Browsing by Author "Pretorius, Arnold"

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    Contributions to quantitative feedback theory design and preliminary application to a variable-pitch quadcopter
    (2023) Pretorius, Arnold; Boje, Edward
    Abstract: Contributions to quantitative feedback theory design and preliminary application to a variable-pitch quadcopter Arnold Pretorius (26/02/2023) This thesis details the mathematical and mechanical modelling and design, state estimation, and preliminary control of a novel variable-pitch quadcopter. The experimental framework is first developed, which includes the physical quadcopter platform, as well as a vision-based motion capture system. Modelling and system identification methods are applied to the quadcopter low-level subsystems, in order to understand and verify the fundamental dynamics of the system. Following this, a state estimation algorithm, based on the extended Kalman filter, is developed, which fuses camera information from the motion capture system to provide pose estimation of the quadcopter platform. A novel rotor thrust observer scheme is also presented, which enables on-board estimation of the quadcopter rotor thrusts during operation. Preliminary low-level control of the rotor speed and thrust is demonstrated, which is facilitated by the on-board rotor speed and thrust estimates. Finally, a simulated demonstration of the position control scheme is provided, which makes use of the previously modelled subsystems and designed control schemes. The simulation environment includes nonlinearities and noise effects that emulate that of the real-world experimentation, and the variable-pitch quadcopter is shown to perform as expected. This thesis investigates the state of the art of quantitative feedback theory, with a particular focus on reducing feedback controller design conservatism. Starting with the generalised single-input-single-output problem formulation, we introduce a novel means of synthesizing a per-plant closed-loop model specification that caters to the signal and phase limitations of every plant instance in the plant set. This information is then incorporated into a univariate constraint set on the feedback controller element, which is predicated on the existence of a valid, non-empty pre-filter solution space in the arithmethic-complex plane. Using a simple example case, we are able to show that the method is effective in balancing the tracking performance across the entire plant set, subject to the inherent signal limits. In a subsequent contribution, we introduce a new approach to the 2x2 model-error tracking problem that combines a plant-inverting design routine with a novel non plant-inverting method, with the aim of reducing the controller design conservatism. We show that geometric-based existence conditions can be exploited to arrive at a univariate design constraint set on the particular feedback controller element of interest, whilst reducing design conservatism at all pertinent frequencies of interest. This method is shown to substantially outperform traditional plant-inverting 2x2 methods, especially at the gain-phase crossover range. Serving as the main QFT contribution of this thesis, we develop a generalised multi-variable refinement approach to the tracking error problem that is intended to ease the feedback control design at all frequencies. Assuming a valid a priori feedback design exists, a feedforward filter is synthesized using optimisation, with the intention of loosening the strictures on a subsequent differential feedback design. The resulting prototype control solution is then used to provide additional gain and phase information that aids in reducing the design conservatism when applying the triangle inequality. This process can be applied iteratively in order to refine the loop transfer behaviour and reduce the feedback controller gain. The method is shown to surpass current multivariable QFT design routines in specific benchmarking examples in terms of expanding the admissible feedback controller per-frequency solution space, especially in the gain-phase crossover region.
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    Open Access
    Contributions to quantitative feedback theory design and preliminary application to a variable-pitch quadcopter
    (2023) Pretorius, Arnold; Boje, Edward
    Abstract: Contributions to quantitative feedback theory design and preliminary application to a variable-pitch quadcopter Arnold Pretorius (26/02/2023) This thesis details the mathematical and mechanical modelling and design, state estimation, and preliminary control of a novel variable-pitch quadcopter. The experimental framework is first developed, which includes the physical quadcopter platform, as well as a vision-based motion capture system. Modelling and system identification methods are applied to the quadcopter low-level subsystems, in order to understand and verify the fundamental dynamics of the system. Following this, a state estimation algorithm, based on the extended Kalman filter, is developed, which fuses camera information from the motion capture system to provide pose estimation of the quadcopter platform. A novel rotor thrust observer scheme is also presented, which enables on-board estimation of the quadcopter rotor thrusts during operation. Preliminary low-level control of the rotor speed and thrust is demonstrated, which is facilitated by the on-board rotor speed and thrust estimates. Finally, a simulated demonstration of the position control scheme is provided, which makes use of the previously modelled subsystems and designed control schemes. The simulation environment includes nonlinearities and noise effects that emulate that of the real-world experimentation, and the variable-pitch quadcopter is shown to perform as expected. This thesis investigates the state of the art of quantitative feedback theory, with a particular focus on reducing feedback controller design conservatism. Starting with the generalised single-input-single-output problem formulation, we introduce a novel means of synthesizing a per-plant closed-loop model specification that caters to the signal and phase limitations of every plant instance in the plant set. This information is then incorporated into a univariate constraint set on the feedback controller element, which is predicated on the existence of a valid, non-empty pre-filter solution space in the arithmethic-complex plane. Using a simple example case, we are able to show that the method is effective in balancing the tracking performance across the entire plant set, subject to the inherent signal limits. In a subsequent contribution, we introduce a new approach to the 2x2 model-error tracking problem that combines a plant-inverting design routine with a novel non plant-inverting method, with the aim of reducing the controller design conservatism. We show that geometric-based existence conditions can be exploited to arrive at a univariate design constraint set on the particular feedback controller element of interest, whilst reducing design conservatism at all pertinent frequencies of interest. This method is shown to substantially outperform traditional plant-inverting 2x2 methods, especially at the gain-phase crossover range. Serving as the main QFT contribution of this thesis, we develop a generalised multi-variable refinement approach to the tracking error problem that is intended to ease the feedback control design at all frequencies. Assuming a valid a priori feedback design exists, a feedforward filter is synthesized using optimisation, with the intention of loosening the strictures on a subsequent differential feedback design. The resulting prototype control solution is then used to provide additional gain and phase information that aids in reducing the design conservatism when applying the triangle inequality. This process can be applied iteratively in order to refine the loop transfer behaviour and reduce the feedback controller gain. The method is shown to surpass current multivariable QFT design routines in specific benchmarking examples in terms of expanding the admissible feedback controller per-frequency solution space, especially in the gain-phase crossover region.
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    Open Access
    Design of a low-cost optical motion capture system using a multi-camera configuration and an asynchronous extended kalman filter
    (2025) Meyer, Zakariya; Pretorius, Arnold; Hepworth, James
    Motion capture technology, originating from the entertainment industry, has expanded its applications to various fields including robotics, medical and healthcare, the automotive industry, and virtual and augmented reality. Despite its versatility, the high cost of off-the-shelf commercial motion capture systems makes this technology inaccessible for many smaller institutions and businesses. This dissertation presents the design and development of a low-cost optical motion capture system using a multi-camera setup and a novel algo-rithm that embeds the camera model within an extended Kalman filter (EKF) for precise tracking of a robot's pose. The goal of this dissertation is to reduce the cost of an optical motion capture system by a factor of 7, targeting a total system cost of approximately $900. In comparison, off-the-shelf commercial optical motion capture systems currently cost over $6,600. The methodology includes initial simulation of the system in MATLAB, which is enhanced by real-world experimentation using affordable cameras programmed to track predefined features on a rigid-body robot. These cameras use image processing techniques to transmit pixel coordinate locations to a local base station, where the EKF algorithm processes the data to estimate the robot's pose. Experimental testing results demonstrates the system's ability to achieve a position and orientation accuracy of less than 1 cm and 2◦, respectively, within a 2 × 2 × 2 m capture space, at a cost of $883,34, which is significantly lower when compared to off-the-shelf commercial systems. The development revealed significant challenges in balancing cost and performance, pri-marily due to the limitations of low-cost cameras. The accuracy of motion capture is heavily dependent on camera specifications such as resolution and refresh rate. As cam-era performance improves, costs rise dramatically. The implications of this research are broad, offering a foundation for future explorations into cost-effective motion capture so-lutions. The current work is completely opensource and offered as an invitation to share and collaborate with other institutes of interest.
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    Open Access
    The Creation of Motion Planning Software for a Car-like Robot
    (2023) McAlpine, Liam; Pretorius, Arnold
    As the demand for robots and vehicles capable of autonomously navigating through obstacle-rich environments increases, so will the need for software capable of achieving such functionality. This dissertation presents software capable of finding a solution to the motion planning problem for a car-like robot in an obstacle-rich, static environment and then controlling a real-world robot such that it closely tracks the prevailing best solution while the software continually searches for better solutions. The insights gained from this dissertation may be used to develop motion planning software for industrial robots and autonomous vehicles. To achieve the objective of this dissertation, the use of planning, control, perception and localisation software is necessary. The planning software uses an online Rapidly-exploring Random Tree Star (RRT*) algorithm, which attempts to improve upon the solutions generated by the RRT* algorithm while the robot is tracking the prevailing best solution. The control software uses a trajectory tracking control method that, through a Lyapunov-like analysis using Barbălat's Lemma, is shown to provide global asymptotic stability. The perception software relays wheel encoder and onboard camera information to the localisation software, which uses the extended Kalman filter to estimate the configuration and configuration covariance of the robot. The software is based on the Robot Operating System (ROS) and is tested using a Duckiebot (DB21M). Because the Duckiebot does not possess a rear-facing camera, the mathematical model used for the car-like robot is the Dubins car, where a compact set of closed-form equations that describe the set of Dubins paths is presented in this dissertation. These equations are derived by modelling the Dubins car as an underactuated system on the special Euclidean group in dimension 2 and then solving an associated set of inverse kinematics problems. A similar set of equations that describe the set of Reeds-Shepp paths is also presented in this dissertation. To demonstrate the efficacy of the software, three tests are conducted. In the first, second and third tests, the environment contains no obstacles, one obstacle and three obstacles, respectively, where the obstacles are wooden stands with AprilTags attached, which are used to help localise the Duckiebot. The results of these tests show the software searching for and finding a solution to the motion planning problem in these environments and then controlling the Duckiebot such that it closely tracks the prevailing best solution while the software continually searches for better solutions.