Nonlinear optimal control and its application to a two-wheeled robot

This research studies two advanced nonlinear optimal control techniques, i.e., the freezing control and the iteration scheme, and their associated applications, such as a single inverted pendulum (IP) on a cart system and a two-wheeled robot (TWR) system. These techniques are applied to stabilise the highly unstable nonlinear systems in the vertical upright position when facing different initial pitch angles. Different linear optimal controllers (linear quadratic regulator and linear quadratic Gaussian) and nonlinear optimal controllers are designed and applied to the models for concurrent control of all state variables. The controlled systems are tested in simulation and the best performing control design is eventually implemented on a robot prototype built with an educational kit – the LEGO EV3, after practical factors such as motor voltage limitation, gyro sensor drift and model uncertainties have been considered, analysed and dealt with. Simulations and experiments on the TWR robot prototype demonstrate the superiority of the nonlinear freezing optimal control technique, showing larger operation ranges of the robot pitch angle and better response performances (i.e., shorter rise time, less overshoot and reduced settling time) than the linear optimal control methods. In particular, a novel mixing method to create a new nonlinear model (Model AB) from two different models on the same physical prototype with an increased controllable region of the TWR system is introduced, for the first time, for the calculations of optimal feedback gains for the system. Significantly, the utilisation of this mixed model, combined with the nonlinear freezing controller, achieves true global control of the TWR, even from an initial pitch angle of 90° (i.e., the horizontal position), when a motor with a saturated voltage of 48V and nominal torque of 298 mNm is adopted in simulation tests. This is wider than the angle achievable from the primary model (Model A) and any other single feedback control method on TWR reported in the literature. Robustness tests when introducing model uncertainties by adding mass and height on the TWR also illustrate excellent control performances from the nonlinear optimal control in both simulations and hardware implementations

Complex Fractional-Order LQIR for Inverted-Pendulum-Type Robotic Mechanisms: Design and Experimental Validation

This article presents a systematic approach to formulate and experimentally validate a novel Complex Fractional Order (CFO) Linear Quadratic Integral Regulator (LQIR) design to enhance the robustness of inverted-pendulum-type robotic mechanisms against bounded exogenous disturbances. The CFO controllers, an enhanced variant of the conventional fractional-order controllers, are realised by assigning pre-calibrated complex numbers to the order of the integral and differential operators in the control law. This arrangement significantly improves the structural flexibility of the control law, and hence, subsequently strengthens its robustness against the parametric uncertainties and nonlinear disturbances encountered by the aforementioned under-actuated system. The proposed control procedure uses the ubiquitous LQIR as the baseline controller that is augmented with CFO differential and integral operators. The fractional complex orders in LQIR are calibrated offline by minimising an objective function that aims at attenuating the position-regulation error while economising the control activity. The effectiveness of the CFO-LQIR is benchmarked against its integer and fractional-order counterparts. The ability of each controller to mitigate the disturbances in inverted-pendulum-type robotic systems is rigorously tested by conducting real-time experiments on Quanser single-link rotary pendulum system. The experimental outcomes validate the superior disturbance rejection capability of the CFO-LQIR by yielding rapid transits and strong damping against disturbances while preserving the control input economy and closed-loop stability of the system

Design of high-performance legged robots: A case study on a hopping and balancing robot

The availability and capabilities of present-day technology suggest that legged robots should be able to physically outperform their biological counterparts. This thesis revolves around the philosophy that the observed opposite is caused by over-complexity in legged robot design, which is believed to substantially suppress design for high-performance. In this dissertation a design philosophy is elaborated with a focus on simple but high performance design. This philosophy is governed by various key points, including holistic design, technology-inspired design, machine and behaviour co-design and design at the performance envelope. This design philosophy also focuses on improving progress in robot design, which is inevitably complicated by the aspire for high performance. It includes an approach of iterative design by trial-and-error, which is believed to accelerate robot design through experience. This thesis mainly focuses on the case study of Skippy, a fully autonomous monopedal balancing and hopping robot. Skippy is maximally simple in having only two actuators, which is the minimum number of actuators required to control a robot in 3D. Despite its simplicity, it is challenged with a versatile set of high-performance activities, ranging from balancing to reaching record jump heights, to surviving crashes from several meters and getting up unaided after a crash, while being built from off-the-shelf technology. This thesis has contributed to the detailed mechanical design of Skippy and its optimisations that abide the design philosophy, and has resulted in a robust and realistic design that is able to reach a record jump height of 3.8m. Skippy is also an example of iterative design through trial-and-error, which has lead to the successful design and creation of the balancing-only precursor Tippy. High-performance balancing has been successfully demonstrated on Tippy, using a recently developed balancing algorithm that combines the objective of tracking a desired position command with balancing, as required for preparing hopping motions. This thesis has furthermore contributed to several ideas and theories on Skippy's road of completion, which are also useful for designing other high-performance robots. These contributions include (1) the introduction of an actuator design criterion to maximize the physical balance recovery of a simple balancing machine, (2) a generalization of the centre of percussion for placement of components that are sensitive to shock and (3) algebraic modelling of a non-linear high-gravimetric energy density compression spring with a regressive stress-strain profile. The activities performed and the results achieved have been proven to be valuable, however they have also delayed the actual creation of Skippy itself. A possible explanation for this happening is that Skippy's requirements and objectives were too ambitious, for which many complications were encountered in the decision-making progress of the iterative design strategy, involving trade-offs between exercising trial-and-error, elaborate simulation studies and the development of above-mentioned new theories. Nevertheless, from (1) the resulting realistic design of Skippy, (2) the successful creation and demonstrations of Tippy and (3) the contributed theories for high-performance robot design, it can be concluded that the adopted design philosophy has been generally successful. Through the case study design project of the hopping and balancing robot Skippy, it is shown that proper design for high physical performance (1) can indeed lead to a robot design that is capable of physically outperforming humans and animals and (2) is already very challenging for a robot that is intended to be very simple

Development of U-model enhansed nonlinear systems

Nonlinear control system design has been widely recognised as a challenging issue where the key objective is to develop a general model prototype with conciseness, flexibility and manipulability, so that the designed control system can best match the required performance or specifications. As a generic systematic approach, U-model concept appeared in Prof. Quanmin Zhu’s Doctoral thesis, and U-model approach was firstly published in the journal paper titled with ‘U-model based pole placement for nonlinear plants’ in 2002.The U-model polynomial prototype precisely describes a wide range of smooth nonlinear polynomial models, defined as a controller output u(t-1) based time-varying polynomial models converted from the original nonlinear model. Within this equivalent U-model expression, the first study of U-model based pole placement controller design for nonlinear plants is a simple mapping exercise from ordinary linear and nonlinear difference equations to time-varying polynomials in terms of the plant input u(t-1). The U-model framework realised the concise and applicable design for nonlinear control system by using such linear polynomial control system design approaches.Since the first publication, the U-model methodology has progressed and evolved over the course of a decade. By using the U-model technique, researchers have proposed many different linear algorithms for the design of control systems for the nonlinear polynomial model including; adaptive control, internal control, sliding mode control, predictive control and neural network control. However, limited research has been concerned with the design and analysis of robust stability and performance of U-model based control systems.This project firstly proposes a suitable method to analyse the robust stability of the developed U-model based pole placement control systems against uncertainty. The parameter variation is bounded, thus the robust stability margin of the closed loop system can be determined by using LMI (Linear Matrix Inequality) based robust stability analysis procedure. U-block model is defined as an input output linear closed loop model with pole assignor converted from the U-model based control system. With the bridge of U-model approach, it connects the linear state space design approach with the nonlinear polynomial model. Therefore, LMI based linear robust controller design approaches are able to design enhanced robust control system within the U-block model structure.With such development, the first stage U-model methodology provides concise and flexible solutions for complex problems, where linear controller design methodologies are directly applied to nonlinear polynomial plant-based control system design. The next milestone work expands the U-model technique into state space control systems to establish the new framework, defined as the U-state space model, providing a generic prototype for the simplification of nonlinear state space design approaches.The U-state space model is first described as a controller output u(t-1) based time-varying state equations, which is equivalent to the original linear/nonlinear state space models after conversion. Then, a basic idea of corresponding U-state feedback control system design method is proposed based on the U-model principle. The linear state space feedback control design approach is employed to nonlinear plants described in state space realisation under U-state space structure. The desired state vectors defined as xd(t), are determined by closed loop performance (such as pole placement) or designer specifications (such as LQR). Then the desired state vectors substitute the desired state vectors into original state space equations (regarded as next time state variable xd(t) = x(t) ). Therefore, the controller output u(t-1) can be obtained from one of the roots of a root-solving iterative algorithm.A quad-rotor rotorcraft dynamic model and inverted pendulum system are introduced to verify the U-state space control system design approach for MIMO/SIMO system. The linear design approach is used to determine the closed loop state equation, then the controller output can be obtained from root solver. Numerical examples and case studies are employed in this study to demonstrate the effectiveness of the proposed methods

Dynamical systems : control and stability

Proceedings of the 13th Conference „Dynamical Systems - Theory and Applications" summarize 164 and the Springer Proceedings summarize 60 best papers of university teachers and students, researchers and engineers from whole the world. The papers were chosen by the International Scientific Committee from 315 papers submitted to the conference. The reader thus obtains an overview of the recent developments of dynamical systems and can study the most progressive tendencies in this field of science

Contributions to the modelling and control of two-wheeled mobile robots

Wheeled mobile robots moving on uneven terrain are attracting interest at an impressive pace. In the work reported here two distinct architectures of two-wheeled mobile robots are proposed.The first architecture, corresponding to the case where the two wheels linked by a frame lie in a vertical plane, constitutes the material of our earlier research and is laid out in the appendix. The system being unilaterally constrained by the environment, slipping or losing contact with the ground can occasionally occur. Therefore, nine distinct topologies are identified and accounted for in the model describing all the possible motion modes of the system. The mathematical model is formulated using the Natural Orthogonal Complement (NOC) and takes into account the terrain geometry. Additionally, the model includes necessary conditions for the switching between the distinct topologies.The second architecture pertains to an Anti-Tilting Outdoor Mobile robot, ATOM, composed of two spherical wheels and a cylindrical central body. The spherical shape of the wheels allows the robot to restore its posture after flipping over, thus giving it the anti-tilting property. Moreover, this particular shape ensures pure-rolling motion on uneven terrain without resorting to any adaptive structure; i.e., without increasing the complexity of the system. Here, also, the mathematical model is developed using the NOC, while taking into account the terrain geometry. Moreover, constraints on the terrain curvatures are derived in order to ensure pure rolling. Although the design of ATOM is simple, this brings about new challenges in terms of control. According to its structure, ATOM pertains to the class of Mobile Wheeled Pendulums (MWP). A feature common to MWPs, that is not encountered in other wheeled robots, is that their central body, which constitutes the platform of the robot, can rotate about the wheel axis. Therefore, aside the nonholonomy aspects encountered in conventional wheeled robots; the central body stabilization problem is pointed out here and rigorously treated in order to avoid unstable zero-dynamics. For that, an intrinsic dynamcal property, referred to as the natural behaviour of the system, is brought forward and employed to control the heading velocity of the robot using the inclination of the central body. Moreover, a particular selection of the generalized coordinates and the system outputs allows a global stabilization of the system without resorting to any linearization. Furthermore, a posture control (preceded by a velocity and orientation control) is proposed based on sliding mode and Lyapunov function for navigation. Finally, the robust aspect of the controller is underlined by showing the control behaviour versus an over/under estimation of system parameters