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Analyzing Frequency Response of Inertial MEMS in Stabilization Systems
Introduction to Stabilization Systems
Figure 1. Basic platform stabilization system.
Since many stabilization systems require more than one axis of active correction, inertial measurement units (IMUs) often include at least three axes of gyroscopes (measuring angular velocity) and three axes of accelerometers (measuring acceleration and angular orientation) to provide the feedback sensing function. The ultimate goal of the feedback sensor is to provide accurate measurements of the platform’s orientation, even when it is in motion. Since there is no “perfect” sensor technology that can provide accurate angle measurements under all conditions, the IMUs in platform stabilization systems often employ two or three sensor types on each axis.
An accelerometer responds to both static and dynamic acceleration in the direction of each of its axes. “Static acceleration” may seem like a strange term, but it encompasses an important sensor behavior: response to gravity. Assuming that no dynamic acceleration exists, and that sensor errors have been removed through calibration, each accelerometer output will represent the orientation of its axis, with respect to gravity. To determine the actual average orientation in the presence of the vibration and rapid acceleration often experienced in stabilization systems, filters and fusion routines (combining readings from multiple sensor types to obtain a best estimate) are often applied to the raw measurements.
Another type of sensor is the gyroscope, which provides angular rate measurements. Gyroscope measurements contribute to the angle measurements through integration of the angular-rate over finite time periods. When performing integration, bias errors will cause a proportional angle drift that accumulates with respect to time. Therefore, gyroscope performance often relates to the sensitivity of a device’s bias to different environmental factors, such as temperature variation, supply variation, off-axis rotation, and linear acceleration (linear-g and rectified-g × g). A calibrated high-quality gyroscope, with high rejection of linear acceleration, enables these devices to provide wideband angle information to complement the low-frequency information provided by accelerometers.
A third type of sensor is the 3-axis magnetometer, which measures magnetic field intensity. Magnetic field measurements from three orthogonal axes enable estimates of orientation angle, with respect to the local direction of the earth’s magnetic field. When the magnetometer is near motors, monitors, and other sources of dynamic field disturbance, managing its accuracy can be challenging, but in the right circumstances its angular data can augment the measurements from accelerometers and gyroscopes. While many systems use only accelerometers and gyroscopes, magnetometers can improve measurement accuracy in some systems.
The generic block diagram of Figure 2 shows how gyroscope and accelerometer measurements can be employed in a manner that uses their basic strengths but minimizes the impact of their weaknesses. The pole locations of the low-pass accelerometer and high-pass gyroscope filters are typically application-dependent, with accuracy goals, phase delay, vibration, and “normal” motion expectations, all contributing to these decisions. System-dependent behaviors will also affect the weighting factors, which also have an impact on how these two measurements are combined. The extended Kalman filter is one example of an algorithm that combines the filtering and weighting functions to calculate dynamic angle estimates.
Figure 2. Combining single-axis sensor outputs.
MEMS IMU Frequency Response Analysis
A strategy for evaluating IMU bandwidth is determining what information is available in product documentation, analyzing the impact of this information on the system’s response to inertial motion, and stabilizing the system’s response. This analysis, and any corrective actions it entails, will become the basis for preliminary testing.
Frequency response is often represented as “bandwidth” in specification tables for IMUs and gyroscopes. As a performance parameter, it represents the frequency at which the output magnitude drops to about 70% (–3 dB) of the actual magnitude of motion that the sensor is experiencing. In some cases, bandwidth may also be defined by the frequency at which the output response lags the actual motion by 90 degrees (for a 2-pole system). Both of these metrics can directly impact an important stability criterion for a control loop: unity-gain phase margin—the difference between the actual phase angle of the loop response and –180° at a loop gain of 1. Understanding the frequency response of the feedback sensor is a key factor in optimizing the trade-off between stability assurance and system response. In addition to managing stability criteria, the frequency response also has a direct impact on vibration rejection and establishing a sampling strategy that allows all critical transient information on an inertial platform to be measured.
Analyzing frequency response in a system starts with a high-level, “black box” view, which describes the system’s response to inputs over the entire frequency range of interest. In electronic circuits, where the input and output are defined in common terms, such as signal level (volts), this typically involves developing a transfer function, using s-domain representation and circuit-level relationships, such as Kirchhoff’s voltage and current laws. For an inertial MEMS system, the input is the inertial motion that the IMU experiences, and the outputs are often represented by digital codes. While s-domain analysis techniques are valuable, developing a complete transfer function for this type of system often requires additional techniques and consideration.
The analysis process starts with understanding all of the components associated with a sensor signal chain. Figure 3 offers an overall diagram of the typical functions. The signal chain starts with a core sensor element, which translates the inertial motion into a representative electrical signal. If the bandwidth is not limited in the sensor element, it is often limited by filters in the signal-conditioning circuit preceding the ADC. After the signals are digitized, a processor typically applies correction (calibration) formulas and digital filtering. The secondary digital filters reduce the bandwidth and sample rates that the feedback systems use in their control routines. All of these stages can influence the gain and phase of the sensor signal, with respect to frequency. Figure 3 provides an example of an IMU that has multiple filters in a mixed-signal processing system. This system will serve as an example for illustrating some useful analysis techniques.
Figure 3. ADIS16488 sensor in a signal chain for frequency analysis.
Core MEMS Sensor Element
Interface Circuit/Analog Filter
The accelerometer’s single-pole (f1) transfer function is.
These formulas provide the basis for numerical analysis in programs that can manage the complex numbers associated with the
Fmax = 9840/2; % one-half of the sample rate
For a quick assessment of the time delay associated with these filters, notice that the phase delay of a single-pole filter is equal to 45° at its −3-dB frequency, or 1/8 of the corner frequency’s period. In this case, the time delay of the accelerometer’s filter is approximately equal to 0.38 ms. For the gyroscope, the delay is equal to the sum of the time delays of the two stages, or about 0.47 ms.
Averaging/Decimating Filter Stage
When using MATLAB for analysis, use a sample rate (fs) of 9.84 kSPS and four taps (N), along with the same frequency array (f) used to analyze the analog filter. Using a common frequency array will make it easier to combine the results of each stage. Use the following code to analyze this first stage:
Fmax = 9840/2; % one-half of the sample rate
Analyzing the second averaging/decimating filter will require prior knowledge of the control system’s sample rate but will use the same relationships. For example, if a control loop requires a sample rate that is close to 400 SPS, the second filter’s average and decimation rate would be equal to six (for a sample rate of 410 SPS and four samples, 9840/[410 × 4] = 6). Use the same
Figure 4. Analog filter and first decimation stage.
Figure 5. Composite response for 410-SPS data rate.
Programmable FIR Filter Analysis
Fortunately, many modern programs contain specific tools or commands for this type of analysis, based on these basic relationships. It is still useful to understand them when verifying results of the automatic assessment tools and in developing an intuitive feel for when to question the outputs of a FIR design tool. The MATLAB “fdatool” command launches its filter analysis and design package, which helps design and analyze the system FIR filter implementation.
Inertial Frequency Response Test Methods
For early analysis validation without a rate table, measuring the spectral noise over the frequency band of interest can provide useful insights. This simplified approach does not require sophisticated test equipment but only a secure mechanical connection to a stable platform and data collection instrumentation. However, it does rely on the mechanical noise having a “flat” noise magnitude with respect to frequency.
Figure 6 illustrates two examples that both use the same 2-pole, low-pass filter. The first example (ADIS16375) uses a gyroscope that has a flat response over its usable frequency range. The second example (ADIS16488) uses a gyroscope that has a modest amount of peaking at 1.2 kHz, which actually extends the –3-dB frequency to approximately 380 Hz. Recognizing this resonant behavior can be valuable for those in the process of modeling and simulating a control loop. Identifying this behavior in a simple test can also help explain noise levels that are higher than expected when performing a more thorough system characterization. When understood and identified early in a project, these behaviors can normally be managed with adjustments to the filter poles.
When measuring noise density, make sure that the sample rate is at least twice the highest frequency of interest to meet the Nyquist criterion. Also, take enough data samples to reduce the uncertainty of the measurements. The plots in Figure 6 were derived from FFT analysis of a time record with a length of 256k samples at a maximum rate of 2.46 kSPS.
Figure 6. Noise density comparison.
Another approach uses a gyroscope’s self-test function. The self-test function provides an opportunity to stimulate the sensor’s mechanical structure, using an electrical signal, without requiring the device to be subjected to external inertial motion. The self-test function forces a change in the sensor core that simulates its response to actual motion, producing a corresponding change in the electrical output. Not all products provide real-time access to this, but it can be a useful tool when available, or if the manufacturer can provide data from this type of frequency-response test. In the simplest approach, the self-test, which simulates response to a step, is compared with the analytical expectation. Repeating the self-test assertion at specific frequencies provides a direct method for studying the magnitude of the sensor response at each frequency. Consider the two different responses in Figure 7. At the lower frequency, the gyroscope output looks like a square wave, with the exception of the transient response at each transition. The transient response follows the expectation of a “step response” for the filter network in the sensor signal chain. In the second example, where the frequency of the self-test is high enough to prevent full settling, a decrease in magnitude occurs. Notice the difference in magnitudes between the blue and black-dotted responses, on the bottom signal in this figure. There are a number of methods for estimating the magnitude of these time records. A discrete Fourier transform (DFT) separates the primary frequency content (self-test frequency) from the harmonic content, which can contribute errors to the magnitude/frequency response.
Figure 7. Self test.
Developing an understanding of an IMU’s bandwidth and its role in system stability should employ analysis, modeling, test data, and iteration of these factors. Start by quantifying the information available, make assumptions to close any gaps, and then develop a plan to refine these assumptions.
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