Virtual Nonholonomic Constraints: A Geometric Approach

Virtual constraints are functional relations (i.e., constraints) on the state variables of a robot's model that are achieved through the action of actuators and feedback control instead of physical contact forces. They are called virtual because they can be re-programmed on the fly without modifying any physical connections among the links of the robot or its environment. Previous analytical and experimental work has established that vector relative degree two virtual holonomic (i.e., only configuration dependent) constraints are a powerful means to synchronize the links of a bipedal robot so as to achieve walking and running motions over a variety of terrain profiles. This paper introduces a class of virtual nonholonomic constraints that depend on velocity through (generalized) angular momentum while maintaining the property of being relative degree two. This additional freedom is shown to yield control solutions that handle a wider range of gait perturbations arising from terrain variations and exogenous forces. Moreover, including angular momentum in the virtual constraints allows foot placement control to be rigorously designed on the basis of the full dynamic model of the biped, instead of on the basis of an inverted pendulum approximation of its center of mass, as is commonly done in the bipedal robotics literature. This new class of control laws is shown in simulation to be robust to a variety of common gait disturbances.

This paper applies a recently developed geometric PID controller to stabilize a three-link planar bipedal hybrid dynamic walking model. The three links represent the robot torso and two kneeless legs, with an independent control torque available at each hip joint. The geometric PID controller is derived for fully actuated mechanical systems, however in the swing phase the three-link biped robot has three degrees of freedom and only two controls. Following the bipedal walking literature, underactuation is addressed by choosing two "virtual constraints" to enforce, and verifying the stability of the resulting two-dimensional zero dynamics. The resulting controlled dynamics do not have the structure of a mechanical system, however this structure is restored using "feedback regularization," following which geometric PID control is used to provide robust asymptotic regulation of the virtual constraints. The proposed method can tolerate significantly greater variations in inclination, showing the value of the geometric methods, and the benefit of integral action.

Model-Following Control of Nonlinear Systems based on Virtual Constraints

We employ two adaptive strategies for motion control of uncertain nonholonomic mechanical systems with chained form kinematics. Chained form systems possess sufficient structure to permit very effective control strategies to be developed for them. For example, these systems are differentially flat, which can be shown to be quite useful for motion control, and they are also easier to stabilize than some other classes of nonholonomic systems; both of these properties are exploited in the paper. As a consequence of the adaptive nature of the control laws and the properties of chained form systems, the employed schemes are simple and modular, are easily implemented with a wide range of mechanical systems, and provide a reasonable starting point for addressing such practical problems as collision avoidance and sensor-based planning and control. The first controller combines an efficient trajectory generation algorithm with an adaptive tracking strategy to obtain the desired motion control, while the second scheme achieves this control objective through the use of a stabilizing control law developed utilizing adaptation and homogeneous system theory. Each of the control strategies is computationally efficient, ensures accurate motion control, and is implementable in the presence of considerable uncertainty concerning the mechanical system model. The efficacy of the approach is illustrated through computer simulations with a single chain nonholonomic system.

In this paper, which is largely review, I will discuss dissipative behavior in mechanical systems which preserve energy. The paper will encompass both the classical and quantum domains. In the classical context I will consider almost Poisson systems; systems which which have a Poisson braket which fails the Jacobi identity. This class of systems includes nonholonomic mechanical systems: systems with nonintegrable constraints such as rolling constraints. Such systems may either preserve or fail to preserve a natural measure. I will discuss also pure Hamiltonian systems such as the Toda lattice which can exhibit dissipative behavior in certain contexts as well as infinite-dimensional systems exhibiting radiative damping. In the quantum context I will discuss systems of quantum oscillators coupled to a heat bath, which also exhibit natural dissipative behavior.

In his works, V.V. Kozlov proposed a mathematical model for the dynamics of a mechanical system with nonintegrable constraints, which was called vakonomic. In contrast to the then conventional nonholonomic model, trajectories in the vakonomic model satisfy necessary conditions for a minimum in a variational problem with equality constraints. We consider the so-called irregular case of this variational problem, which was not covered by Kozlov, when the trajectory is a singular point of the constraints and the necessary minimum conditions based on the Lagrange principle make no sense. This situation is studied using the theory of abnormal problems developed by the first author. As a result, the classical necessary minimum conditions are strengthened and developed to this class of problems.

Nonholonomic systems provide an important class of mechanical control systems. One reason for this importance is that nonintegrability is essential to both the mechanics and the control: Nonintegrable constraint distributions are the essence of nonholonomic systems, while a nonintegrable distribution of control vector fields is the key to controllability of nonlinear systems. We will learn how these two different types of nonintegrability work together when we study control of nonholonomic mechanical systems.

This paper discusses the averaging, control authority, and vibrational control of mechanical control-affine systems with high-frequency, high-amplitude inputs. The inputs have different frequencies of the same order. This work is an extension of the existing averaging method for high-frequency mechanical systems with single-frequency inputs. Vibrational control authority of mechanical control-affine systems is introduced, and the effects of inputs' waveform and frequency on vibrational control authority are investigated. The results show that, in general, using multifrequency inputs may result in lower control authority of mechanical systems compared to single-frequency inputs, especially when using harmonic inputs. The results on vibrational control authority of the systems with multifrequency inputs are demonstrated using vibrational control of a horizontal pendulum with two inputs. This paper also discusses the averaging of multiple-time-scale control systems.

Underactuated mechanical systems are well-known for the challenges they pose to motion planning and control. In particular, the motion planner for these systems must take into account their dynamics, while the controller has to stabilize all degrees of freedom with a fewer number of actuators. One common way to overcome these hurdles is by employing the concept of virtual (holonomic) constraints for planning and transversal stabilization for control. This paper investigates the use of piecewise virtual constraints to improve motion planning for the systems with one degree of underactuation and demonstrates that periodic motions planned with piecewise constraints can be stabilized using existing tools for orbital stabilization. We have conducted extensive experimental evaluation of the algorithm on a practical pendubot robot, and the results suggest that the proposed method yields considerably smaller cost function values than other approaches, while exhibiting faster convergence.

Traditionally, robot control has been done typically by “highly precise control algorithms”: their joint angle trajectories are accurately determined at any time with vast amount of computation. This, however, causes serious problems, particularly in terms of adaptability and energy efficiency. On the other hand, an extreme approach has been gaining a lot of attention recently. A good instantiation is the passive dynamic walker, driven only by exploiting the intrinsic dynamics of its mechanical system. However, mechanical system is not everything as well as control system is not everything: “well-balanced” coupling between control and mechanical systems should be considered. In addition, the “meeting point” between the two systems should be varied according to the environment encountered. In light of these facts, this study particularly focuses on the stiffness of robots’ joints, since this strongly contributes to tuning the dominance relationship between control and mechanical systems. More specifically, the aim of this study is to develop a “Real-time tunable spring” that can smoothly change its elasticity without changing its natural length, allowing robot’s joints to change their position and stiffness independently.

Traditionally, robot control has been done typically by "highly precise control algorithms": the position of each movable body part is accurately determined at any time with vast amount of computation. This, however, causes serious problems, particularly in terms of adaptability and energy efficiency. On the other hand, an extreme approach has been gaining a lot of attention recently. A good instantiation is the passive dynamic walker, driven only by exploiting the intrinsic dynamics of its mechanical system. However, the mechanical system is not everything, just as the control system is not everything; "well-balanced" coupling between control and mechanical systems should be considered. In addition, the "meeting point" between the two systems should be flexibly varied according to the environment encountered. In light of these facts, this study intensively focuses on the stiffness of robots' joints, since this effectively influences the dominance relationship between control and mechanical systems. More specifically, the aim of this study is to develop a "real-time tunable spring" that can smoothly change its elasticity without changing its natural length, allowing robot's joints to change their position and stiffness independently

The control and mechanical systems of an embodied agent should be tightly coupled so as to emerge useful functionalities such as adaptivity. This indicates that the mechanical system as well as the control system should be responsible for a certain amount of computation for generating the behavior. However, there still leaves much to be understood about how such “computational offloading” from the control system to the mechanical system can be achieved. In order to intensively investigate this, here we particularly focus on the “softness” of the body, and show how the computational offloading derived from this property is exploited to simplify the control system and to increase the degree of adaptivity. To this end, we employ a two-dimensional amoeboid robot as a practical example, consisting of incompressive fluid (i.e. protoplasm) covered with an outer skin composed of a network of real-time tunable springs. Preliminary simulation results show that the exploitation of the “long-distant interaction” stemming from “the law of conservation of protoplasmic mass” allows us to simplify the control mechanism; and that adaptive amoeboid locomotion can be realized without the need of a central controller. The results obtained are expected to shed light on how control and mechanical systems should be coupled, and what the “brain-body-interaction” carefully designed brings to the resulting behavior.

In previous studies, information dynamics methods such as Von Neumann entropy and Rényi entropy played an important role in many fields, covering both macroscopic and microscopic studies. They have a solid theoretical foundation, but there are few reports in the field of mechanical nonlinear systems. So, can we apply Von Neumann entropy and Rényi entropy to study and analyze the dynamic behavior of macroscopic nonlinear systems? In view of the current lack of suitable methods to characterize the dynamics behavior of mechanical systems from the perspective of nonlinear system correlation, we propose a new method to describe the nonlinear features and coupling relationship of mechanical systems. This manuscript verifies the above hypothesis by using a typical chaotic system and a real macroscopic physical nonlinear system through theory and practical methods. The nonlinear vibration correlation in multi-body mechanical systems is very complex. We propose a full-vector multi-scale Rényi entropy for exploring the chaos and correlation between the dynamic behaviors of mechanical nonlinear systems. The research results prove the effectiveness of the proposed method in modal identification, system dynamics evolution and fault diagnosis of nonlinear systems. It is of great significance to extend these studies to the field of mechanical nonlinear system dynamics.

Many interesting control are mechanical control systems. In spite of this, there has not been much effort to develop methods which use the special structure of mechanical to obtain analysis tools which are suitable for these systems. In this dissertation we take the first steps towards a methodical treatment of mechanical control systems. First we develop a framework for analysis of certain classes of mechanical control systems. In the Lagrangian formulation we study mechanical control systems whose Lagrangian is kinetic energy minus potential energy. We propose a new and useful definition of controllability for these and obtain a computable set of conditions for this new version of controllability. We also obtain decompositions of simple mechanical in the case when they are not controllable. In the Hamiltonian formulation we study whose control vector fields are Hamiltonian. We obtain decompositions which describe the controllable and uncontrollable dynamics. In each case, the dynamics are shown to be Hamiltonian in a suitably general sense. Next we develop intrinsic descriptions of Lagrangian and Hamiltonian mechanics in the presence of external inputs. This development is a first step towards a control theory for general Lagrangian and Hamiltonian control systems. Systems with constraints are also studied. We first give a thorough overview of variational methods including a comparison of the nonholonomic and vakonomic methods. We also give a generalised definition for a constraint and, with this more general definition, we are able to give some preliminary controllability results for constrained systems.

We have completed a major Wide Field Upgrade (WFU) of the Hobby-Eberly Telescope (HET) and reentered scheduled queue science operations in mid-2016. This paper assesses the performance of the various mechanical systems which were upgraded, or added to HET, including the Telescope Structure, the star Tracker (aka, WFU Tracker), Prime Focus Instrument Package (PFIP), and VIRUS Support Structure (VSS). The upgrades were required to increase the field of view of HET, from 4 arc-minutes, to 22 arc-minutes, increasing the observed area of sky by 30 times the original FoV. In the process, the total weight of the system increased from 100 tons, to 153 tons, requiring a complete overhaul of most of the mechanical, servo, and control systems. The new 13-axis Tracker and control system was tested extensively prior to shipping and installation, and followed up with laser tracker measurements, which brought the tracking system to within 1 arc-minute RMS pointing, and followed up with on-sky derived mount-models, which has improved the pointing and guiding to within 12 arc-seconds RMS, and 0.1 arc-seconds, respectively. A completely new structural support system was implemented to house and connect a total of 156 VIRUS spectrographs, plus 4 new Low-Resolution Spectrographs (LRS2). The VIRUS units are arrayed in two large enclosures mounted to either side of the telescope. Each enclosure is approximately 1.3m deep x 6.7m wide x 6.2m tall and weighs 38 tons fully loaded. This structure is attached to HET in a way that allows it to be positioned by, but stand independent of, the HET during observation. As commissioning has transitioned to phased-science/engineering operations, and subsequently, to science operation, the tracking software and mechanical performance of the Tracker and VSS have been improved to meet specification. Performance data and lessons learned are provided.