. Because input shaping has similarities to notch filtering, it is compared here with a variety of FIR and IIR filters. Several types of input shapers are presented and shown to be more effective than any of the conventional filters. Introduction Control of machine vibration becomes very important as designers attempt to push the state of the art with faster, lighter machines. Many researchers have examined different controller configurations in order to control machines without exciting resonances. Even with a sophisticated controller it is difficult to rapidly move flexible machines without deflections and vibrations. A more achievable goal is to eliminate residual vibration once the machine has achieved a desired setpoint. Input shaping is a command generation technique that reduces residual vibration when a machine is moved from one setpoint to another. Input shaping works like a notch filter that is designed to eliminate decaying sinusoidal responses. Input shaping is implemented by convolving a sequence of impulses, otherwise 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 shaping process is demonstrated in Figure 1. Because of the similarity to conventional filtering techniques, this paper will compare input shaping to a wide range of traditional lowpass and notch filters. First, a brief survey of vibration reduction techniques is presented. Second, several command shaping techniques are discussed. Third, input shaping is reviewed and several types of input shapers are presented. The remainder of the paper will then compare the effectiveness of input shaping to that of conventional filtering. In a previous paper [31] one type of input shaper was compared to conventional filtering. This paper extends the earlier work by introducing several new types of shapers for comparison. Furthermore, the comparison between shapers and filters is expanded by combining performance measures for a better overall evaluation. Vibration Reduction Many researchers have examined feedback approaches to the control of flexible systems. Cannon and Schmitz [7], and Hollars [11] have examined the use of endpoint position feedback. Book [6] and Alberts [1] have examined feedback of strain gage measurements. Another approach to vibration control is to increase structural damping with additional actuators. Plump, Hubbard, and Bailey [26] examined the use of piezoresistive polymer films. Crawley [9] used a distributed array of piezoelectric devices for structural actuation. A survey of vibration control technologies for large space structures was given in [13]. Feedforward or Command Shaping Command shaping is a method of reducing vibration that does not require additional sensors or actuators. The earliest form of command shaping was the use of high-speed cam profiles as motion templates. These input shapes were generated so as to be continuous throughout one cycle. Cam profiles were also shaped to decrease energy at system resonances [44]. Another early form of command shaping was the use of posicast control [39, 41]. This technique breaks a step of a certain magnitude into two smaller steps, one of which is delayed in time. The posicast formulation has been extended to generate low-vibration bang-bang commands [17, 23, 34]. Optimal control approaches have also been used to generate input profiles to flexible systems. Considerable progress has been made toward practical solutions of the optimal control formulation for flexible systems [8, 16]. Frequency shaping terms have been included in the optimal formulation. The derivative of the control input can be included in the penalty function so that, as with cam profiles, the resulting functions are smooth. Farrenkopf [10] and Swigert [40] demonstrated that velocity and torque shaping can be implemented on systems that modally decompose into second order harmonic oscillators. They showed that inputs in the form of the solutions for the decoupled modes can be added so as not to excite vibration in any of the modes. Another technique is based on the concept of the computed torque approach. The system is first modeled in detail. The model is then inverted – the desired output trajectory is specified and the required input needed to generate that trajectory is computed [2, 25].