David Evan Zlotnik

Nonlinear Estimator Design on the Special Orthogonal Group Using Vector Measurements Directly

The convergence properties of popular nonlinear attitude estimators can be traced to the choice of an attitude error function. This paper considers a nonlinear deterministic direction cosine matrix estimator whose form is derived from an alternate attitude error function. While the resulting estimator shares several properties with those previously presented in the literature, the careful selection of an attitude error function results in an estimator with superior convergence properties. The attitude estimate is propagated using a rate gyroscope measurement and corrected using two or more vector measurements. Simulation and experimental results are presented that highlight the desirable properties of the proposed estimator.

Continuous-time norm-constrained Kalman filtering

This paper considers continuous-time state estimation when part of the state estimate or the entire state estimate is norm-constrained. In the former case continuous-time state estimation is considered by posing a constrained optimization problem. The optimization problem can be broken up into two separate optimization problems, one which solves for the optimal observer gain associated with the unconstrained state estimates, while the other solves for the optimal observer gain associated with the constrained state estimates. The optimal constrained state estimate is found by projecting the time derivative of an unconstrained estimate onto the tangent space associated with the norm constraint. The special case where the entire state estimate is norm-constrained is briefly discussed. The utility of the filtering results developed are highlighted through a spacecraft attitude estimation example. Numerical simulation results are included.

Exponential convergence of a nonlinear attitude estimator

A nonlinear attitude estimator is presented. The estimator uses a minimum of two vector measurements and a rate-gyro to estimate the attitude and the rate-gyro bias. The estimator can be implemented using a low-cost inertial measurement unit such as those typically used on inexpensive aerial vehicles. Estimator design is based on an alternate attitude error function that yields an estimator with faster convergence properties than similar estimators previously considered in the literature. A proof of exponential stability about the desired equilibrium point is provided. Simulation and experimental results demonstrate the desirable properties of the proposed estimator.

A nonlinear attitude estimator with desirable convergence properties

A nonlinear attitude estimator is considered in this paper. The estimator shares several properties with estimators previously considered including ease of implementation and guaranteed asymptotic stability. The proposed algorithm is based on Poissons equation and propagates the attitude estimate using measurements taken by typical low-cost sensing systems. Rate gyro bias estimation is also considered. The estimator structure is derived using an alternate attitude error function, which results in desirable convergence properties. These properties are demonstrated in simulation where the proposed attitude estimator is shown to achieve a faster convergence rate than an attitude estimator considered in the literature.

Attitude sensing and control of a stratospheric balloon platform

A fundamental question yet to be answered is how fast the universe is expanding. In order to answer this exciting question physicists must be able to accurately determine the magnitude of various astronomical objects, such as stars and supernovae. In order to precisely calibrate the magnitudes of such objects, ground-based astronomical observatories are relied on. However, the observatories must be calibrated before use. To calibrate aground-based observatory, a known microwave source is pointed at the observatory from far away. The purpose of this paper is to outline how the McGill University High Altitude Balloon (McHAB) team intend to use a high-altitude balloon to carry the required light source to altitudes of 15km to 20km. The main challenge, and the novelty of this work, is the design, construction, and implementation of a low-cost attitude control and estimation system that will enable pointing of the microwave source at ground-based observatories. The McHAB team has designed an attitude control system to point a payload to within an accuracy of ±1◦. This platform will be actuated using a single reaction wheel that will control the platform yaw angle. A reaction wheel is chosen due to its simplicity. A PID controller is used to control the reaction wheel, and hence point the scientific payload. A difficult challenge is accurately estimating the attitude of the platform. A rotation-matrix-based complementary filter will be used to fuse accelerometer, gyroscope, and magnetometer measurements from an onboard inertial measurement unit. Hardware consists of an inertial measurement unit containing a 16MHz ATmega328 microcontroller for data acquisition and a single-board computer, the Raspberry Pi, containing a 700MHz ARMv6 processor for data processing and control. The first flight of the system is documented in this paper.

Saturated control of flexible-joint manipulators using a Hammerstein strictly positive real compensator

In this paper the control of flexible-joint manipulators while explicitly avoiding actuator saturation is considered. The controllers investigated are composed of a bounded proportional control term and a Hammerstein strictly positive real angular rate control term. This control structure ensures that the total torque demanded of each actuator is bounded by a value that is less than the maximum torque that each actuator is able to provide, thereby disallowing actuator saturation. The proposed controllers are shown to render the closed-loop system asymptotically stable, even in the presence of modeling uncertainties. The performance of the controllers is demonstrated experimentally and in simulation.

Rotation-matrix-based attitude control without angular velocity measurements

Rigid-body attitude control without angular velocity measurements is considered. The error rotation matrix describing the attitude error is used directly within the control algorithm. A controller composed of a proportional control term and an angular velocity control term is first considered. The proportional control term is a function of the error rotation matrix, while the angular velocity control term is realized by a strictly positive real system. Asymptotic stability of a desired closed-loop equilibrium point is shown, and a derivative free implementation of the controller is identified for use when angular velocity measurements are not available. Numerical simulation results demonstrate successful set-point regulation and noise mitigation.

Sigma Point Kalman Filtering on Matrix Lie Groups Applied to the SLAM Problem

This paper considers sigma point Kalman filtering on matrix Lie groups. Sigma points that are elements of a matrix Lie group are generated using the matrix exponential. Computing the mean and covariance using the sigma points via weighted averaging and effective use of the matrix natural logarithm, respectively, is discussed. The specific details of estimating landmark locations, and the position and attitude of a vehicle relative to the estimated landmark locations, is considered.

MPC for coupled station keeping, attitude control, and momentum management of GEO satellites using on-off electric propulsion

This paper develops a model predictive control (MPC) policy for simultaneous station keeping, attitude control, and momentum management of a nadir-pointing geostationary satellite equipped with three reaction wheels and four on-off electric thrusters mounted on two boom assemblies attached to the anti-nadir face of the satellite. A closed-loop pulse-width modulation (PWM) scheme is implemented in conjunction with the MPC policy in order to generate on-off commands to the thrusters. The MPC policy is shown to satisfy all station keeping and attitude constraints while managing stored momentum, enforcing thruster constraints, and minimizing required delta-v.

Exteroceptive measurement filtering embedded within an SO(3)-based attitude estimator

A nonlinear deterministic attitude estimator is developed. The estimator evolves directly on the special orthogonal group of rigid-body rotations. Similar estimators in the literature are characterized by constant estimator gains that amplify noise across all frequencies. The proposed method accounts for undesirable noise in exteroceptive measurements by incorporating a strictly positive real (SPR) filter in the estimator structure. By tuning the “embedded” SPR filter over a frequency bandwidth undesirable noise corrupting exteroceptive measurements may be rejected. Rate-gyro bias estimation is also considered. The proposed attitude estimator is shown to asymptotically converge to the desired equilibrium point (i.e., when the estimated attitude and bias are equal to their true counterparts), provided a restriction on the set of initial conditions is satisfied. Moreover, exponential convergence is shown in a certain region about the desired equilibrium point. The proposed method is demonstrated in simulation.