ACS has participated in the design and analysis of pointing control systems for several Earth imaging and interplanetary spacecraft and has experience in many areas of GN&C design, implementation, test, and operation. Some important areas of GN&C are discussed below. The discussion is not intended to be complete or comprehensive, and much is omitted for brevity. See the articles listed in the boxes on the right for more detailed discussions of particular areas of spacecraft GN&C.

### GN&C Design, and Sensor and Actuator Sizing, Selection, Modeling, and Simulation

Sensor and actuator sizing and selection can often be done by back-of-the-envelope calculations and with the aid of software tools, taking into consideration a set of GN&C and system requirements and trades. Prior designs (aka "heritage") can be used as reference designs, *provided that* the designs, assumptions, and limitations are well understood, particularly in the context of the new spacecraft and its operating environment. Back-of-the-envelope calculations aided by software tools yield a rough first-order cut for sensor and actuator selection and for a conceptual design. A modern design approach will include modeling and simulation to obtain a second-order cut at the design.

Many science missions are unique and rely more on modeling and simulation. High performance systems require interactive inter-disciplinary and multi-disciplinary design approach. This requires the GN&C engineer to be more of a systems engineer and to have a thorough understanding of factors that affect pointing and payload performance.

### Attitude Determination and Sensor Calibration and Alignment

A quick overview of attitude determination and sensor calibration and alignment is given here. There is a detailed discussion of this topic in an article listed in the box on the right. Although the discussion there is on high-performance attitude determination, some of the recommendations apply to lower performance requirements as well.

Performance issues in attitude determination systems are due to hardware problems, software problems, estimation algorithm deficiencies or errors, and implementation problems. Performance issues occur when systems are designed with an incomplete understanding of the performance of attitude sensors (such as star trackers and gyros) under actual operating conditions, including a variety of environmental effects, field of view and field of regard, and angular rate and acceleration. Attitude sensors have to be chosen with consideration of their operating conditions. Performance predictions must be based on error models that are valid for the expected operating conditions. The fidelity of the models depends not only details of the models themselves, but also on knowledge of their parameters. Even if a nominal set of parameters is available, they vary over time and operating conditions, which must be taken into account.

Atitude sensors have to be calibrated and aligned for optimal performance, and calibration and alignment parameters, along with other performance measures, should be monitored and trended. I developed the RADICAL software to estimate attitude along with calibration and alignment parameters for an attitude determination system with star trackers and gyros. The RADICAL filter can be used in flight software, in desktop processing to evaluate performance from simulation data or flight telemetry, and it can be used as a ground-based attitude data processor.

### Pointing Performance and Image Motion OTF analysis

The performance of an optical payload depends in part on image motion and structural perturbations in the optical path, as measured by an image motion optical transfer function (IM OTF) and wavefront error. The IM OTF can be evaluated directly from time-domain data, but this requires significant amount of computation and does not reveal information about individual contributions from disturbance sources. Alternatively, the statistical IM OTFs are based on poointing performance metrics (PPM). The PPM are means and covariances obtained from the frequency response and broadband noise response of the system. The IMOTF and PPM rigorously define displacement, smear, and jitter, in contrast to traditional definitions of stability and jitter. For more information, see in the box on the right for articles on pointing performance and Image Motion OTFs.

### Dynamics Modeling and Simulation

Dynamics modeling and simulation are an integral part of pointing performance analysis to obtain the pointing performance metrics. Dynamics modeling includes disturbance modeling and structural modeling. Disturbances include forces and torques from actuators, mechanisms, thermal ping, and sensor errors. Structural modeling is discussed in the next topic below. The pointing performance metrics are easily evaluated by using frequency response and covariance calculations. However, the covariance calculation assumes stationary inputs. A frequency sweep over a range of reaction wheel speeds is sufficient, provided the wheel speeds are essentially constant over the exposure interval of an optical payload. Gyro noise is stationary, except for flicker noise being non-stationary. Star tracker noise may be stationary or have a slow-changing bias during inertial hold and very small angular rate, but it can be non-stationary at higher angular rates. A covariance analysis for pointing performance due to star tracker noise requires a combination of time-domain and frequency-domain analysis. A time-domain simulation requires very small time steps and incurs numerical error when the stroctural modes cover a wide range of frequencies. Frequency-domain anaysis is much faster and more accurate, and more amenable to Monte Carlo simulation.

### Control-Structure Interaction (CSI) design and analysis

An attitude control system has to be designed to 1) avoid unstable control-structure interaction (CSI) between the feedback controller and structural modes, and 2) to achieve pointing and payload performance requirements following a slew maneuver or sudden thermal change on the structure. CSI is avoided by either gain stabilization or phase stabilzation of structural modes.

Gain stabilization does not actually stabilize structural modes, rather the objective is to avoid putting energy into the modes due to feedback and to use the inherent damping in the structure to remove energy from the structural modes. This is accomplished through the use of an appropriate structural mode filter in the feedback loop. I favor a particular type of structural mode filter that has minimum low-frequency phase lag and good high-frequency attenuation and roll-off. Under certain conditions, low-frequency modes in or near the control loop bandwidth will have increased closed-loop damping due to favorable phase in the feedback loop. Inevitably the damping of some modes, at frequencies just above the control loop bandwidth, may increase due to feedback. The structural mode stability margin for gain stabilization is a gain margin, and also a phase margin for modes in and near the control loop bandwidth. An attitude control system uses multivariable feedback (three rate and three attitude channels). The stability margins can be evaluated by a variety of classical and modern methodologies.

Phase stabilization is designed to remove energy from structural modes and increase their closed-loop damping. This can be accomplished with collocated sensors and actuators.Inevitably at some frequency near the bandwidth of sensors and actuators, and with a physical limitation on collocation, there is a transition from phase stabilization to gain stabilization. The structural mode stability margin for phase stabilization can be expressed as a phase margin. A time-delay margin is needed as well, since inevitable latency in a control loop causes phase lag.

It is important to ensure that a structural model received from the mechanical engineering is a free-free model in the appropriate configuration needed for analysis (e.g., stowed, partially deployed, deployed, various solar array rotation angles). It is useful to separate the rigid-body modes from the flexible-body modes in the model. I derived a numerical method to do this operation. Next, the location of sensors, actuators, and other points of interest have to be identified, and only those mode shapes and mode slopes of interest need to be retained in the mode shape/slope matrix. Various criteria are used to reduce the order of a structural dynamics model to facilitate control system design and simulation. Care should be taken in retaining modes in and near the control loop bandwidth because low-ampltude modes can cause phase reversals, which affect stability. It is important to choose realistic modal damping ratios and model uncertainty factors (MUF) from with the aid of historical in-flight data, established guidelines, and an experienced structural engineer. One should *not* assume a highly conservative value (say, 0.001) for all modes. Modal frequency uncertainty has to be considered. Performance simulation should always be performed with a higher-order model because several low-amplitude modes can make a non-negligible contribution to pointing performance.

When plotting a the frequency response of a structural model, it is essential to capture the peak responses as well as the zeros. This can be hit-or-miss if a grid of frequencies is used, even if a dense grid is used, which can be a computational burden. Note that the peaks are often not exactly at the modal frequencies. I use a phase-gain tracker, which adaptively adjusts the frequency step.

### Attitude Trajectory Design

Pointing command and control algorithms are designed for pointing, scanning, targeting, tracking, and maneuvering. Attitude trajectories are designed to meet pointing control objectives subject to constraints on actuator saturation and power, structural vibration, thermal effects, and stray light from the Earth, Moon, and Sun affecting optical payloads and optical attitude sensors.

### GPS-based precision orbit determination

I am knowledgable in GPS-based orbit determination, I led the development of the GOODS GPS-based orbit determination software, which is now a standard software item at Northrop-Grumman (formerly Orbital Sciences and CTA Space Systems). GOODS is a major improvement over the GEODE orbit determination software that was developed for NASA/GSFC. Later I investigated Runge-Kutta numerical integration algorithms with automatic step size adjustment and interpolators. These numerical integration algorithms seem to have become popular for orbit integration since then. Accurate short-term orbit prediction is needed on some spacecraft for real-time guidance and geolocation.

### Implementation, Integration, and Test

ACS understands algorithm and software development, implementation, and test, including GN&C telemetry specification and testing. It is essential when designing test procedures to take into account the limitations of not only simulation, but also software-in-the-loop (SWIL) and hardware-in-the-loop (HWIL) testing.

**Contact me if you need engineering support in attitude control system design, analysis, and simulation.**

**Copyright © 2021 by Mark E. Pittelkau and Aerospace Control Systems, LLC**