In this paper, we consider flocking of multiple inertial agents with second-order dynamics. For a single agent, its state is therefore velocity and position. There exists both velocity coupling and position coupling between agents that are not necessarily equal. The nonequal couplings are distinguished here, instead of the common assumption of the equal velocity coupling and position coupling in most literature, since they play different roles in the system dynamics. The main contributions are in two aspects: first, we provide a general geometric decomposition approach for analyzing such systems of the agents. Then, we provide some stability results via Lyapunov analysis as a function of couplings and gains that generalize existing results in this area, which are especially useful when the couplings are partially known. Although the stability results are only sufficient, they serve as important complements of the eigenvalue analysis, which generally requires complete knowledge of the couplings in the eigenvalue analysis.

The preservation of connectivity in mobile robot networks is critical to the success of existing algorithms designed to achieve various goals. The available connectivity control algorithms mainly work through preservation of existing edges in the network. A link may be deleted if distributed decision making determines that the edge is not a cut-bridge. A controller is presented which allows edges to be broken in a continuous manner without higher-level decision making. The controller is based on maximization of the second smallest eigenvalue of the graph Laplacian. The controllers are designed for holonomic robots, and are extended for implementation on non-holonomic wheeled mobile robots. Finally, the performance of the extended controllers are demonstrated experimentally.

Bipedal walking robots are inherently hybrid systems due to their intermittent, switching dynamics resulting from the impact between the robot foot and the ground as the robot foot lands on the ground. It is well known that stable (passive) limit cycles for the biped robots can be induced on shallow slopes without actuation. Recently the studies in passive dynamic walking have considered the robots with knee and point or curved feet. In this paper, we study the passive dynamic walking for biped robots with knee and fixed flat feet, which includes heel and toe rocking motions and the effect of foot length on the passive limit cycles. We derive the dynamic equations of motion for this model. We show by simulation that the proposed robot model can walk down a slope passively and also verify the stability of this walking by calculating the eigenvalues of the Jacobian of the Poincarè map. By using a numerical search method, we find the initial conditions of the stable limit cycles for various slope angles and foot lengths.

In this paper, we present a hierarchical framework that enables motion planning for asymptotically stable 3-D bipedal walking in the same way that planning is already possible for zero moment point walking. This framework is based on the construction of asymptotically stable gait primitives for a class of hybrid dynamical systems with impacts. Each primitive corresponds to an asymptotically stable hybrid limit cycle that admits rules a priori for sequential composition with other primitives, reducing a high-dimensional feedback motion planning problem into a low-dimensional discrete tree search. As a constructive example, we develop this planning framework for the 3-D compass-gait biped, where each primitive corresponds to walking along an arc of constant curvature for a fixed number of steps. We apply a discrete search algorithm to plan a sequence of these primitives, taking the 3-D biped stably from start to goal in workspaces with obstacles. We finally show how this framework generalizes to more complex models by planning walking paths for an underactuated five-link biped.

This paper applies geometric reduction-based control to derive a set of asymptotically stable dynamic walking gaits for a 3-D bipedal robot, each corresponding to walking along a nominal arc of constant curvature for a fixed number of steps. We show that any such set of asymptotically stable gait primitives may be composed in arbitrary order without causing the robot to fall, so any walking path that is a sequence of these gaits may be followed by the robot. This result enables motion planning for bipedal dynamic walkers, which are fast and energetically efficient, in a similar manner to what is already possible for biped locomotion based on Zero Moment Point (ZMP) equilibrium constraints.

In this paper we study the problem of passive walking for a compass-gait biped with gait asymmetry. In particular, we identify and classify bifurcations leading to chaos caused by gait asymmetries due to unequal leg masses. We present bifurcation diagrams showing step period versus the ratio of leg masses at various walking slopes. The cell mapping method is used to find stable limit cycles as the parameters are varied. It is found that a variety of bifurcation diagrams can be grouped into six stages that consist of three expanding and three contracting stages. The analysis of each stage shows that passive dynamic walking has multiple attractors depending on initial conditions, and marginally stable limit cycles exhibit not only period doubling, but also period remerging, disconnecting, and disappearing. We also show that the rate of convergence of period doubling sequences is in good agreement with the Feigenbaum constant.

The planar compass-gait biped has been extensively studied in the dynamic walking community, motivated by the gravity-based pendular efficiencies of human walking. These results can be extended to three dimensions using controlled geometric reduction for open-chain robots, by which stable 3-D walking gaits are built from known sagittal-plane limit cycles. We apply this method to the standard and with-torso compass-gait (hipless) bipeds, showing straight-ahead walking gaits (i.e., stable 1-step periodic limit cycles) as well as h-step turning in full circles (i.e., stable h-periodic limit cycles). These constant-curvature maneuvers are composed of stable 1-periodic turning gaits modulo heading change, demonstrating two types of gaits for directional dynamic walking in three dimensions.

We applied partial feedback linearization to the unicycle model to stabilize part of the state at a desired position on the plane and extend it to design formation control for teleoperation. Further, using output synchronization results we derive a Single Master Multiple Slave bilateral teleoperation system robust to constant unknown, possible different time delays between master and formation and among mobile robots and formation. We show our simulation's results to illustrate the performance of the derived control law.

Among the still existing issues in bilateral teleoperation, there is the inability by force-feedback control schemes to guarantee delay-independent stability and achieve both position coordination and force reflection independently of the remote environmental dynamics. Particularly, most bilateral control frameworks fail to address position coordination when interacting with rigid environments. In this paper we present a novel control strategy that aims to passively compensate for position errors that arise during contact tasks and, in general, achieve stability and transparency when alternating between unobstructed (free) and obstructed (contact) environments. The proposed control framework exploits the wave impedance independent passivity property of the scattering transformation to guarantee stability and transparency by gradually switching between a low wave impedance, ideal for free motion, and a sufficiently large impedance, suitable for contact tasks. The validity of the control framework is verified through simulations and experiments on a pair of nonlinear robots.

A planar compass-like biped on a shallow slope is one of the simplest models of a passive walker. It is a 2-degree-of-freedom (DOF) impulsive mechanical system that is known to possess periodic solutions reminiscent of human walking. Finding such solutions is a challenging computational task that has attracted many researchers who are motivated by various aspects of passive and active dynamic walking. We propose a new approach to find stable as well as unstable hybrid limit cycles without integrating the full set of differential equations and, at the same time, without approximating the dynamics. The procedure exploits a time-independent representation of a possible periodic solution via a virtual holonomic constraint. The description of the limit cycle obtained in this way is useful for the analysis and characterization of passive gaits as well as for design of regulators to achieve gaits with the smallest required control efforts. Some insights into the notion of hybrid zero dynamics, which are related to such a description, are presented as well.

A detailed experimental comparison study of several published algorithms for motion and force control of bilateral teleoperators, with emphasis on Internet-based teleoperation, is presented. The study investigates the effects of data losses, communication delays, and environmental constraints on a teleoperation system for different control techniques, which are based on wave variables, Smith predictors, and recent algorithms on synchronization. The controllers are compared on stability, transparency, and complexity using two identical nonlinear robots coupled via a stochastic network model that allowed transmission round-trip delays and data-loss rates to range from 8 to 1088 ms and 0% to 50%, respectively. A total of 18 subjects, which were distributed among 26 experiments with the aims of regulating the effects of the operators learning process and dynamic properties, participated in this study. Overall, the comparison study reports a deteriorating effect in the performance (i.e., larger position errors and lower fidelity of contact information) from delays and data losses. Yet, the effect of data losses is less critical when compared with time delays. In addition, the preference for a particular control framework is shown to strongly depend on the operational conditions of the system, such as the characteristics of the coupling channel, the specifics of the remote task, and the computational capabilities of the manipulators.

This paper presents a control law that time-scales reference trajectories of dynamical systems, yielding arbitrary velocities in arbitrary time. It is shown that, for unforced (passive) reference trajectories, constant time-scaling results in a potential energy-shaping control. Application to walking trajectories of bipedal robots is shown, extending the use of the control beyond purely continuous dynamical systems to a class of hybrid dynamical systems with discontinuities that are linear in velocity. Two biped models are used to demonstrate the control law: for the compass-gait biped, we illustrate time-scaling of a passive reference trajectory; for the biped with a torso, we show time-scaling of a semi-passive reference trajectory.

This paper presents a novel decentralized control scheme that achieves dynamic formation control and collision avoidance for a group of nonholonomic robots. First, we derive a feedback law using Lyapunov-type analysis that guarantees collision avoidance and tracking of a reference trajectory for a single robot. Then, we extend this result to the case of multiple nonholonomic robots, and show how different classes of multi-agent problems involving an interacting group of nonholonomic robots such as formation control can be addressed in this framework. Finally, we combine the above results to address the problem of driving a group of robots according to a given trajectory while maintaining a specific formation.

We consider bilateral teleoperation of a wheeled mobile robot over communication channels with constant delays. Our main objective is to enable humans to control the mobile robot much as they drive a car: i.e. by operating a master haptic joystick, they can control the linear velocity and heading angle of the mobile robot, much like they do so with the gas pedal and steering wheel. Passivity of the closed-loop system is also enforced so that, even with communication delays, humans can stably and safely teleoperate the wheeled mobile robot with force-reflection. A semi-experiment (i.e. real master/simulated slave) is performed to validate the proposed framework

We propose a novel control framework for bilateral teleoperation of a pair of multi-degree-of-freedom (DOF) nonlinear robotic systems under constant communication delays. The proposed framework utilizes the simple proportional-derivative (PD) control, i.e. the master and slave robots are directly connected via spring and damper over the delayed communication channels. Using the controller passivity concept, the Lyapunov-Krasovskii technique, and Parseval's identity, we can passify the combination of the delayed communication and control blocks altogether robustly, as long as the delays are finite constants and an upper-bound for the round-trip delay is known. Having explicit position feedback through the delayed P-action, the proposed framework enforces master-slave position coordination which is often compromised in the velocity-based schemes (e.g. conventional scattering-based teleoperation). The proposed control framework provides humans with extended physiological proprioception so that s/he can affect and sense the remote slave environments mainly relying on her/his musculoskeletal systems. Experiments are performed to validate the proposed control framework

In a previous paper we proposed a bilateral teleoperation framework of a wheeled mobile robot over communication channel with constant time delay. In this paper we present experimental results. Our goal is to illustrate and validate the properties of the proposed scheme as well as to present practical implementation issues and the adopted solutions. In particular, the bilaterally teleoperated system is passive and the system is stable in the presence of time delay. Internet has been used as the communication channel and a buffer has been implemented to maintain a constant time delay and to handle packet order

This paper addresses the problem of steady-state position and force tracking in bilateral teleoperation. Passivity-based control schemes for bilateral teleoperation provide robust stability against network delays in the feedback loop and velocity tracking, but do not guarantee steady-state position and force tracking in general. Position drift due to data loss and offset of initial conditions is a well-known problem in such systems. In this paper, we introduce a new architecture, which builds upon the traditional passivity-based configuration by using additional position control on both the master and slave robots, to solve the steady-state position and force-tracking problem. Lyapunov stability methods are used to establish the range of the position control gains on the master and slave sides. Experimental results using a single-degree-of-freedom master/slave system are presented, showing the performance of the resulting system

We propose a novel control framework for bilateral teleoperation of a pair of multi-degree-of-freedom nonlinear robotic systems under constant communication delays. The proposed framework uses the simple proportional-derivative control, i.e., the master and slave robots are directly connected via spring and damper over the delayed communication channels. Using the controller passivity concept, the Lyapunov-Krasovskii technique, and Parseval's identity, we can passify the combination of the delayed communication and control blocks altogether robustly, as long as the delays are finite constants and an upper bound for the round-trip delay is known. Having explicit position feedback through the delayed P-action, the proposed framework enforces master-slave position coordination, which is often compromised in the conventional scattering-based teleoperation. The proposed control framework provides humans with extended physiological proprioception, so that s/he can affect and sense the remote slave environments mainly relying on her/his musculoskeletal systems. Simulation and experiments are performed to validate and highlight properties of the proposed control framework

In a companion paper [1], we propose a control framework for the bilateral teleoperation between a single master robot and multiple cooperative slave robots over delayed communication network. In this paper, we perform simulation and semi-experiment (i.e. real master and simulated slaves) to illustrate and validate properties of the proposed control scheme. In particular, the three key properties of the proposed control framework are highlighted in this paper: 1) cooperative fixtureless grasping and manipulation of inertial and deformable objects by multiple slave robots; 2) passive teleoperation of the overall behavior of the multiple slave robots (and the grasped object) over the delayed communication with force reflection; and 3) grasping safety (i.e. secure grasping) and interaction stability regardless of the communication delay and human command.

We propose a control framework for the bilateral teleoperation between a single master robot and multiple cooperative slave robots with communication-delay in the master-slave communication channel. Using passive decomposition, we first decompose the dynamics of multiple slaves into two decoupled systems while preserving energetic passivity: the shape system describing cooperative grasping aspect, and the locked system representing overall behavior of the multiple slaves. Then, by locally controlling the decoupled shape system with the cancellation of disturbances on it, secure and tight (and possibly fixtureless) cooperative grasping can be achieved regardless of the communication-delay and human command. We also construct a bilateral teleoperation loop between the master and the locked system s.t., by operating the master, a human operator can control the overall behavior of the multiple slaves and the grasped object while perceiving environmental forces acting on them. Scattering-based communication is used to passify the master-slave communication-delay. By exploiting the passivity property of the decomposition and scattering-based communication, energetic passivity of the closed-loop system can be ensured, thus, interaction stability and safety are improved significantly. In a companion paper [1], simulation and semi-experiment (i.e. real master and simulated slaves) are performed to illustrate properties of this proposed framework.

In this paper, we investigate issues in the discrete-time implementation of passivity based control of bilateral teleoperators. The usual scattering formalism which, in continuous time, guarantees passivity for any constant delay, is extended in several important ways to the discrete domain, in particular to the case where communication between the master and slave robots occurs over a packet-switched network. We first show that passivity can be maintained in the face of varying delay and packet loss but that it depends fundamentally on the mechanism used to handle missing packets. Passivity alone is not sufficient to guarantee good performance. Therefore, we also introduce a novel buffering and interpolation scheme which not only preserves passivity but has been shown through simulation and experiments to improve tracking performance and transparency in a single-degree-of-freedom teleoperator system.

This paper continues our investigations into the passivity-based control of the compass gait biped. It was shown in [Spong, M.W., 1999] for the compass gait biped, and in [Spong, M.W., and Bullo, F., 2002] for the general case, that a passive limit cycle for a given constant slope can be made slope invariant via potential energy shaping control. In this paper we consider the application of total energy shaping to this problem. Using a control law that shapes the total energy of the biped we are able to influence both the exponential rate of convergence to the limit cycle and its basin of attraction. This increases robustness and variation in the ground slope. As illustrations we show stable walking gaits both in the presence of external disturbances and on varying slopes that do not exist without the total energy shaping control.

This work is an initial step toward combining control theory with computer vision for the case of uncalibrated monocular (or single-camera) three-dimensional manipulation. We investigate an achievable control goal using a novel image measure and derive a sampled-data control system that results in quick convergence of the end-effector trajectory to a depth-invariant velocity subspace in the camera coordinate frame. We also discuss how this uncalibrated behavior can be used to perform online calibration.

The presence of right half plane (RHP) zeros in the transfer function relating the base torque to the tip position has long been the source of difficulty in noncollocated control of a single flexible link (SFL). In this paper, the authors suggest an alternative noncollocated measurement called the virtual angle of rotation. The authors then show the following. (A) The transfer function with this output does not have any zeros in the open RHP. (B) The zero dynamics arising from this measurement are stable. (C) This distributed system can be stabilized by a finite-dimensional controller in the presence of Rayleigh damping.

We present a new controller to solve the visual servo control problem for planar robots in the fixed-camera configuration. For a static target we characterize the global closed loop attractor using the full robot dynamics, and prove local asymptotic stability of the end-effector tracking error. Neither the inverse kinematics nor the inverse Jacobian are used in the controller. Experimental results on a 2-DOF direct drive manipulator are presented.

Discusses the use of computer vision for real-time state estimation in feedback control systems. To this end, the authors construct a system for visual state estimation of simple state vectors and study the effects of various real-world disturbances on the state estimates. Simulations are performed using a detailed camera model to study the performance of an image plane position estimation algorithm for a single circular feature. Various disturbances, such as lens distortion, noise, defocus, and blurring are simulated and analyzed with respect to this estimation routine and visual state estimation in general.>

This paper presents algorithms for continuous-time quadratic optimization of impedance control. Explicit solutions to the Hamilton-Jacobi equation for optimal control of rigid-body motion are found by solving an algebraic matrix equation. System stability is investigated according to Lyapunov function theory, and it is shown that global asymptotic stability holds. The solution results in design parameters in the form of square weighting matrices or impedance matrices as known from linear quadratic optimal control. The proposed optimal control is useful both for motion control and force control.>

Investigates the problem of swing up control of the Acrobot, a two-link, underactuated robot that is a useful vehicle to study problems in nonlinear control. The author develops a swing up strategy based on partial feedback linearization. The algorithm works by creating "unstable zero dynamics" which drives the first link of the Acrobot away from its open loop stable equilibrium toward the inverted position. Control is switched to a linear controller, designed to balance the arm about the inverted configuration, whenever the swing up controller moves the Acrobot into the near vertical position. Simulation results are presented showing the performance of the system.>

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.>

The authors extend the integral manifold approach for the control of flexible joint robot manipulators from the known parameter case to the adaptive case. The resulting adaptive law, referred to as the corrective control law, consists of a fast component to damp the fast dynamics and a slow component which is designed based on the integral manifold theory and which consists of a rigid based component along with additional corrective terms. The authors give a detailed derivation of the corrective control law and present tracking results. They illustrate the implementation of the control law using simulation examples and study tracking performance and robustness with respect to an allowable range of stiffness and high adaptation gains.>

The author uses tools from Hamiltonian mechanics and Riemannian geometry to illustrate some properties of robot dynamics that are useful both for robot control and for the design of robotic manipulators. Several authors have noted that if the robot inertia matrix D(q) can be factored as N/sup T/(q)N(q), where N(q) is the Jacobian of a function Q(q), then Q and P=N(q)q define a canonical transformation relative to which the robot dynamics are particularly simple. In the present work, the author gives necessary and sufficient conditions for the existence of such a factorization and discusses their implications for robot control.>

The authors validate their previously published (1988) algorithm for controlling a force-reflecting teleoperator with time delay. The time-delay compensation is shown to result in a control law which ensures asymptotic stability of manipulator velocities when contacting arbitrary passive environments. The results are based on the full n-DOF (degrees of freedom) nonlinear system and allow for the inclusion of power gain between the human operator and the environment.>

The authors present an adaptive control result for flexible-joint robot manipulators. Under the assumption of weak joint elasticity, a singular perturbation argument is used to show that recent adaptive control results for rigid robots can be used to control flexible-joint robots, provided a simple correction term is added to the control law to damp out the elastic oscillations at the joints. In this way, fundamental properties of rigid robot dynamics can be used to design robust adaptive control laws for flexible-joint robots. The implementation of the full controller requires only joint position and velocity information. Thus, robustness to parametric uncertainty is achieved without the need for acceleration and jerk measurements.>

The inclusion of force information in the control of robots increases their adaptability to uncertain environments, such as are found in deburring, grinding, and assembly tasks. The authors present a foundation for force control strategies, in view of the fact that the type of control strategy that is employed depends fundamentally on the characteristics of the environment. A general control approach is introduced, called hybrid impedance control which in its simplest forms reduces the operational space control of O. Khatib and J. Burdick (1986), or to N. Hogan's (1985) impedance control. The control law is formulated in a general enough fashion, however, to allow for higher order controllers.>

This work presents a new algorithm for controlling the contact forces of a robotic manipulator. The algorithm, to be called hybrid impedance control (HIC), combines two fundamental force control strategies; hybrid position/force control, and impedance control. A duality condition is defined which specifically determines how the manipulator should respond for a given environment. A system-theoretic approach is utilized to motivate the algorithm, and specific examples are developed to illustrate it.

The control problem for robot manipulators with flexible joints is considered. The results are based on a recently developed singular perturbation formulation of the manipulator equations of motion where the singular perturbation parameter µ is the inverse of the joint stiffness. For this class of systems it is known that the reduced-order model corresponding to the mechanical system under the assumption of perfect rigidity is globally linearizable via nonlinear static-state feedback, but that the full-order flexible system is not, in general, linearizable in this manner. The concept of integral manifold is utilized to represent the dynamics of the slow subsystem. The slow subsystem reduces to the rigid model as the perturbation parameter µ tends to zero. It is shown that linearizability of the rigid model implies linearizability of the flexible system restricted to the integral manifold. Based on a power series expansion of the integral manifold around µ = 0, it is shown how to approximate the feedback linearizing control to any order in µ. The result is then an approximate feedback linearization which, assuming stability of the fast variables, linearizes the system for all practical purposes.

The motion control of robotic manipulators is investigated using a recently developed approach to linear multivariable control known as the stable factorization approach. Given a nominal model of the manipulator dynamics, the control scheme consists of an approximate feedback linearizing control followed by a linear compensator design based on the stable factorization approach. Using a multiloop version of the small gain theorem, robust trajectory tracking is shown under the assumption that the deviation of the model from the true system satisfies certain norm inequalities. In turn, these norm inequalities lead to quantifiable bounds on the tracking error.

In this paper we study the nonlinear control problem for a single link manipulator with joint elasticity. We illustrate several recent nonlinear control techniques, namely feedback linearization, integral manifold and composite control. We work out in detail both a global feedback linearizing control law and a nonlinear composite control law based on a singular perturbation formulation of the manipulator dynamics and the concept of integral manifold. The two solutions are then compared and contrasted.

In this paper we examine a recently developed singular perturbation formulation of the equations of motion for a robot manipulator with flexible joints, where the fast variables are the elastic forces at the joints and their time derivatives. The concept of an invariant manifold is utilized to represent the dynamics of the slow subsystem. The dynamics of the system restricted to this manifold reduce to the usual rigid body dynamics as the perturbation parameter ε tends to zero. Based on a power series expansion of the exact manifold around ε = 0, higher order corrections of the manifold are obtained. This leads to reduced order models of the full system which may prove more useful for control system design than either the full model or the rigid model. The case of a single link with joint flexibility is worked out in detail.

In this paper we investigate the application to the motion control of n-link robotic manipulators of the recently developed stable factorization approach to tracking and disturbance rejection. Given a nominal model of the manipulator dynamics, the control scheme consists of an approximate feedback linearizing control followed by a linear compensator design based on the stable factorization approach to achieve optimal tracking and disturbance rejection. Using a multi-loop version of the small gain theorem [17], the applicability of the linear design techniques and the stability of the closed loop system are rigorously demonstrated.