Underactuated robotics: A review <s> The classification of UR <s> A discussion is presented of the control of legged-robot body orientation in flight using the internal motion of the leg. The angular momentum constraint (nonholonomic) is used to recast the problem into a nonholonomic motion planning problem. Chow's theorem is then applied to verify that the system is controllable, and the concept of holonomy is introduced for constructing an optimal path. Finally, linearization control is used in the internal motion space to realize the planned path. An additional degree of control to dynamically balance a legged robot that runs is provided with this strategy. > <s> BIB001 </s> Underactuated robotics: A review <s> The classification of UR <s> Recent developments in the theory of geometric nonlinear control provide powerful methods for controller design for a large class of nonlinear systems. Many systems, however, do not satisfy the restrictive conditions necessary for either full state linearization [6, 5] or input-output linearization with internal stability [2]. In this paper, we present an approach to controller design based on finding a linearizable nonlinear system that well approximates the true system over a desirable region. ::: We outline an engineering procedure for constructing ::: the approximating nonlinear system given the true system. We ::: demonstrate this approach by designing a nonlinear controller for a simple mechanical system patterned after a gymnast performing on a single parallel bar. <s> BIB002 </s> Underactuated robotics: A review <s> The classification of UR <s> A method of controlling the position of a manipulator composed of active and passive joints is described. The active joints have actuators and position sensors. The passive joints have holding brakes instead of actuators. While the brakes are released, the passive joints are indirectly controlled by the motion of the active joints using the coupling characteristics of manipulator dynamics. While the brakes are engaged, the passive joints are fixed and the active joints are controlled. The position of the manipulator is controlled by combining these two control modes. The basic principle of the control method and the conditions that ensure the controllability of the passive joints are described. An algorithm for point-to-point control of the manipulator is presented. The feasibility of the method is demonstrated by simulations for a manipulator with two degrees of freedom. > <s> BIB003 </s> Underactuated robotics: A review <s> The classification of UR <s> A class of inherently nonlinear control problems has been identified, the nonlinear features arising directly from physical assumptions about constraints on the motion of a mechanical system. Models are presented for mechanical systems with nonholonomic constraints represented both by differential-algebraic equations and by reduced state equations. Control issues for this class of systems are studied and a number of fundamental results are derived. Although a single equilibrium solution cannot be asymptotically stabilized using continuous state feedback, a general procedure for constructing a piecewise analytic state feedback which achieves the desired result is suggested. > <s> BIB004 </s> Underactuated robotics: A review <s> The classification of UR <s> The attitude stabilization problem of a rigid spacecraft using control torques supplied by gas jet actuators about only two of its principal axes is considered. If the uncontrolled principal axis of the spacecraft is not an axis of symmetry, then the complete spacecraft dynamics are small time locally controllable. However, the spacecraft cannot be asymptotically stabilized to any equilibrium attitude using time-invariant continuous feedback. A discontinuous stabi- lizing feedback control strategy is constructed which stabilizes the spacecraft to any equilibrium attitude. If the uncontrolled principal axis of the spacecraft is an axis of symmetry, the complete spacecraft dynamics are not even assessible. However, the spacecraft dynamics are strongly accessible and small time locally controllable in a reduced sense. The reduced spacecraft dynamics cannot be asymptotically stabilized to any equilibrium attitude using time-invariant continuous feedback, but again a discon- tinuous stabilizing feedback control strategy is constructed. In both cases, the discontinuous feedback controllers are constructed by switching between several feedback functions which are selected to accomplish a sequence of spacecraft maneuvers. The results of the paper show that although standard nonlinear control techniques are not applicable, it is possible to construct a nonlinear discontinuous control law based on the dynamics of the particular physical system. <s> BIB005 </s> Underactuated robotics: A review <s> The classification of UR <s> A mechanism and a control method of a two-link brachiation robot are described. The robot can generate motions by a heuristic method and locomotes with the trajectory and arm-direction feedback control. It controls the switching amplitude by a method based on parametric excitation. Experimental results of the swing and locomotion control of the robot using these methods are given. They enable the robot to catch its target and to continue locomotion from any initial states. > <s> BIB006 </s> Underactuated robotics: A review <s> The classification of UR <s> Provides a summary of recent developments in control of nonholonomic systems. The published literature has grown enormously during the last six years, and it is now possible to give a tutorial presentation of many of these developments. The objective of this article is to provide a unified and accessible presentation, placing the various models, problem formulations, approaches, and results into a proper context. It is hoped that this overview will provide a good introduction to the subject for nonspecialists in the field, while perhaps providing specialists with a better perspective of the field as a whole. The paper is organized as follows: introduction to nonholonomic control systems and where they arise in applications, classification of models of nonholonomic control systems, control problem formulations, motion planning results, stabilization results, and current and future research topics. <s> BIB007 </s> Underactuated robotics: A review <s> The classification of UR <s> Underactuated mechanical systems are those possessing fewer actuators than degrees of freedom. Examples of such systems abound, including flexible joint and flexible link robots, space robots, mobile robots, and robot models that include actuator dynamics and rigid body dynamics together. Complex internal dynamics, nonholonomic behavior, and lack of feedback linearizability are often exhibited by such systems, making the class a rich one from a control standpoint. In this article the author studies a particular underactuated system known as the Acrobot: a two-degree-of-freedom planar robot with a single actuator. The author considers the so-called swing up control problem using the method of partial feedback linearization. The author gives conditions under which the response of either degree of freedom may be globally decoupled from the response of the other and linearized. This result can be used as a starting point to design swing up control algorithms. Analysis of the resulting zero dynamics as well as analysis of the energy of the system provides an understanding of the swing up algorithms. Simulation results are presented showing the swing up motion resulting from partial feedback linearization designs. > <s> BIB008 </s> Underactuated robotics: A review <s> The classification of UR <s> We consider the problem of feedback stabilization of translational oscillations by a rotational actuator (TORA) system. The main obstacle to controller design is nonlinear coupling from the rotational to the translational motion through a sinusoidal term. We present several controller designs based on the cascade and passivity paradigms. <s> BIB009 </s> Underactuated robotics: A review <s> The classification of UR <s> Control of a manipulator with a passive joint which has neither an actuator nor a holding brake is investigated. The manipulator has three degrees of freedom in a horizontal plane, with the third joint being passive. The dynamic constraint on the free link is shown to be second-order nonholonomic. Controllability of the system is proved by constructing examples of the input trajectories from arbitrary initial states to arbitrary desired states, considering the motion of the center of percussion of the link. Trajectories for positioning are composed of simple translational and rotational trajectory segments. The trajectory segments are stabilized by nonlinear feedback control. Simulations and experimental results show the effectiveness of the planned trajectory and the feedback control law. <s> BIB010 </s> Underactuated robotics: A review <s> The classification of UR <s> This paper presents a theoretical framework for the dynamics and control of underactuated mechanical systems, defined as systems with fewer inputs than degrees of freedom. Control system formulation of underactuated mechanical systems is addressed and a class of underactuated systems characterized by nonintegrable dynamics relations is identified. Controllability and stabilizability results are derived for this class of underactuated systems. Examples are included to illustrate the results; these examples are of underactuated mechanical systems that are not linearly controllable or smoothly stabilizable. <s> BIB011 </s> Underactuated robotics: A review <s> The classification of UR <s> This paper studies motion planning from one zero velocity state to another for a three-joint robot in a horizontal plane with a passive revolute third joint. Such a robot is small-time locally controllable on an open subset of its zero velocity section, allowing it to follow any path in this subset arbitrarily closely. However, some paths are “preferred” by the dynamics of the manipulator in that they can be followed at higher speeds. In this paper we describe a computationally efficient trajectory planner which finds fast collision-free trajectories among obstacles. We are able to decouple the problem of planning feasible trajectories in the robot’s six-dimensional state space into the computationally simpler problems of planning paths in the three-dimensional configuration space and time scaling the paths according to the manipulator dynamics. This decoupling is made possible by the existence of velocity directions, fixed in the passive link frame, which can be executed at arbitrary speeds. We have demonstrated the motion planner on an experimental underactuated manipulator. To our knowledge, it is the first implementation of a collision-free motion planning algorithm for a manipulator subject to a second-order nonholonomic constraint. <s> BIB012 </s> Underactuated robotics: A review <s> The classification of UR <s> Abstract The paper presents a dynamic modelling technique for a manipulator with multiple flexible links and flexible joints, based on a combined Euler–Lagrange formulation and assumed modes method. The resulting generalised model is validated through computer simulations by considering a simplified case study of a two-link flexible manipulator with joint elasticity. Controlling such a manipulator is more complex than controlling one with rigid joints because only a single actuation signal can be applied at each joint and this has to control the flexure of both the joint itself and the link attached to it. To resolve the control complexities associated with such an under-actuated flexible link/flexible joint manipulator, a singularly perturbed model has been formulated and used to design a reduced-order controller. This is shown to stabilise the link and joint vibrations effectively while maintaining good tracking performance. <s> BIB013 </s> Underactuated robotics: A review <s> The classification of UR <s> This note develops a nonlinear output-feedback controller to force a nonminimum phase, underactuated vertical take-off and landing aircraft to globally asymptotically track a reference trajectory generated by a reference model. The control development is based on a global exponential observer, some global coordinate transformations, Lyapunov's direct method and an extension of the backstepping technique. Interestingly, the proposed methodology also yields new results for the previously studied problems of stabilization and output tracking or regulation. Numerical simulations illustrate the effectiveness of the proposed controller. <s> BIB014 </s> Underactuated robotics: A review <s> The classification of UR <s> This paper presents a control strategy for a two-link manipulator with revolute joints (2R) with an unactuated second joint, wherein the motion of the system is confined to a horizontal plane. The model takes into account the frictional forces present in the system. The control objective is to move the end-effector from a given position to a target point. The methodology involves two stages. In the first stage a finite-time controller is used to move the passive link to its desired position. In the second stage, the first link is moved to its desired position keeping the second link at rest, using friction as a “brake” and subject to the constraint that the cross-coupling torque acting on the second joint does not exceed the static friction. <s> BIB015 </s> Underactuated robotics: A review <s> The classification of UR <s> In this paper, a nonlinear prediction-based control approach is proposed for stabilization of planar vertical take-off and landing (PVTOL) aircraft. This system has fewer control inputs than degrees of freedom (i.e. under-actuated) and has unstable zero dynamics (i.e. non-minimum phase). The proposed control approach is based on partial feedback linearization, which allows the emergence of a completely linearized sub-state and internal dynamics. Then, optimal trajectories are proposed for the linearized variables, where the optimization objective is to enhance the behaviour and the stability of the internal dynamics. Stability analysis of the closed-loop system is performed using a graphical tool based on Poincare's section. The performance of the proposed scheme is illustrated through simulations. Copyright © 2007 John Wiley & Sons, Ltd. <s> BIB016 </s> Underactuated robotics: A review <s> The classification of UR <s> We clarified that the common necessary condition for generating a dynamic gait results from the requirement to restore mechanical energy through studies on passive dynamic walking mechanisms. This paper proposes a novel method of generating a dynamic gait that can be found in the mechanism of a swing inspired by the principle of parametric excitation using telescopic leg actuation. We first introduce a simple underactuated biped model with telescopic legs and semicircular feet and propose a law to control the telescopic leg motion. We found that a high-speed dynamic bipedal gait can easily be generated by only pumping the swing leg mass. We then conducted parametric studies by adjusting the control and physical parameters and determined how well the basic gait performed by introducing some performance indexes. Improvements in energy efficiency by using an elastic-element effect were also numerically investigated. Further, we theoretically proved that semicircular feet have a mechanism that decreases the energy dissipated by heel-strike collisions. We provide insights throughout this paper into how zero-moment-point-free robots can generate a novel biped gait. <s> BIB017 </s> Underactuated robotics: A review <s> The classification of UR <s> INTRODUCTION: Brief History. Multifingered Hands and Dextrous Manipulation. Outline of the Book. Bibliography. RIGID BODY MOTION: Rigid Body Transformations. Rotational Motion in R3. Rigid Motion in R3. Velocity of a Rigid Body. Wrenches and Reciprocal Screws. MANIPULATOR KINEMATICS: Introduction. Forward Kinematics. Inverse Kinematics. The Manipulator Jacobian. Redundant and Parallel Manipulators. ROBOT DYNAMICS AND CONTROL: Introduction. Lagrange's Equations. Dynamics of Open-Chain Manipulators. Lyapunov Stability Theory. Position Control and Trajectory Tracking. Control of Constrained Manipulators. MULTIFINGERED HAND KINEMATICS: Introduction to Grasping. Grasp Statics. Force-Closure. Grasp Planning. Grasp Constraints. Rolling Contact Kinematics. HAND DYNAMICS AND CONTROL: Lagrange's Equations with Constraints. Robot Hand Dynamics. Redundant and Nonmanipulable Robot Systems. Kinematics and Statics of Tendon Actuation. Control of Robot Hands. NONHOLONOMIC BEHAVIOR IN ROBOTIC SYSTEMS: Introduction. Controllability and Frobenius' Theorem. Examples of Nonholonomic Systems. Structure of Nonholonomic Systems. NONHOLONOMIC MOTION PLANNING: Introduction. Steering Model Control Systems Using Sinusoids. General Methods for Steering. Dynamic Finger Repositioning. FUTURE PROSPECTS: Robots in Hazardous Environments. Medical Applications for Multifingered Hands. Robots on a Small Scale: Microrobotics. APPENDICES: Lie Groups and Robot Kinematics. A Mathematica Package for Screw Calculus. Bibliography. Index Each chapter also includes a Summary, Bibliography, and Exercises <s> BIB018
Since the beginning of the 21st century, great attentions have been paid to mechanical systems with nonholonomic constraints. Non-holonomic constraints could not be expressed as time derivatives of functions in generalized coordinates. A complete constraint could only be represented by a configuration variable. The literature on the formulation of equations of motion and the dynamics of non-holonomic systems is enormous. The book written by Neȋmark and Fufaev 9 is a good reference. There are many classic examples of non-holonomic constraints on speed, such as diver movements, pure rolling of wheels, and so on.  BIB018 Non-integrable speed constraints are also known as first-order non-holonomic constraints. Non-holonomic systems that satisfy incompatible kinematic or speed relationships (such as wheeled mobile robots (WMRs)) have been studied deeply. The non-holonomic control system comes from the formulation of a non-holonomic system containing control inputs. BIB007 In recent years, non-holonomic systems with non-integrable acceleration constraints have special benefits. The mechanical system of UR has second-order nonholonomic constraints. The following are some examples of the non-holonomic systems listed. From the perspective of non-holonomic constraints, non-holonomic systems usually appear in systems of the following categories 10,12 : 1. No-slip constraint includes sledges or knife edge systems sliding on a plane and simple wheels rolling without slipping on a plane, BIB004 which are shown in Figure 2 . 2. Conservation of angular momentum includes a freefloating body,  legged robots in flight phase, BIB001 rigid underactuated spacecraft with only two control torques, BIB005 and aircrafts. BIB016 BIB014 Yan and Yue 20 proved that the discontinuous feedback control algorithm using only two control torques as input could stabilize the angular velocity and attitude angle of the spacecraft to the equilibrium position. A guidance vector field based on path-following control method for an underactuated airship was presented.  3. Robots (mechanisms) with non-actuated joints or flexible joints include passive joint robot,  BIB003 BIB012 BIB015 two-link gymnastic robot called Acrobot, BIB002 BIB008  and special devices or mechanisms such as a brachiating robot, BIB006 walking robots, BIB017 flexible manipulators (with elastic joints),  flexible-link robots, BIB013 cartpole, translational oscillations by a rotational actuator (TORA), BIB009 and fully actuated mechanisms with actuator failure. BIB010 As a mechanical system, the size of the configuration space of underactuated system exceeds the size of the control input space. In other words, the control input is less than the DOF. Underactuated systems are widely used in robots, aerospace systems, and ocean systems. There are three reasons for the characteristics of underactuated systems: 1. Systems dynamics. To accomplish a specific task in a particular environment, the dynamics of the system itself is the lack of drivers, such as spacecraft, helicopters, underwater ships, and so on. Or when designing some systems, in order to improve the flexibility of the system and reduce the cost, it is necessary to artificially reduce the drive, such as two propeller satellites and flexible-link robots. The flexible one-link robot system is shown in Figure 3 . Flexible-link robot is an important class of underactuated mechanical systems for simulating space applications. 2. Deliberately designed to create complex nonlinear systems for knowledge of controls and robots. The IP system is known as an underactuated reference mechanical system and generally classified into many categories, such as a first-order IP system, a second-order IP system, a rotary pendulum system, and other high-order IP systems. The first-order IP system is also called vehicle pendulum system, as shown in Figure 4 (a). And it is the most common and important IP system. It has important research significance in aerospace and civil engineering, such as upright control of rocket system and seismic design of high-rise buildings. According to the different driving joints, the second-order IP system could be divided into two types: Acrobot and Pendubot. Among them, Acrobot is the second-order IP system of the driver in the "elbow," which is shown in Figure 4 (b). Pendubot is the second-order IP system of the driver in the "shoulder," which is shown in Figure 4 (c). The rotating IP system was proposed by the Japanese Fukuta, so it is also known as the Fukuta IP system that is shown in Figure 5 . Its flat drive arm is motor-driven and the cantilever that needs to be inverted is underactuated. 3. Actuator fault. As stated before, it could be known that many non-holonomic systems are naturally suitable for underactuated systems. Underactuated systems belong to non-holonomic systems. The link between non-holonomic systems and underactuated systems has not been fully studied. BIB007 Underactuated systems are usually derived from non-holonomic systems with integrable constraints: first-order and/ or second-order non-holonomic constraints.  BIB011 UR is very practical in certain environments. Underactuated devices are expected to be more efficient, plain, and dependable than full-actuated devices. But the control of underactuated devices is more complex than full-actuated devices in theory. Non-holonomic systems and/or underactuated systems have been the common concern of many researchers in different disciplines. A lot of studies have been done on the control of these systems. Murray BIB018 outlined a non-holonomic control system. Experts in control and robotics have good research papers on dynamics and control of non-holonomic and/ or underactuated mechanical systems. For instance, experts summarized some of the major advances in control of non-holonomic systems, which mainly studied non-holonomic systems with first-order non-holonomic constraints. BIB007 Wen 10 mainly studied the kinematic control problems. These studies and introductions show that the non-holonomic robot problem is the fusion of robots and mechanics, mathematics, nonlinear control theory, and other disciplines, which makes scholars of many different disciplines pay more attention to the research of non-holonomic robots. The research of non-holonomic robots has gradually become a research hotspot in the field of robotics. The underactuated mechanism is a non-holonomic system, so the classification of UR could draw on the theory of non-holonomic systems. "Complete" and "non-integrity" are mechanical classification concepts of constraints and systems, while machinery is a mechanism and machine based on mechanical principles. The foundation of machinery is mechanics. In mechanics, the restriction on the shape and motion of the system is called constraint. The system with only complete constraints is a complete system, and the system with non-holonomic constraints is called a nonholonomic system. A robot with a non-holonomic constraint is called a non-holonomic robot, that is, a UR. From the point of view of analytical mechanics, non-holonomic robots are underactuated mechanical systems whose number of DOF is less than the dimension of their configuration spaces. How to exploit the "non-integrity" of constraints to develop a controllable UR is also a very meaningful work. The characteristics of non-holonomic robots are mainly reflected in the non-holonomic constraint equations. Different non-holonomic constraints directly affect the motion planning and control methods of UR. According to the form of non-holonomic constrained equations, UR is classified into three types: first-order UR, second-order UR, and high-order (n > 2) UR, which is shown in Figure 6 . Nonholonomic constraints of WMRs, space robots, and other robots generally appear as first-order differential equations, which are first-order UR. The joint of the robot with a UR arm (such as an IP) is subjected to a non-holonomic constraint, which is generally characterized by a second-order differential equation, that is, a second-order UR. The motion characteristics of the second-order UR system are different from those of the first-order UR system. Therefore, the research on the characteristics of non-integrity, controllability, and state feedback stabilization could not be used in the analysis design method of the first-order UR system. The motion of the first-order UR system could be reduced to a symmetric affine system. The motion of the second-order and high-order (n > 2) UR systems corresponds to the asymmetric affine system. In fact, the symmetric affine system is a special case when the deviation term of the asymmetric affine system is zero. They could all be regarded as nonlinear control systems.