Adaptive Correction and Look-up Table Based Interpolation of Quadrature Encoder Signals

This paper presents a new method to increase the available measurement resolution of quadrature encoder signals. The proposed method features an adaptive signal correction phase and an interpolation phase. Typical imperfections in the encoder signals including amplitude difference, mean offsets and quadrature phase shift errors are corrected using recursive least squares with exponential forgetting and resetting. Interpolation of the corrected signals are accomplished by a quick access look-up table formed offline to satisfy a linear mapping from available sinusoidal signals to higher order sinusoids. The position information can be derived from the conversion of the high-order sinusoids to binary pulses. With the presented method, 10nm resolution is achieved with an encoder having 1μm of original resolution. Further increase in resolution can also be satisfied with minimizing electrical noises. Experiment results demonstrating the effectiveness of the proposed method for a single axis and two axis slider systems are given. INTRODUCTION There is growing interest for precision motion control with increasing demand for micro/nano-technology related equipment [1]. Precision positioning is required in many industrial applications including micro/nano-scale manufacturing and assembly, optical component alignment systems, scanning microscopy applications, cell/tissue engineering etc. [2-4]. Micro/nano-scale applications require micro/nano-positioning devices with high precision and resolution. However, the performance characteristics of the positioning devices depend highly on the precision and resolution that can be obtained from the encoders. Hence, in order to achieve high performance with the overall positioning system, it is crucial to increase the resolution of the encoders. Yet, achievable resolution is limited by the available manufacturing technologies used for the encoders [5, 6]. As an example, with the current available manufacturing technologies, commercially available linear optical encoders can have 0.512 micrometers scale grating in pitch satisfying 0.128 micrometers of optical resolution. However, further development in resolution with decreasing the pitch of scale grating is rather limited. Signal processing techniques for interpolation of the available encoder signals serves further improvement of the encoder resolution by deriving intermediate position values out of the sinusoidal encoder signals. Although it is possible to achieve high resolution values using various kinds of interpolation approaches, both hardware and software interpolation methods require ideal encoder signals with a quadrature phase difference between them. However, the encoder signal pairs usually contain some noise and errors due to encoder scale manufacturing tolerances, assembly problems, operation environment conditions, and electrical grounding problems. Interpolation errors will occur while extracting intermediate position values from the distorted pair of sinusoidal encoder signals. Therefore, these errors and noises have to be compensated before the interpolation method is applied. So far, many different approaches have been developed to correct the distorted encoder signals containing amplitude errors, mean offsets, and quadrature phase shift errors. The first introduced method was proposed by Heydemann [7]. In this method, the errors in the encoder signal pairs are determined effectively using least squares minimization. Then, the correction is done based on the calculated error values. However, since the correction parameters are calculated offline, this method does not offer an effective compensation when the errors are changing dynamically through the motion. Applications of this correction method can be found in [5] and [8]. In order to compensate the dynamic errors in encoder signals, several online compensation methods are developed. In [9], Balemi used gradient search method to calculate the correction parameters online, but performance of this method is not effective in low frequencies and noisy signals as mentioned in [10]. Another online error compensation method proposed in [6] corrects the sinusoidal signals obtained from a linear optical encoder by making use of an adaptive approach based on radial basis functions. Then, authors use the similar procedure to increase the resolution of the encoder. Although high-resolutions are achieved with this method, it requires a training period for every new encoder. Also, changes in the environmental conditions may require a training period. In addition to this method, some other interpolation methods applied on optical encoders to increase the resolution are discussed in [11] and [12]. Cheung [11] proposed a sine-cosine interpolation method using logic gates and comparators. In [12], interpolation of encoder signals is accomplished by using the digital signal processing (DSP) algorithms following the digitization of sinusoidal encoder signals with analog-to-digital converters (ADC). However, these interpolation approaches require external hardware such as high precision ADCs and DSPs to obtain high resolution from the encoder. Hence, their applicability to typical servo controller with a digital incremental encoder interface is limited [6]. Another interpolation approach used so far is based on look-up tables. Tan et al. [5] obtained high-order sinusoids from original encoder signals and stored them in a look-up table for online mapping of encoder signals. With this method, they managed to achieve high resolution. Some other hardware and software based interpolation methods are also applied on magnetic encoders and resolver sensors [10, 13, 14]. This paper presents a new method to obtain high-resolution position values out of the original encoder signals. Our motivation here is to obtain high-order sinusoids from original encoder signals. Mapping of original signals to high-order ones is accomplished by a look-up table. Signal conditioning before the interpolation is achieved using an adaptive approach. Important aspects of the work presented here can be given as the adaptive characteristics of the correction method as well as the simplicity of the interpolation method (i.e. for real time performance). Requirement for additional hardware is also eliminated. Moreover, applicability of the presented interpolation method is examined on single and two axis positioning devices. Experimental results obtained with the application of the proposed method on a linear optical encoder are provided in the related sections of the paper. OVERVIEW OF THE PROPOSED APPROACH Proposed method features two main steps: correction of signal errors and interpolation of corrected signals. For the correction step, an adaptive correction method is adopted to compensate the encoder signal errors including amplitude difference, mean offsets, and quadrature phase shift errors. The adaptation is satisfied by the recursive least squares (RLS) with exponential forgetting and resetting. The reason for adopting an adaptive correction technique is due to the dynamic characteristics of the errors. For high precision positioning applications, assembly and alignment of the encoder is very important to be able to obtain required accuracy and precision. However, for closed systems or long range positioning systems, it may not be possible to align the encoder to obtain perfect quadrature signals. Characteristics of the resulting signal may change through the motion. Hence, adaptive approach used in this paper is more suitable for the systems where the signal errors change dynamically. In the second step of the proposed method, interpolation of the corrected signals is satisfied by a look-up table based method. In this method, the basic idea is to obtain high-order sinusoids from the original encoder signals by mapping the original signals to high-order ones online with the help of a quick access look-up table. Since the look-up table is formed offline, the computational effort is considerably less compared to the previously mentioned online interpolation methods. At the end of the interpolation step, position information can be derived from the conversion of the high-order sinusoids to binary pulses. This conversion is accomplished without using an additional hardware such as high precision ADCs. An overall flow diagram for the proposed approach is shown in Fig. 1. In this figure, signal correction and interpolation steps are labeled as step 1 and step 2, respectively. The correction step takes the encoder signals u1 and u2 and generates signals û1 and û2 as corrected quadrature signals. In order to compensate the errors in u1 and u2, a set of correction parameters θ is calculated using RLS with exponential forgetting and resetting in the parameter adjustment block and fed into the signal correction block. Here, our parameter adjustment rule uses the current encoder signals u1 and u2 and corrected signals û1 and û2 from previous iteration. λ is the forgetting factor or discounting factor. When the correction step is completed, index calculator generates index, i, for signals to obtain the corresponding high-order sinusoid values, u1n and u2n, from the look-up table. The look-up table is constructed using the correct values of high-order sinusoids since the corrected signals coming from the Step 1 are calculated with sufficient precision through the adaptive correction scheme. Therefore, the look-up table can be easily generated offline without requiring high computational effort. Using high-order sinusoids, pulse generator generates quadrature binary pulses A and B without requiring high-precision ADCs. Finally, position value is calculated by detecting zero crossings of high-order sinusoids. FIGURE 1. GENERAL FLOW DIAGRAM OF THE PROPOSED APPROACH Step 1: Adaptive Encoder Signal Correction Before the interpolation step, it is crucial to correct the errors in the original encoder signals to prevent high interpolation errors. Common errors affecting the quadrature encoder signals are amplitude difference, mean offsets and quadrature phase shift errors. In this paper, an adaptive approach is used to correct these errors. In some applications, it is possible to have similar error characteristics through the motion. In these cases, errors can be compensated with offline correction methods given in [5] and [7]. On the other hand, in some cases where the encoder alignment cannot be performed effectively or systems have long range of movement track, the errors change throughout the motion. In these cases, in order to obtain high-resolution, adaptive approaches may be adopted to track the errors better. For this purpose, RLS with exponential forgetting and resetting method is developed to adjust correction parameters adaptively. In order to develop the mathematical foundation (i.e. Eq. (1) Eq. (14)) for our proposed method, we will start with the formulation given in [7]. An ideal set of quadrature encoder signals with amplitude of A, u1i and u2i, can be expressed as