I would also like to thank my co-mentor, Gunnar Ristroph, for his support during the course of the research and for inspiring me to tackle the conic scanning problem. Although algorithms capable of tracking a mobile antenna satellite have been developed and described in detail, instabilities in the system and cross-axis effects degrade performance. Since the increase in tracking performance would be most beneficial if it did not require extensive hardware changes, this project focuses on improvements to the cone scan algorithm.
An algorithm which is one of the oldest and most common implementations of the mobile satellite tracking system.
Problem
Background
Motivation
The most commonly used RSS tracking method is known as conical scanning, which rotates the antenna around the current best estimate of the satellite's location known as the boresight axis. By convolving the RSS with the position displacement wave in the x and y directions over a full cycle, it is possible to extract the phase shift, which can be used to estimate the position along each axis.
Literature Review
Conical Scanning
This process of keeping the signal "high" is called tracking and consists of combining received signals with antenna disturbances. One of the most complete analyzes of cone scanning was done as a technical report issued by NASA/JPL in 1976 [3]. The report describes how the system was perturbed and the effect that scan rate as well as scan radius had on performance without taking a shape for signal strength.
A detailed analysis of the noise performance was performed together with a comparison with experimental results.
Parameter Estimation using Assumed Functions
This report documents the work that was done to demonstrate the feasibility of using conical scanning on the 64 meter diameter paraboloid antenna in Goldstone, California. Instead of using a naive correlation, these more advanced architectures use a Kalman filter formulation for the estimator.
Extended Kalman Filter
I took a theoretical approach to solving this problem, starting with the fundamental method of trying to estimate an (x, y) position, using nothing but a measurement of distance from the origin. For a real system the following sensors and actuators will be present which can be converted to a Cartesian coordinate system with the following values. For now, it is assumed that the Received Signal Strength can be modeled as r2 and the controller tries to minimize it.
The following methods examine the advantages and disadvantages of conical scanning, as well as newly proposed alternative methods for RSS detection.
Variable Parameters
Analysis Methodology
Simulink Conventions
Stabilized Dynamics Block with Noise
- Dynamics
- Stabilizing Controller
- Noise
- Simulink Model
- Advantages
- Disadvantages
Sensor Noise: Sensor Noise is modeled to simulate noise in the output signals from gyroscopes. The power spectral density of this noise was chosen so that the random displacement in position caused by the noise was approximately 1 milliradian displacement per 2 seconds. Noise in the received signal: The pattern of the received signal has been disturbed by atmospheric effects over time.
We tuned the RSS noise to have an RMS error of 40 micro-radians in the estimated radius. A simulation of the traditional method of conical scanning was used on the JPL Deep Space Network [3]. As can be seen in figure 3.1, the reference velocity is chosen to be the derivative of the position offset.
Conical scan compares the relative magnitude of RSS when the antenna is pointing to the right of the boresight axis to when the antenna is pointing to the left. Along the y-axis, the offset wave is Asin(ωt), which is positive when the antenna points above the boresight axis and negative when it points below. The main disadvantage of conical scanning is the increase in error in real-time registration, because the additional positional displacement is added to the inherent noise in the system.
When the target and receiver are stationary, scanning can be periodic, and the antenna can be pointed back to the boresight axis after the highest signal strength is found if the disturbances slowly accumulate. This causes the movement in one of the two axes to be significantly faster than the other, which becomes visible in the simulation as a curved path to the peak in signal strength.
Predicted Gradient Conical Scanning
Simulink Model
Advantages
Disadvantages
Measured Gradient Conical Scanning
Simulink Model
Advantages
Disadvantages
Measured Gradient Tracking
Simulink Model
Advantages
Disadvantages
Preliminary Results
Based on the preliminary results, a more detailed mathematical analysis was performed and provided as follows to compare the traditional cone scan estimator with the newly proposed gradient cone scan estimator.
Overview of Approach
Performance Metrics
Top Level Block Diagram
System Dynamics Equations
In the steady state with no noise, adding these perturbations to our initial position (x0, y0) gives us equations 3.3 and 3.4 showing the location the antenna is pointing to for x(t) all the time. 3.4) Combining these two equations gives. In the traditional method of conical scanning, the signal strength R2 is convolved with the position perturbation (xperturb,yperturb) and integrated over a full scan to obtain a position estimate (ˆx, ˆy). If we maintain the assumption that the system is in a steady state with no noise, it becomes possible to substitute equations 3.5, 3.1 and 3.2 to get equations 4.3 and 4.4.
Step Response
For analysis purposes, φ= 3π2 was chosen as it shows one of the worst possible step and ramp responses for a traditional taper estimator. This happens because the step response constantly overestimates the actual position due to the convolution method. While the problems discussed in the following sections occur over a wide range of values for φ, the differences are most easily observed at this particular value.
A = 1 is the same as setting the radius of the scan equal to the size of the step taken. Although the estimator works well for large values of A, the transient response becomes worse as A becomes smaller. After looking at the step response of the system, it is important to look at the step response across the axes.
This figure shows that the estimates of x and y are combined which can create instability in the system. It is this feature of the traditional estimator that causes asymmetric tracking motion and generates a curved trace at the origin increasing the amount of accumulated error.
Ramp Response
Noise Response
Problems
A new estimator was designed to help eliminate some of the problems encountered using the traditional method, many of which are due to the fact that the R2 term is unbounded and increases rapidly as one moves away from the origin. Instead of using the value of the received signal strength R2(t) directly, the new conical scan estimator uses the derivative of the received signal strength dtd[R(t)2] and convolves it with the velocity perturbation signal (dtd[xperturb) ], dtd[yperturb]) over one scan to estimate the current position. The most general form of the gradient conical scan estimator is given by Equations 5.1 and 5.2 with Eqs.
As with the traditional conical scan estimator, we can substitute equations 3.5, 3.1, and 3.2 into the estimator by making the assumption that the system is in a steady state with no noise. Aωcos(ωτ)) dτ =y0 (5.4) shows that this estimator is also able to perfectly estimate position after a single cycle. Therefore, the estimator is able to perfectly predict the position while in a steady state without noise.
Step Response
While the step response has good features, it is important to keep track of what is happening with the other rater. This can be done by looking at the transverse axis step response shown in Figure 5.3, which shows how ˆy varies with a step input inx. Unlike the cross-axis step response given by figure 4.4 which showed an infinite increase in the cross-axis step response by reducing the scan radius.
Since the response is independent of A, the step size is no longer important. The error is not only limited, but significantly smaller than the off-axis step response of the traditional cone scan estimator.
Ramp Response
Noise Response
The noise response of the estimator is significantly worse, as most of the high-frequency noise escapes. Although the noise response of the estimator is significantly worse, this may not have a dramatic effect on the closed-loop system. The closed loop system acts as a low pass filter for the noise and most of the higher frequency noise coming through the estimator will be at frequencies above the control loop bandwidth so the system will not be able to track them.
This will effectively limit the effect of noise and prevent the system from becoming unstable by not tracking the noise.
Problems
In addition to analyzing the estimators separately, a simulation of the closed loop system was performed, adding the estimator back and feeding the output back into the system dynamics via a PI feedback control. The closed loop system for the conical gradient scanning estimator is similar and shown previously in Figure 6.1. The gains were chosen by modeling the system as a spring mass system and determining the parameters required for critical damping.
Conclusions
The closed loop system is capable of higher bandwidth with the gradient estimator allowing for better tracking performance. The gradient conic scan estimator is shown to be a theoretical improvement over the traditional estimator, and in cases without excessive amounts of noise, it should outperform the traditional estimator. The gradient conical scan estimator is the recommended estimator for us on mobile satellite communication antenna.
Future Work
- Signal Saturation
- Parameter Optimization
- Measured Gradient Scanning and Tracking Systems
- Hardware Implementation
In addition to using the measurements, the effect of the internal noise on the observability is another topic that could be investigated. I showed through simulation that noise has sufficient observability to trace to the origin, but limits to noise are not specified. While the theoretical basis for the new estimators shows that they will perform better under certain conditions than the traditional conical scan estimator, it is important that a hardware test is performed under real conditions to determine if the assumptions in the thesis can be validated.
Theoretical problem setup
Simulink model of the stabilized dynamics with noise
Current implementation of the conical scanning system used for the JPL
Simulink model of the predicted gradient conical scanning method
Simulink model of the measured gradient conical scanning method
Simulink model of the measured gradient tracking method
Overall system design diagram for the conical scanning estimators to track
The range measurement (signal strength) is convolved with the position
Step response of the traditional conical scanning estimator with ω = 2π and
Step response of the traditional conical scanning estimator with ω = 2π and
Cross-axis step response of the traditional conical scanning estimator with
Ramp response of the traditional conical scanning estimator with ω = 2π
Cross-axis ramp response of the traditional conical scanning estimator with
The range measurement (signal strength) is differentiated with respect to
Step response of the gradient conical scanning estimator with ω = 2π and
Cross-axis step response of the gradient conical scanning estimator with
