Underactuated robotics: A review <s> b. Optimization based on fuzzy control <s> Computed torque algorithms are used to compensate for the changing dynamics of robot manipulators in order to ensure that a constant level of damping is maintained for all configurations. There are three significant problems with existing computed torque algorithms. First, they are nonpassive and, as shown, can lead to unstable behavior; second, they make poor use of actuator capability; and third, they cannot be used to maintain a constant end-effector stiffness for force control tasks. A new class of passive computed torque controller which have guaranteed stability properties, utilize actuators effectively, and maintain constant end-effector stiffness is introduced. Network models are used to motivate and illustrate the approach. > <s> BIB001 </s> Underactuated robotics: A review <s> b. Optimization based on fuzzy control <s> Previously obtained results on L2-gain analysis of smooth nonlinear systems are unified and extended using an approach based on Hamilton-Jacobi equations and inequalities, and their relation to invariant manifolds of an associated Hamiltonian vector field. On the basis of these results a nonlinear analog is obtained of the simplest part of a state-space approach to linear H/sub infinity / control, namely the state feedback H/sub infinity / optimal control problem. Furthermore, the relation with H/sub infinity / control of the linearized system is dealt with. > <s> BIB002 </s> Underactuated robotics: A review <s> b. Optimization based on fuzzy control <s> In this paper we discuss the partial feedback linearization control of underactuated mechanical systems. We consider an n degree of freedom system having m actuated, or active, degrees of freedom and l=n-m unactuated, or passive, degrees of freedom. It is known that the portion of the dynamics corresponding to the active degrees of freedom may be linearized by nonlinear feedback. In this paper we show, alternatively, that the portion of the dynamics corresponding to the passive degrees of freedom may be linearized by nonlinear feedback under a condition that we call strong inertial coupling. We derive and analyze the resulting zero dynamics which are crucial to an understanding of the response of the overall system. Simulation results are presented showing the performance of two link underactuated robots under partial feedback linearization control. > <s> BIB003 </s> Underactuated robotics: A review <s> b. Optimization based on fuzzy control <s> Abstract In this paper we discuss some design methods for the control of a class of underactuated mechanical systems. This class of systems includes gymnastic robots, such as the Acrobot, as well the classical cart-pole system. The systems considered here are nonminimum phase which greatly complicates the control design issues. Our design techniques combine nonlinear partial feedback linearization with Lyapunov methods based on saturation functions, switching and energy shaping. <s> BIB004 </s> Underactuated robotics: A review <s> b. Optimization based on fuzzy control <s> From the Publisher: ::: This book gives a unified treatment of classical input output stability theory and recent developments in nonlinear robust and passivity-based control. The synthesis between these areas is provided by the theory of dissipative systems. Specifically, the small-gain and passivity theorems and their implications for nonlinear stability and stabilization are discussed from this vantage-ground. The connection between L - gain and passivity via scattering is detailed." "The passivity concepts are enriched by a generalised Hamiltonian formalism, emphasising the close relations with modeling and control by interconnection. Feedback equivalence to a passive system and resulting stabilization strategies are discussed." "The potential of L[subscript 2] - gain techniques in nonlinear control is demonstrated, including a compact treatment of the nonlinear H[subscript [infinity]] optimal control problem. <s> BIB005 </s> Underactuated robotics: A review <s> b. Optimization based on fuzzy control <s> The authors present an approach for constructing optimal feedback control laws for regulation of a rotating rigid spacecraft. They employ the inverse optimal control approach which circumvents the task of solving a Hamilton-Jacobi equation and results in a controller optimal with respect to a meaningful cost functional. The inverse optimality approach requires the knowledge of a control Lyapunov function and a stabilizing control law of a particular form. For the spacecraft problem, they are both constructed using the method of integrator backstepping. The authors give a characterization of (nonlinear) stability margins achieved with the inverse optimal control law. <s> BIB006 </s> Underactuated robotics: A review <s> b. Optimization based on fuzzy control <s> Bounded feedback control laws are designed for stabilization and tracking of underactuated spacecraft. The flat outputs of the system are computed and are used to generate reference trajectories for the tracking problem. <s> BIB007 </s> Underactuated robotics: A review <s> b. Optimization based on fuzzy control <s> This paper studies the global tracking problem for an underactuated ship with only two propellers. Under sufficient conditions of persistent excitation, two constructive solutions are proposed by application of Lyapunov's direct method. Both solutions exploit the inherent cascade interconnected structure of the ship dynamics and generate explicit Lyapunov functions. A common feature of the second solution and the recent cascade approach of Lefeber (Tracking control of nonlinear mechanical systems, Ph.D. Thesis, University of Twente, 2000) is that global exponential tracking is achieved, but at the expense of transient performance. Extension to unmeasured thruster dynamics is also considered. Simulation results validate the proposed tracking methodology. <s> BIB008 </s> Underactuated robotics: A review <s> b. Optimization based on fuzzy control <s> 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. <s> BIB009 </s> Underactuated robotics: A review <s> b. Optimization based on fuzzy control <s> Discusses a modification of the Newton algorithm applied to nonholonomic motion planning with energy optimization. The energy optimization is performed either by optimizing motion in the null space of the Jacobian matrix derived from the nonholonomic system or coupling this motion with movement toward the goal. Resulting controls are smooth and easily generated by motors or thrusters. The two methods can be used, when kinematics are considered, to steer any driftless nonholonomic systems particular free-floating objects, underactuated manipulators, mobile robots (with trailers). Similarities and differences are also discussed in the Newton algorithm for holonomic and nonholonomic systems. <s> BIB010 </s> Underactuated robotics: A review <s> b. Optimization based on fuzzy control <s> To swing up art acrobot, a two-link under-actuated robotic system, a fuzzy neural network (FNN) controller with limited torque based on genetic algorithm (GA) is presented and tested. The acrobot is composed of two links rotating in a same plane around two joints, only the second of which is actuated. The upswing process is a complex problem to rotate the two links from the low to the top balance position by single torque on the actuated joint. In a real application system, motor output is often limited. Therefore, an INN controller is proposed to generate limited torque on the actuated joint according to different states of the two links. The antecedents and consequences of the FNN controller are determined with an improved GA. Simulation results on the acrobot system verify the feasibility and generalization of the proposed GA-based FNN control scheme. <s> BIB011 </s> Underactuated robotics: A review <s> b. Optimization based on fuzzy control <s> A Lyapunov direct method is proposed for a class of underactuated, mechanical systems. The direct method is derived in general for systems having n degrees of freedom of which only m < n are actuated. The applications consist of a class of systems where the elements of the mass/inertia matrix and the gravitational forces/torques are either constants or functions of a single generalized position variable and where n is two and m is one. The time derivative of the candidate Lyapunov function produces a relation that is solved via a matching method. Some of the matching equations consist of linear differential and partial differential equations. It is shown for this class of systems, that the solutions of these linear differential and partial differential equations necessary for assuring asymptotic stability can be evaluated numerically as part of the feedback process. Examples are presented involving an inverted pendulum cart and an inertia wheel pendulum. <s> BIB012 </s> Underactuated robotics: A review <s> b. Optimization based on fuzzy control <s> This brief addresses the problem of controlling the planar position and orientation of an autonomous underactuated surface vessel. Under realistic assumptions, we show, first, that there exists a natural change of coordinates that transforms the whole dynamical system into a cascade nonlinear system, and, second, the control problem of the resulting system can be reduced to the stabilization of a third-order chained form. A time-invariant discontinuous feedback law is derived to guarantee global uniform asymptotic stabilization of the system to the desired configuration. The construction of such a controller is based on the backstepping design approach. A simulation example is included to demonstrate the effectiveness of the suggested approach <s> BIB013 </s> Underactuated robotics: A review <s> b. Optimization based on fuzzy control <s> The purpose of this paper is to present a robust tracking control algorithm for underactuated biped robots capable of self-balancing in the presence of external disturbances. The biped is modeled as a five-link planar robot with four actuators located at hip and knee joints. A sliding mode control law has been developed for the biped to follow a human-like gait trajectory while keeping the torso nearly upright. The control forces are calculated by defining four first-order sliding surfaces as a linear combination of the torso and the four joint tracking errors. The control approach is shown to guarantee that all trajectories will reach and stay on these surfaces during each step, while the walking cycle stability is maintained through a Lyapunov function. The criteria for asymptotic stability of the surfaces are presented and a numerical search method is implemented for the selection of the corresponding surface parameters. The paper further investigates the robustness of the controller in response to disturbances. Numerical simulations demonstrate the tracking stability of the biped's multistep walk and its human-like response to an external disturbance. <s> BIB014 </s> Underactuated robotics: A review <s> b. Optimization based on fuzzy control <s> In this paper, the stability analysis of the GA-based adaptive fuzzy sliding model controller for a nonlinear system is presented. First, an uncertain and nonlinear plant for the tracking of a reference trajectory is well approximated and described via the reference model and the fuzzy model involving fuzzy logic control rules. Next, the difficulty in designing a fuzzy sliding mode controller (FSMC) capable of rapidly and efficiently controlling complex and nonlinear systems is how to select the most appropriate initial values for the parameter vector. The initial values of the consequent parameter vector are decided via the genetic algorithm. After this, a modified adaptive law can be adopted to find the best high-performance parameters for the fuzzy sliding model controller. The adaptive fuzzy sliding model controller is derived to simultaneously stabilize and control the system. The stability of the nonlinear system is ensured by the derivation of the stability criterion based upon Lyapunov's direct method. Finally, a numerical simulation is provided as an example to demonstrate the control methodology. <s> BIB015 </s> Underactuated robotics: A review <s> b. Optimization based on fuzzy control <s> Abstract In this paper, we implement the recently developed energy- and entropy-based hybrid control framework for stabilization of Lagrangian systems on the experimental testbed of the Rotational/Translational Proof-Mass Actuator (RTAC) system. In addition, on the same experimental platform, we implement the sliding mode control framework for stabilization of underactuated dynamical systems and compare the performances of all three controllers. The concept of an energy-based hybrid controller involves a hybrid controller that exploits the feature that the states of the dynamic controller may be reset to enhance the overall energy dissipation in the closed-loop system. We give a detailed description of the hardware layout for the testbed and present the experimental results. The real-time data indicate that the energy- and entropy-based hybrid controllers result in almost the same closed-loop system behavior with the same control effort. However, hybrid controllers use significantly less control effort and stabilize the system in a shorter period of time than the sliding mode controller. <s> BIB016 </s> Underactuated robotics: A review <s> b. Optimization based on fuzzy control <s> Fast human walking includes a phase where the stance heel rises from the ground and the stance foot rotates about the stance toe. This phase where the biped becomes underactuated is not present during the walk of current humanoid robots. The objective of this study is to determine whether the introduction of this phase for a 3-D bipedal robot is useful to reduce the energy consumed in the walking. In order to study the efficiency of this new gait, two cyclic gaits are presented. The first cyclic motion is composed of successive single-support phases with a flat stance foot on the ground, and the stance foot does not rotate. The second cyclic motion is composed of single-support phases that include a subphase of rotation of the supporting foot about the toe. The single-support phases are separated by a double-support phase. For simplicity, this double-support phase is considered as instantaneous (passive impact). For these two gaits, optimal motions are designed by minimizing the torques cost. The given performances of actuators are taken into account. It is shown that, for a fast motion, a foot-rotation subphase is useful to reduce the cost criterion. These gaits are illustrated with simulation results. <s> BIB017 </s> Underactuated robotics: A review <s> b. Optimization based on fuzzy control <s> This study presents a model predictive control (MPC) motion planning-based sliding mode control (SMC) for a wheeled pendular-like suspension (WPS) vehicle by using Olfati transformation. To improve the tracking efficiency and enhance the control performance of the vehicle system, the mobile platform is required to follow the reference trajectory fast enough, while the swing of the suspension needs to be within an acceptable domain. To achieve these multi-objectives, a two-step design strategy consisting of a motion planning stage and a velocity tracking control design stage is proposed to control such an underactuated system. Specifically, a novel MPC, which satisfies various physical constraints of the WPS vehicle, is presented by which the nonholonomic constraint is dealt with as well. The SMC is then constructed in the second step to make the vehicle track the desired velocities generated by motion planning. As far as the steering subsystem is concerned, the global terminal SMC is used to ensure the fast convergence of the steering tracking; as for the forward subsystem, a composite sliding mode manifold is successfully introduced thanks to the underactuated dynamics decoupling by Olfati transformation. The numerical simulation validates the effectiveness of proposed control approaches even in the presence of sophisticated disturbance and physical limitations. <s> BIB018
Inappropriate control parameter causes instability in underactuated systems relative to other nonlinear systems. Based on expert knowledge, the optimal control parameters could not be obtained efficiently. Therefore, some other optimization algorithms are needed to improve the control performance. Martínez et al.  designed a trajectory tracking controller taking into account the kinematics and the dynamics of the autonomous mobile robot using type-2 fuzzy logic and genetic algorithms (GAs). GAs are used for the optimization of the constants for the trajectory tracking and optimization of the parameters of membership functions for fuzzy logic control. A simple FNN controller with limited torque is developed for upswing control of the Acrobot. The antecedents and consequences of the FNN controller are optimized simultaneously with an improved GA. BIB011 The main contribution BIB015 is to propose the GA-based fuzzy sliding model controller with a modified adaptive law for robust control of an uncertain and nonlinear plant. Partial feedback linearization. A useful technique for controlling a UR system is the so-called PFL. BIB003 Spong proposed a PFL method for underactuated systems, which implements system transformation by defining a new control quantity. It is finally linearized locally. This method is widely used for the control of underactuated linkage systems. Based on the mechanical model of the bicycle underactuated by the front wheel, Huang et al.  designed the motion controller based on the PFL principle. It is worth noting that the system obtained by PFL is still nonlinear. And its coupling is stronger than the transformation. This has led to a more complex controller structure. Collocated PFL. The collocated PFL BIB004 shows the process of actuating the configuration variable linearization, which globally converts all underactuated systems to a fully actuated form.  The new control input appears in the dynamics of the two subsystems. Olfati-Saber 210 addressed this stabilization problem for important classes of nontriangular cascade nonlinear systems in non-triangular linear-quadratic normal form. This includes normal forms for class-II, class-IV, and class-VII underactuated systems. Non-collocated PFL. The non-collocated PFL  presents a linearization process for the unstarted configuration variables, and it is applied to the restricted underdrive system class. Olfati-Saber 211 introduced cascade normal forms for underactuated mechanical systems that are convenient for control design. Olfati-Saber studies noncollocated PFLs with symmetrical underactuated systems and converted non-triangular quadratic formal forms into cascading nonlinear systems with strict feedforward forms. Energy-based approach. The robust control method based on the passive theory maintains the passiveness of the closedloop system when the parameters of the robot are uncertain. Desoer and Vidyasagar 212 presented the feedback system input and output theory. Combined with the skew symmetry of the robot itself, a passive mapping of composite state variables and output torque could be defined. Then the compound variable is progressively convergent. Anderson BIB001 demonstrated that some feedback linearization-based controllers are not passive. The uncertainty will lead to system instability. The PD controller is designed by the passive torque-based computational moment method. Even the moment of inertia is uncertain, the closed-loop system could still maintain its stability. The main disadvantage is that the singular value of the inertia matrix needs to be calculated, and the calculation amount is large. There are many energy-based control methods, such as L 2 -gain method BIB002 BIB005 and direct Lyapunov method. BIB008 BIB012 Sliding mode control. SMC is a strong and robust variable structure control method developed from the 1950s. With the higher-order SMC theory, a sliding mode tracking control method is presented for an underactuated surface vessel in presence of parameter variations and external disturbances.  For an underactuated spacecraft with two control actuators, an attitude controller is proposed based on hierarchical SMC.  It is essentially a nonlinear control. The control structure changes over time. Sliding mode variable structure control is an analytical method commonly used in nonlinear systems. Its significant advantage is strong robustness to uncertain parameters and external disturbances. The sliding mode is the state in which the system is constrained to move on a sub-manifold. In general, the initial state of the system is not necessarily on the submanifold. Rather, through the action of the variable structure controller, the state trajectory of the system could be driven to and maintained on the sub-manifold over a certain time range. This process is called reachability. The state trajectory of the system moves in the sliding mode and eventually approaches the origin. This process is called sliding mode motion. SMC makes full use of the model of the object. BIB018 But its requirements for the model are very low, that is, no precise model of the object is needed. As long as the upper bounds of the parameters and disturbance uncertainty in the model are known, a progressively stable controller could be designed. However, the SMC faces the adverse effects of buffeting. The buffeting comes from the conservative estimation of uncertainty and disturbance, the sign switching of the approach law on both sides of the sliding surface, the limited switching frequency, and the time lag of the control actuator. At present, there are mainly the following methods for weakening the shaking: (1) In filtering method, the filter is used to smooth the signal; BIB009 in observer method, using the observer to eliminate interference and uncertainty, thereby reducing the source of jitter; and (3) reduce the switching gain. The shaking is caused by the switching of the discontinuous controller, so reducing the gain of the switching could effectively weaken the shaking. SMC has successfully owed satellites, 221 underactuated fuel cell systems,  TORA, BIB016 underactuated biped robot, BIB014 and overhead crane.  At the same time, many researchers are also working on designing generalpurpose SMC for UR systems with the same dynamic characteristics. The robot system is essentially a nonlinear system with many unknown disturbances, which provides a typical application environment for sliding mode variable structure control. SMC based on backstepping is a common method. Based on the uniform global stability and tracking of the control problem of underactuated surface ships, Prof. Ghommam et al. BIB013 used the backstepping method to derive the discontinuous feedback control law. The Backstepping design method could decompose complex nonlinear systems into subsystems, which do not exceed the system order and then design them separately. The distinguishing feature is its system and simple framework for designing positional strategies. Backstepping could design a virtual controller for each subsystem based on Lyapunov theory. The control law of the MIMO system is designed by combining with the SMC. Once the backstepping design method was proposed, it gained widespread attention. And it has been extended to areas such as adaptive control and robust control. Optimal control. Optimal control theory is a major branch of modern control theory. Emphasis is placed on the basic conditions and integrated methods for optimizing the performance metrics of control systems. Currently, optimization methods are moving toward online optimization methods and intelligent optimization methods. The optimal control is mainly divided into two categories: time optimal and energy optimal. Duleba and Sasiadek, BIB010 Plestan et al.,  Tlalolini et al., BIB017 and Yang et al.  studied the study of time optimal control. Most control methods are based on a class of non-holonomic rigid systems. Krstic and Tsiotras, BIB006 Tsiotras and Luo, BIB007 and Sharma and Tewari 233 studied the energy optimal control. There is currently no general method for controlling UR systems. Through the research on the UR control method and the above review, the control methods and research literature of the UR are classified and summarized. Control methods and related studies are shown in Table 1 .