In this paper we introduce a new method to design control laws for non-linear underactuated systems. Our method produces an infinite dimensional family of control laws, whereas most control techniques only produce a finite dimensional family. These control laws each come with a natural Lyapunov function. The inverted pendulum cart is used as an example. In addition, we construct an abstract system which is open loop unstable and cannot be stabilized using any linear control law, and demonstrate that our method produces a stabilizing control law.
In this paper we will discuss problems and techniques related to underactuated systems. We give a mathematical formulation of several problems arising from applications, review some standard and new techniques, and pose some interesting and challenging open questions.
This note describes two problems related to the digital implementation of control laws in the infinite dimensional family of matching control laws, namely state estimation and sampled data induced error. The entire family of control laws is written for an inverted pendulum cart. Numerical simulations which include sampled data and a state estimator are presented for one of the control laws in this family.
In this paper, we present a global state feedback tracking controller for underactuated surface marine vessels. This controller is based on saturated control inputs and, under an assumption on the reference trajectory, the closed-loop system is globally asymptotically stable (GAS). It has been designed using a 3 Degree of Freedom benchmark vessel model used in marine engineering. The main feature of our controller is the boundedness of the control inputs, which is an essential consideration in real life. In absence of velocity measurements, the controller works and remains stable with observer…
In this paper, the control problem for underactuated systems in the presence of external disturbances and model uncertainties is considered. An adaptive fuzzy sliding mode controller (AFSMC) is proposed to solve the problem, satisfying the robustness properties against uncertainties and disturbances. The designed controller can be applied to a wide class of underactuated systems and have fewer restrictions, compared with many pervious works. To evaluate the performance of proposed algorithm, it is applied to an underactuated inverted pendulum and a Translational Oscillator/Rotational Actuator …
We consider tracking control for multibody systems which are modelled using holonomic and nonholonomic constraints. Furthermore, the systems may be underactuated and contain kinematic loops and are thus described by a set of differential-algebraic equations that cannot be reformulated as ordinary differential equations in general. We propose a control strategy which combines a feedforward controller based on the servo-constraints approach with a feedback controller based on a recent funnel control design. As an important tool for both approaches we present a new procedure to derive the interna…
This paper solves the rendezvous problem for a network of underactuated rigid bodies such as quadrotor helicopters. A control strategy is presented that makes the centres of mass of the vehicles converge to an arbitrarily small neighborhood of one another. The convergence is global, and each vehicle can compute its own control input using only an on-board camera and a three-axis rate gyroscope. No global positioning system is required, nor any information about the vehicles' attitudes.
This paper derives nonlinear feedback control synthesis for general control affine systems using second-order actions---the second-order needle variations of optimal control---as the basis for choosing each control response to the current state. A second result of the paper is that the method provably exploits the nonlinear controllability of a system by virtue of an explicit dependence of the second-order needle variation on the Lie bracket between vector fields. As a result, each control decision necessarily decreases the objective when the system is nonlinearly controllable using first-orde…
This paper derives nonlinear feedback control synthesis for general control affine systems using second-order actions---the needle variations of optimal control---as the basis for choosing each control response to the current state. A second result of the paper is that the method provably exploits the nonlinear controllability of a system by virtue of an explicit dependence of the second-order needle variation on the Lie bracket between vector fields. As a result, each control decision necessarily decreases the objective when the system is nonlinearly controllable using first-order Lie bracket…
This work concerns the internal stabilization of underactuated linear systems of $m$ heat equations in cascade, where the control is placed internally in the first equation only and the diffusion coefficients are distinct. Combining the modal decomposition method with a recently introduced state-transformation approach for observation problems, a proportional-type stabilizing control is given explicitly. It is based on a transformation for the ODE system corresponding to the comparatively unstable modes into a target one, where calculation of the stabilization law is independent of the arbitra…
Motion planning and control are key problems in a collection of robotic applications including the design of autonomous agile vehicles and of minimalist manipulators. These problems can be accurately formalized within the language of affine connections and of geometric control theory. In this paper we overview recent results on kinematic controllability and on oscillatory controls. Furthermore, we discuss theoretical and practical open problems as well as we suggest control theoretical approaches to them.
In this paper we use an affine connection formulation to study an optimal control problem for a class of nonholonomic, under-actuated mechanical systems. In particular, we aim at minimizing the norm-squared of the control input to move the system from an initial to a terminal state. We consider systems evolving on general manifolds. The class of nonholonomic systems we study in this paper includes, in particular, wheeled-type vehicles, which are important for many robotic locomotion systems. The two special aspects of this optimal control problem are the nonholonomic constraints and under-actu…
We deal with regular Lagrangian constrained systems which are invariant under the action of a symmetry group. Fixing a connection on the higher-order principal bundle where the Lagrangian and the (independent) constraints are defined, the higher-order Lagrange-Poincaré equations of classical mechanical systems with higher-order constraints are obtained from classical Lagrangian reduction. Higher-order Lagrange-Poincaré operator is introduced to characterize higher-order Lagrange-Poincaré equations. Interesting applications are derived as, for instance, the optimal control of an underactuated E…
Optimal control problems for underactuated mechanical systems can be seen as a higher-order variational problem subject to higher-order constraints (that is, when the Lagrangian function and the constraints depend on higher-order derivatives such as the acceleration, jerk or jounces). In this paper we discuss the variational formalism for the class of underactuated mechanical control systems when the configuration space is a trivial principal bundle and the construction of variational integrators for such mechanical control systems. An interesting family of geometric integrators can be defined…
This paper demonstrates a refined approach to solving dynamic optimization problems for underactuated marine surface vessels. To this end the differential flatness of a mathematical model assuming full actuation is exploited to derive an efficient representation of a finite dimensional nonlinear programming problem, which in turn is constrained to apply to the underactuated case. It is illustrated how the properties of the flat output can be employed for the generation of an initial guess to be used in the optimization algorithm in the presence of static and dynamic obstacles. As an example en…
Control of underactuated dynamical systems has been studied for decades in robotics, and is now emerging in other fields such as neuroscience. Most of the advances have been in model based control theory, which has limitations when the system under study is very complex and it is not possible to construct a model. This calls for data driven control methods like machine learning, which has spread to many fields in the recent years including control theory. However, the success of such algorithms has been dependent on availability of large datasets. Moreover, due to their black box nature, it is…
Stability Analysis of Geometric Control for a Canonical Class of Underactuated Aerial Vehicles with Spurious Forces
Robotics and Mechatronics group, Faculty of Electrical Engineering, Mathematics, and Computer Science (EEMCS), University of Twente（1 机器人与机电组，电气工程、数学与计算机科学学院（EEMCS），代尔夫特理工大学）；Department of Computer, Control and Management Engineering, Sapienza University of Rome（2 计算机、控制与管理工程系，萨皮恩扎罗马大学）
The goal of the present article is to study controllability properties of mixed systems of linear parabolic-transport equations, with possibly non-diagonalizable diffusion matrix, on the one-dimensional torus. The equations are coupled by zero or first order coupling terms, with constant coupling matrices, without any structure assumptions on them. The distributed control acts through a constant matrix operator on the system, so that there might be notably less controls than equations, encompassing the case of indirect and simultaneous controllability. More precisely, we prove that in small ti…
A novel model-based parameters estimation combining local optimization and global optimization of nonlinear ship models with physical experiment dataset
This note describes a method for generating an infinite-dimensional family of nonlinear control laws for underactuated systems. For a ball and beam system, the entire family is found explicitly.
The task of inducing, via continuous static state-feedback control, an asymptotically stable heteroclinic orbit in a nonlinear control system is considered in this paper. The main motivation comes from the problem of ensuring convergence to a so-called point-to-point maneuver in an underactuated mechanical system. Namely, to a smooth curve in its state--control space, which is consistent with the system dynamics and connects two (linearly) stabilizable equilibrium points. The proposed method uses a particular parameterization, together with a state projection onto the maneuver as to combine tw…
