J. Jim Zhu

Omni-directional mobile robot controller design by trajectory linearization

In this paper, modeling and nonlinear controller design for an omnidirectional mobile robot are presented. Based on the robot dynamics model, a nonlinear controller is designed using the trajectory linearization control (TLC) method. Some simulation results of the controller are presented.

Omni-directional mobile robot controller based on trajectory linearization

In this paper, a nonlinear controller design for an omni-directional mobile robot is presented. The robot controller consists of an outer-loop (kinematics) controller and an inner-loop (dynamics) controller, which are both designed using the Trajectory Linearization Control (TLC) method based on a nonlinear robot dynamic model. The TLC controller design combines a nonlinear dynamic inversion and a linear time-varying regulator in a novel way, thereby achieving robust stability and performance along the trajectory without interpolating controller gains. A sensor fusion method, which combines the onboard sensor and the vision system data, is employed to provide accurate and reliable robot position and orientation measurements, thereby reducing the wheel slippage induced tracking error. A time-varying command filter is employed to reshape an abrupt command trajectory for control saturation avoidance. The real-time hardware-in-the-loop (HIL) test results show that with a set of fixed controller design parameters, the TLC robot controller is able to follow a large class of 3-degrees-of-freedom (3DOF) trajectory commands accurately.

Guidance, Navigation, and Control System Design for Tripropeller Vertical-Take-Off-and-Landing Unmanned Air Vehicle

In this paper, we present the design and development of the guidance, navigation, and control system of a small vertical-takeoff-and-landing unmanned air vehicle based on a 6 degrees-of-freedom nonlinear dynamic model. The vertical-takeoff-and-landing unmanned air vehicle is equipped with three propellers for vertical thrust, and thrust differential together with a set of yaw trim flaps are used for 3 degrees-of-freedom attitude and thrust control actuation. The focus is on the 6 degrees-of-freedom flight control algorithm design using the trajectory linearization control method, along with simulation verification and robustness tests. Hardware and software implementation of the flight controller and onboard navigation sensors are also briefly discussed.

Singular Perturbation Analysis for Trajectory Linearization Control

Trajectory linearization control (TLC) is a nonlinear control design method, which combines an open-loop nonlinear dynamic inversion and a linear time-varying feedback stabilization. Singular perturbation theory has been applied in TLC applications to simplify the design procedure. In this paper, TLC design for a general nonlinear system with singular perturbation is illustrated. The stability of such design is analyzed. The analysis is based on the Lyapunov second method and linear time-varying spectra theory, which belongs to Lyapunov first method. It provides a guideline to TLC design with singular perturbation.

Flight Control of Hypersonic Scramjet Vehicles Using a Differential Algebraic Approach

Abstract : Trajectory Linearization Control is applied to the longitudinal hypersonic scramjet vehicle model under development at the Air Force Research Laboratory. The algorithm is based on Differential Algebraic Spectral Theory which features a time-varying eigenvalue concept and avoids the use of so-called frozen-time eigenvalues which can lead to unreliable results when applied to time-varying dynamics systems. A trajectory linearization control was first designed for a non-linear, affine, rigid-body model using an allocation strategy based on trim-condition look-up tables formulated by trimming the model at multiple operating points while varying velocity and altitude. This data was then fitted to a polynomial function, and the lookup tables were replaced by analytical expressions for the effector settings. The TLC design was then verified on the first-principles based, longitudinal, rigid-body hypersonic vehicle model developed at AFRL using both look-up table and curve fit strategies, and simulation testing results are presented. The current design will be further extended to allow adaptive control of time-varying flexible modes using time-varying bandwidth notch filters and a trajectory linearization observer.

Attitude tracking control of a quadrotor UAV in the exponential coordinates

Abstract A new approach to control the attitude of a quadrotor UAV in terms of the exponential coordinates is developed in this paper. The exponential coordinate is a minimal representation of the rotation matrix, but it can avoid singularities. Since the quadrotor UAV can be considered as a rigid body aircraft, the analytic closed-form expressions of a rigid bodys attitude kinematics are derived from differential of exponential on SO(3). Furthermore, based on the exponential expressions of attitude kinematics, the controller of a fully actuated rigid body is designed using trajectory linearization control method. The overall attitude controller contains two loops, which are designed according to the torque equation and the angular velocity equation respectively. In the numerical simulation, the proposed attitude controller is compared to a controller in the Euler angles, showing that singularities induced by Euler angles are avoided by using exponential coordinates. The robustness test of the attitude controller is also demonstrated in the simulation. The simulation results indicate that the proposed method can be applied to the attitude tracking control of an aerial robot especially when the robot needs to make aggressive maneuverings.

Time-varying notch filters for control of flexible structures and vehicles

Vibration due to flexible modes can be attenuated using a notch filter tuned at the known resonant frequencies of a flexible structure or vehicle. However, there are applications in which those particular resonant frequencies are not known a priori. Further complication arises when those frequencies are not constant in time. The next generation reusable launch vehicle poses such as problem. the X33 concept vehicle, for example, burns up to 90% of its fuel during the first few minutes of flight. The significantly alters the mass distribution and therefore the natural modes of the vehicle. The current work investigates the use of time-varying (TV) notch filters based on the PD-eigenvalue theory Both the notch frequency and the bandwidth of the filter can be adjusted in real time to attenuate the time-varying flexmodes whenever they are excited, and to reduce the undesirable phase lag caused by the filter when the flexmodes are quiescent. Real-time identification of resonant frequencies is performed by a bank of linear, time-invariant (LTI) ban pass filters spanning the entire operating spectrum and a flex-mode detection logic unit. The design principle, as well as some promising simulation results, is given in the paper.

A singular perturbation approach for time-domain assessment of Phase Margin

This paper considers the problem of time-domain assessment of the Phase Margin (PM) of a Single Input Single Output (SISO) Linear Time-Invariant (LTI) system using a singular perturbation approach, where a SISO LTI fast loop system, whose phase lag increases monotonically with frequency, is introduced into the loop as a singular perturbation with a singular perturbation (time-scale separation) parameter ε. First, a bijective relationship between the Singular Perturbation Margin (SPM) εmax and the PM of the nominal (slow) system is established with an approximation error on the order of ε2. In proving this result, relationships between the singular perturbation parameter ε, PM of the perturbed system, PM and SPM of the nominal system, and the (monotonically increasing) phase of the fast system are also revealed. These results make it possible to assess the PM of the nominal system in the time-domain for SISO LTI systems using the SPM with a standardized testing system called “PM-gauge,” as demonstrated by examples. PM is a widely used stability margin for LTI control system design and certification. Unfortunately, it is not applicable to Linear Time-Varying (LTV) and Nonlinear Time-Varying (NLTV) systems. The approach developed here can be used to establish a theoretical as well as practical metric of stability margin for LTV and NLTV systems using a standardized SPM that is backward compatible with PM.

6DOF flight control of fixed-wing aircraft by Trajectory Linearization

In this paper, Trajectory Linearization Control (TLC) is used to design a six degree-of-freedom, autonomous, trajectory-tracking controller for a fixed-wing vehicle dynamics model. TLC combines an open-loop dynamic inverse of the plant dynamics with a closed-loop tracking-error regulator that accounts for model mismatch, disturbances, and excitation of internal dynamics. Feedback gains are obtained symbolically as a function of the nominal trajectory, thus avoiding the use of gain scheduling, and enabling operation across the full flight-envelope without the need for mode switching. The design method is presented, and trajectory tracking simulation results are given for a climbing, bank-to turn maneuver.

A spectral lyapunov function for exponentially stable LTV systems

This paper presents the formulation of a Lyapunov function for an exponentially stable linear time-varying (LTV) system using a well-defined PD-spectrum and the associated PD-eigenvectors. It provides a bridge between the first and second methods of Lyapunov for stability assessment, and will find significant applications in the analysis and control law design for LTV systems and linearizable nonlinear time-varying systems.