James Richard Forbes

Steven Dahdah, James Richard Forbes

This paper proposes a method to identify a Koopman model of a feedback-controlled system given a known controller. The Koopman operator allows a nonlinear system to be rewritten as an infinite-dimensional linear system by viewing it in terms of an infinite set of lifting functions. A finite-dimensional approximation of the Koopman operator can be identified from data by choosing a finite subset of lifting functions and solving a regression problem in the lifted space. Existing methods are designed to identify open-loop systems. However, it is impractical or impossible to run experiments on some systems, such as unstable systems, in an open-loop fashion. The proposed method leverages the linearity of the Koopman operator, along with knowledge of the controller and the structure of the closed-loop system, to simultaneously identify the closed-loop and plant systems. The advantages of the proposed closed-loop Koopman operator approximation method are demonstrated in simulation using a Duffing oscillator and experimentally using a rotary inverted pendulum system. An open-source software implementation of the proposed method is publicly available, along with the experimental dataset generated for this paper.

S Sivaranjani, James Richard Forbes, Peter Seiler et al.

We present a conic sector theorem for linear parameter varying (LPV) systems in which the traditional definition of conicity is violated for certain values of the parameter. We show that such LPV systems can be defined to be conic in an average sense if the parameter trajectories are restricted so that the system operates with such values of the parameter sufficiently rarely. We then show that such an average definition of conicity is useful in analyzing the stability of the system when it is connected in feedback with a conic system with appropriate conic properties. This can be regarded as an extension of the classical conic sector theorem. Based on this modified conic sector theorem, we design conic controllers that allow the closed-loop system to operate in nonconic parameter regions for brief periods of time. Due to this extra degree of freedom, these controllers lead to less conservative performance than traditional designs, in which the controller parameters are chosen based on the largest cone that the plant dynamics are contained in. We demonstrate the effectiveness of the proposed design in stabilizing a power grid with very high penetration of renewable energy while minimizing power transmission losses.

David Evan Zlotnik, James Richard Forbes

Nonlinear observer design for systems whose state space evolves on Lie groups is considered. The proposed method is similar to previously developed nonlinear observers in that it involves propagating the state estimate using a process model and corrects the propagated state estimate using an innovation term on the tangent space of the Lie group. In the proposed method, the innovation term is constructed by passing the gradient of an invariant cost function, resolved in a basis of the tangent space, through a linear time-invariant system. The introduction of the linear system completes the extension of linear complementary filters to nonlinear Lie group observers by allowing higher-order filtering. In practice, the proposed method allows for greater design freedom and, with the appropriate selection of the linear filter, the ability to filter bias and noise over specific bandwidths. A disturbance observer that accounts for constant and harmonic disturbances in group velocity measurements is also considered. Local asymptotic stability about the desired equilibrium point is demonstrated. A numerical example that demonstrates the desirable properties of the observer is presented in the context of pose estimation.

Kyungmin Jung, Thomas Hitchcox, James Richard Forbes

The recent development of high-precision subsea optical scanners allows for 3D keypoint detectors and feature descriptors to be leveraged on point cloud scans from subsea environments. However, the literature lacks a comprehensive survey to identify the best combination of detectors and descriptors to be used in these challenging and novel environments. This paper aims to identify the best detector/descriptor pair using a challenging field dataset collected using a commercial underwater laser scanner. Furthermore, studies have shown that incorporating texture information to extend geometric features adds robustness to feature matching on synthetic datasets. This paper also proposes a novel method of fusing images with underwater laser scans to produce coloured point clouds, which are used to study the effectiveness of 6D point cloud descriptors.

Mohammed Ayman Shalaby, Charles Champagne Cossette, Jerome Le Ny et al.

Autonomously retracing a manually-taught path is desirable for many applications, and Teach and Repeat (T&R) algorithms present an approach that is suitable for long-range autonomy. In this paper, ultra-wideband (UWB) ranging-based T&R is proposed for vehicles with limited resources. By fixing single UWB transceivers at far-apart unknown locations in an indoor environment, a robot with 3 UWB transceivers builds a locally consistent map during the teach pass by fusing the range measurements under a custom ranging protocol with an on-board IMU and height measurements. The robot then uses information from the teach pass to retrace the same trajectory autonomously. The proposed ranging protocol and T&R algorithm are validated in simulation, where it is shown that the robot can successfully retrace the taught trajectory with sub-metre tracking error.

Steven Dahdah, James Richard Forbes

Approximating the Koopman operator from data is numerically challenging when many lifting functions are considered. Even low-dimensional systems can yield unstable or ill-conditioned results in a high-dimensional lifted space. In this paper, Extended Dynamic Mode Decomposition (DMD) and DMD with control, two methods for approximating the Koopman operator, are reformulated as convex optimization problems with linear matrix inequality constraints. Asymptotic stability constraints and system norm regularizers are then incorporated as methods to improve the numerical conditioning of the Koopman operator. Specifically, the H-infinity norm is used to penalize the input-output gain of the Koopman system. Weighting functions are then applied to penalize the system gain at specific frequencies. These constraints and regularizers introduce bilinear matrix inequality constraints to the regression problem, which are handled by solving a sequence of convex optimization problems. Experimental results using data from an aircraft fatigue structural test rig and a soft robot arm highlight the advantages of the proposed regression methods.

Charles Champagne Cossette, Mohammed Shalaby, David Saussié et al.

A coordinate system is proposed that replaces the usual three-dimensional Cartesian x,y,z position coordinates, for use in robotic localization applications. Range, azimuth, and elevation measurement models become greatly simplified, and, unlike spherical coordinates, the proposed coordinates do not suffer from the same kinematic singularities and angle wrap-around. When compared to Cartesian coordinates, the proposed coordinate system results in a significantly enhanced ability to represent the true distribution of robot positions, ultimately leading to large improvements in state estimation consistency.

Daniil Lisus, Charles Champagne Cossette, Mohammed Shalaby et al.

It is essential that a robot has the ability to determine its position and orientation to execute tasks autonomously. Heading estimation is especially challenging in indoor environments where magnetic distortions make magnetometer-based heading estimation difficult. Ultra-wideband (UWB) transceivers are common in indoor localization problems. This letter experimentally demonstrates how to use UWB range and received signal strength (RSS) measurements to estimate robot heading. The RSS of a UWB antenna varies with its orientation. As such, a Gaussian process (GP) is used to learn a data-driven relationship from UWB range and RSS inputs to orientation outputs. Combined with a gyroscope in an invariant extended Kalman filter, this realizes a heading estimation method that uses only UWB and gyroscope measurements.

Mitchell R. Cohen, Khairi Abdulrahim, James Richard Forbes

This paper considers optimal control of a quadrotor unmanned aerial vehicles (UAV) using the discrete-time, finite-horizon, linear quadratic regulator (LQR). The state of a quadrotor UAV is represented as an element of the matrix Lie group of double direct isometries, $SE_2(3)$. The nonlinear system is linearized using a left-invariant error about a reference trajectory, leading to an optimal gain sequence that can be calculated offline. The reference trajectory is calculated using the differentially flat properties of the quadrotor. Monte-Carlo simulations demonstrate robustness of the proposed control scheme to parametric uncertainty, state-estimation error, and initial error. Additionally, when compared to an LQR controller that uses a conventional error definition, the proposed controller demonstrates better performance when initial errors are large.

Charles Champagne Cossette, Alex Walsh, James Richard Forbes

The complex-step derivative approximation is a numerical differentiation technique that can achieve analytical accuracy, to machine precision, with a single function evaluation. In this letter, the complex-step derivative approximation is extended to be compatible with elements of matrix Lie groups. As with the standard complex-step derivative, the method is still able to achieve analytical accuracy, up to machine precision, with a single function evaluation. Compared to a central-difference scheme, the proposed complex-step approach is shown to have superior accuracy. The approach is applied to two different pose estimation problems, and is able to recover the same results as an analytical method when available.

Natalia Pavlasek, Alex Walsh, James Richard Forbes

This paper considers the use of two position receivers and an inertial measurement unit (IMU) to estimate the position, velocity, and attitude of a rigid body, collectively called extended pose. The measurement model consisting of the position of one receiver and the relative position between the two receivers is left invariant, enabling the use of the invariant extended Kalman filter (IEKF) framework. The IEKF possesses various advantages over the standard multiplicative extended Kalman filter, such as state-estimate-independent Jacobians. Monte Carlo simulations demonstrate that the two-receiver IEKF approach yields improved estimates over a two-receiver multiplicative extended Kalman filter (MEKF) and a single-receiver IEKF approach. An experiment further validates the proposed approach, confirming that the two-receiver IEKF has improved performance over the other filters considered.

Mohammed Shalaby, Charles Champagne Cossette, Jerome Le Ny et al.

It is often convenient to separate a state estimation task into smaller "local" tasks, where each local estimator estimates a subset of the overall system state. However, neglecting cross-covariance terms between state estimates can result in overconfident estimates, which can ultimately degrade the accuracy of the estimator. Common cascaded filtering techniques focus on the problem of modelling cross-covariances when the local estimators share a common state vector. This letter introduces a novel cascaded and decentralized filtering approach that approximates the cross-covariances when the local estimators consider distinct state vectors. The proposed estimator is validated in simulations and in experiments on a three-dimensional attitude and position estimation problem. The proposed approach is compared to a naive cascaded filtering approach that neglects cross-covariance terms, a sigma point-based Covariance Intersection filter, and a full-state filter. In both simulations and experiments, the proposed filter outperforms the naive and the Covariance Intersection filters, while performing comparatively to the full-state filter.

Steven Dahdah, James Richard Forbes

The regression problem associated with finding a matrix approximation of the Koopman operator from data is considered. The regression problem is formulated as a convex optimization problem subject to linear matrix inequality (LMI) constraints. Doing so allows for additional LMI constraints to be incorporated into the regression problem. In particular, asymptotic stability constraints, regularization using matrix norms, and even regularization using system norms can be easily incorporated into the regression problem.

David Evan Zlotnik, James Richard Forbes

The simultaneous localization and mapping (SLAM) problem is considered in three dimensions. The proposed algorithm, differential geometric SLAM (DG-SLAM), employs methods from differential geometry to propagate the state and map estimates. Unlike EKF SLAM, the proposed filter is provably asymptotically stable under the assumption of no measurement noise or biases. The robustness of the DG-SLAM algorithm is assessed in simulation with measurement noise. The simulation demonstrates successful localization and mapping.