This paper introduces climbing gait for the snake robot by adapting to a flexible net. A net deforms in various ways due to external forces. Therefore, we tilt snake robot’s body to adapt to the deformation of the net. In addition, we can adjust the position of the head of a snake robot passing through the mesh. A snake robot can move not only vertically but also horizontally. We demonstrated the validity of the proposed method through experiments in vertical movement and movement in diagonal direction.

The snake robot, which mimics the mechanism of real snakes, is expected to be used in various environments. This article addresses the challenge of snake robots autonomously adapting to and moving inside and outside pipes having complex shapes using a helical rolling motion. We consider that an irregular helical form can be defined as a combination of three components, namely the axis, cross-sectional shape, and pitch angle, and that it is easier to consider deformations that have compliance only in a specific direction. Our proposed method deforms the cross-sectional shape of the snake robot locally so that it adopts the shape of the pipe accurately. Experiments were conducted using our developed snake robot to verify the effectiveness of the proposed method, and it was demonstrated that the proposed method can be used for a snake robot moving inside and outside of straight pipes and outdoor tree, including those having noncircular and varying cross sections.

This article presents a novel path-following-method-based polynomial fuzzy control design. By examining the stabilization problem, the nonconvex stabilization criterion represented in terms of bilinear sum-of-squares (SOS) constraints is proposed to complement the existing convex stabilization criteria. Based on the polynomial Lyapunov function and considering the operation domain, the stabilization control is designed with a systematic region of attraction (ROA) analysis method. Since the proposed stabilization criterion remains in nonconvex form, the conservativeness caused by the transformation from nonconvex (bilinear SOS) constraints into convex (SOS) constraints can be avoided. Moreover, the restriction on the Lyapunov function candidates for the convex transformation in the literature does not exist in the proposed nonconvex stabilization criterion. The stabilization analysis for polynomial fuzzy control systems is concerned with the double fuzzy summation problem that can be treated as the copositivity problem. Therefore, the SOS-based copositive relaxation technique is applied for the proposed stabilization criterion. Since the proposed nonconvex stabilization criterion is represented in terms of bilinear SOS constraints, the path-following method is employed for solving the bilinear SOS problem. Finally, design examples are provided to demonstrate that the proposed nonconvex stabilization criterion complements the existing convex stabilization criteria.

This article presents both stabilization and disturbance attenuation control synthesis conditions for a class of polynomial fuzzy systems. The sum of squares conditions is relaxed by restricting the domain of the polynomial variables that represent the membership functions as well as by using the line integral polynomial fuzzy Lyapunov function. For the disturbance attenuation problem, we consider the transformation of the Hamilton-Jacobi-Isaacs equation into a set of inequalities and a policy iteration algorithm to approximate its value function. Furthermore, we propose an alternative path following to find the initialL2gain stabilizing control policy. Five examples are provided to demonstrate the effectiveness of the proposed conditions.

A snake robot performs diverse motions. To realize a wide range functions in a complex environment, it is necessary to transition between various motions suited to each environment. In this article, we propose a method of transitioning the motion of a snake robot across different environments to expand the application environment of the robot. We first find that the motion at the connection point between two motions must coincide with the tangential movement during motion transition across different environments. We then design a gait called the circular pedal wave. This circular pedal wave allows a hoop-passing motion in which the whole body moves as if it is passing through a virtual hoop fixed in space in sequence from its head through combination with a proposed shift part. The hoop-passing motion allows motion transition across different environments. We propose three application examples of this hoop-passing motion, namely passing through a hole in a wall, entering an underfloor, and attaching to a ladder. We report on experiments conducted to verify the effectiveness of the proposed method and to realize the described motions.

A method for designing the motion of a snake robot negotiating complicated pipe structures having a constant diameter is presented. For such robots moving inside pipes, there are various "obstacles" such as junctions, bends, shears, and blockages. To surmount these obstacles, we propose a method that enables the robot to adapt to multiple pipe structures of a constant diameter. We designed the target form of the snake robot of two helices connected with an arbitrary shape. This method is applicable to various obstacles by designing a part of the target form conforming to the obstacle. The robot negotiates obstacles under shift control by employing a rolling motion. We demonstrated the effectiveness of the proposed method in various experiments.

A task-space method is presented for the control of a head-raising articulated mobile robot, allowing the trajectory tracking of a tip of a gripper located on the head of the robot in various operations, e.g., picking up an object and rotating a valve. If the robot cannot continue moving because it reaches a joint angle limit, the robot moves away from the joint limit and changes posture by switching the allocation of lifted/grounded wheels. An articulated mobile robot with a gripper that can grasp objects using jamming transition was developed, and experiments were conducted to demonstrate the effectiveness of the proposed controller in operations.

This correspondence paper presents a framework for practical model construction and stable altitude control of an unmanned aerial vehicle with a parafoil-type wing (UAV-PW). To design a stable controller, we first construct a dynamical longitudinal model of the UAV-PW. Since there exist no aerodynamics data of the parafoil shape in our UAV-PW, aerodynamics coefficients balanced at the trimmed equilibrium are employed. The model accuracy is investigated by comparing the model outputs with the real test flight experimental data. Next, stable controller design conditions for the UAV-PW model with uncertainties are derived in terms of linear matrix inequalities (LMIs). By solving the LMI conditions, we design a stable controller that asymptotically stabilizes the UAV-PW model with the uncertainties on a considered operation domain. The experimental results demonstrate the viability of the model construction and the stable altitude control.

This paper presents a method that allows a snake robot to climb a ladder. We propose a ladder climbing method for a snake robot that has a smooth surface shape. We design a novel gait for the snake using a gait design method that configures the target form of the snake robot by connecting simple shapes. The climbing motion is executed via shift control and the corresponding motion required to catch the next step on the ladder. In addition, we developed a snake robot that has a smooth exterior body surface through construction of pectinate-shaped parts of the links. We demonstrated the effectiveness of both the proposed gait and the design of the snake robot experimentally.

In this paper, we present a generalization of the line-integral fuzzy Lyapunov function. Our proposal allows considering a conservative vector field of homogeneous polynomials in the consequent of the IF-THEN fuzzy rules, resulting in a line-integral polynomial fuzzy Lyapunov function. Based on the proposed Lyapunov function and using Positivstellensatz, we derive sum of squares nonconvex control synthesis conditions which are applicable to both Tagaki-Sugeno and polynomial fuzzy systems. Moreover, an initialization step for the path following iterative algorithm is proposed. Finally, the effectiveness of the relaxed conditions is demonstrated by two benchmark examples.

This paper presents a new nonconvex design algorithm for guaranteed cost control of polynomial fuzzy systems that include the Takagi-Sugeno (T-S) fuzzy systems as a special case. We derive a sum-of-squares (SOS) noconvex conditions to achieve guaranteed cost control of polynomial fuzzy systems. Although SOS-based design approaches are regarded as a post LMI-based design approach, there exist some remaining problems to be solved in systems analysis and design. To solve the remaining problems of the SOS-based design approaches, this paper gives new ideas that directly solve nonconvex SOS design conditions in guaranteed cost control using the so-called path-following algorithm. In particular, we bring a new scheme that minimizes the upper bound of a given performance function in parallel with minimizing a parameter of checking non-negativity for SOS design conditions. The two parameters minimization in the path-following algorithm is realized by introducing a double-loop structure. Finally, two complex nonlinear system design examples (a polynomial chaotic system and a complicated nonlinear system) are employed to illustrate the validity and applicability of the proposed nonconvex design algorithm for guaranteed cost control of polynomial fuzzy systems.

This paper presents a method for designing the gait of a snake robot that moves in a complicated environment. We propose a method for expressing the target form of a snake robot by connecting curve segments whose curvature and torsion are already known. Because the characteristics of each combined shape are clear, we can design the target form intuitively and approximate a snake robot configuration to this form with low computational cost. In addition, we propose two novel gaits for the snake robot as a design example of the proposed method. The first gait is aimed at moving over a flange on a pipe, while the other is the crawler gait aimed at moving over rough terrain. We demonstrated the effectiveness of the two gaits on a pipe and rough terrain in experiments.

This paper deals with a sum-of-squares (SOS)-based control Lyapunov function (CLF) design for polynomial fuzzy control of nonlinear systems. The design starts with exactly replacing (smooth) nonlinear systems dynamics with polynomial fuzzy models, which are known as universal approximators. Next, global stabilization conditions represented in terms of SOS are provided in the framework of the CLF design, i.e., a stabilizing controller with nonparallel distributed compensation form is explicitly designed by applying Sontag's control law, once a CLF for a given nonlinear system is constructed. Furthermore, semiglobal stabilization conditions on operation domains are derived in the same fashion as in the global stabilization conditions. Both global and semiglobal stabilization problems are formulated as SOS optimization problems, which reduce to numerical feasibility problems. Five design examples are given to show the effectiveness of our proposed approach over the existing linear matrix inequality and SOS approaches.

This paper presents a new sum-of-squares (SOS, for brevity) design framework for robust control of polynomial fuzzy systems with uncertainties. Two kinds of robust stabilization conditions are derived in terms of SOS. One is global SOS robust stabilization conditions that guarantee the global and asymptotical stability of polynomial fuzzy control systems. The other is semiglobal SOS robust stabilization conditions. The latter is available for very complicated systems that are difficult to guarantee the global and asymptotical stability of polynomial fuzzy control systems. The main feature of all the SOS robust stabilization conditions derived in this paper are to be expressed as nonconvex formulations with respect to polynomial Lyapunov function parameters and polynomial feedback gains. Since a typical transformation from nonconvex SOS design conditions to convex SOS design conditions often results in some conservative issues, the new design framework presented in this paper gives key ideas to avoid the conservative issues. The first key idea is that we directly solve nonconvex SOS design conditions without applying the typical transformation. The second key idea is that we bring a so-called copositivity concept. These ideas provide some advantages in addition to relaxations. To solve our SOS robust stabilization conditions efficiently, we introduce a gradient algorithm formulated as a minimizing optimization problem of the upper bound of the time derivative of an SOS polynomial that can be regarded as a candidate of polynomial Lyapunov functions. Three design examples are provided to illustrate the validity and applicability of the proposed design framework. The examples demonstrate advantages of our new SOS design framework for the existing linear matrix inequality approaches and the existing convex SOS approach.

This paper proposes sum-of-squares (SOS) methodologies for stability analysis and region-of-attraction (ROA) estimation for nonlinear systems represented by polynomial fuzzy models via piecewise polynomial Lyapunov functions (PPLFs). At first, two SOS-based global stability criteria are proposed by applying maximum-type and minimum-type PPLFs, respectively. It is known that less-conservative results can be obtained by reducing global stability to local stability, since it is usually the case for nonlinear systems that the stability cannot be reached globally. Therefore, based on the two types of PPLFs, two local stability criteria are further proposed with the algorithms that enlarge the estimated ROA as much as possible. The constraints for checking (global and local) stability and enlarging the estimated ROA are represented in terms of bilinear SOS problems. Hence, the path-following method is applied to solve the bilinear SOS problems in the proposed methodologies. Finally, some examples are provided to illustrate the utility of the proposed methodologies.

This paper proposes control method for a snake robot to ascend and descend steps. In a multiplane step environment, it is necessary for locomotion to transfer from one plane to another. When a snake robot moves, it touches several planes as its body is long and thin. In this paper, we propose a control method to track the trajectory of a snake robot in a step environment. We decomposed the 3-D motion of the robot into two simple models by introducing an assumption that simplifies the model and controller, and derive a model of the robot as a hybrid system with switching. The control method consists of a tracking controller, a method for shifting the robot's part connecting the planes, and active lifting to control the shape of the robot. Ascent and descent experiments confirm the effectiveness of the proposed controller and the method for shifting the connecting part of the robot's body.

This paper presents a sum-of-squares (SOS) based methodology to obtain inner bounds on the region-of-attraction (ROA) for nonlinear systems represented by polynomial fuzzy systems. The methodology searches a polynomial Lyapunov function to guarantee the local stability and the invariant subset of the ROA is presented as the level set of the polynomial Lyapunov function. At first the methodology checks whether the considered system can be guaranteed to be locally asymptotically stable. After confirming that the system is guaranteed to be locally asymptotically stable, the methodology enlarges the invariant subset of the ROA as much as possible. The constraints for both of checking stability and enlarging contractively invariant set are represented in terms of bilinear SOS optimization problems. The path-following method is applied to solve the bilinear SOS optimization problems in the methodology.

This paper presents a novel method for stabilization analysis of the single-input polynomial fuzzy systems using a sum of squares (SOS) approach to construct control Lyapunov functions (CLFs) for the systems. First, we represent a nonlinear system as a polynomial fuzzy system. Then, sufficient conditions in SOS terms for a positive definite function to become a CLF is presented. We solve the sufficient conditions using SOS optimization technique to construct the CLF. Finally, after a CLF is constructed, the controller for the system is designed using Sontag's formula. To illustrate the validity of the proposed approach, a design example is provided.

In this paper, we consider “sinus-lifting motion” of a living snake, in which a snake lifts up some parts of its body from the ground, and switches the lifted parts dynamically. It is not clear whether imitating the sinus-lifting motion is the best locomotion or not for a snake like robot. The aim of this paper is to propose an appropriate motion pattern to a snake like robot considering the optimality of the sinus-lifting motion. We introduce two physical parameters, constraint forces and energy efficiency, as cost functions to optimize and propose switching strategies for generating optimal motion patterns of a snake like robot.

This paper presents development of an autonomous flying robot (unmanned aerial vehicle) and its verification via flight control experiment. The autonomous flying robot has a global positioning system (GPS) sensor to control its altitude and also a wireless camera to gather visual information from the air. First, we construct a longitudinal model of the autonomous flying robot to design a stable controller using a common Lyapunov approach. The longitudinal model has three degree of freedom, i.e., the altitude, the horizontal position and the pitch angle. A stable controller is designed so as to guarantee the stability of the trajectory tracking control system for the three-degree-of-freedom model. The experimental result demonstrates the viability of the stable trajectory tracking control.

This paper proposes a snake robot control method for climbing and descending a step. On a multi-plane step environment, it is necessary for locomotion to transfer from one plane to another. A snake robot can move with touching several planes as its body is long and thin. In this paper, we propose a control method for accomplishing trajectory tracking of a snake robot on a step environment. The control method consists of the tracking controller, the shifting method of the robot's part connecting between the planes, and the active lifting for controlling the shape of the robot. Simulations and experiments demonstrated the effectiveness of the proposed controller and the shifting method of the connecting part of the robot's body.

In this paper, we develop a new type of snake-like robot using screw-drive units that are connected by active joints. The screw-drive units enable the robot to generate propulsion on any side of the body in contact with environments. Another feature of this robot is the omnidirectional mobility by combinations of screws' angular velocities. We also derive a kinematic model and apply it to trajectory tracking control. Furthermore, we design a front-unit-following controller, which is suitable for manual operations. In this control system, operators are required to command only one unit in the front; then, commands for the rest of the units are automatically calculated to track the path of the preceding units. Asymptotic convergence of the tracking error of the front-unit-following controller is analyzed based on a Lyapunov approach for the case of constant curvature. The effectiveness of the control method is demonstrated by numerical examples and experiments.

This paper presents a sum-of-squares (SOS) approach to polynomial fuzzy observer designs for three classes of polynomial fuzzy systems. The proposed SOS-based framework provides a number of innovations and improvements over the existing linear matrix inequality (LMI)-based approaches to Takagi-Sugeno (T-S) fuzzy controller and observer designs. First, we briefly summarize previous results with respect to a polynomial fuzzy system that is a more general representation of the well-known T-S fuzzy system. Next, we propose polynomial fuzzy observers to estimate states in three classes of polynomial fuzzy systems and derive SOS conditions to design polynomial fuzzy controllers and observers. A remarkable feature of the SOS design conditions for the first two classes (Classes I and II) is that they realize the so-called separation principle, i.e., the polynomial fuzzy controller and observer for each class can be separately designed without lack of guaranteeing the stability of the overall control system in addition to converging state-estimation error (via the observer) to zero. Although, for the last class (Class III), the separation principle does not hold, we propose an algorithm to design polynomial fuzzy controller and observer satisfying the stability of the overall control system in addition to converging state-estimation error (via the observer) to zero. All the design conditions in the proposed approach can be represented in terms of SOS and are symbolically and numerically solved via the recently developed SOSTOOLS and a semidefinite-program solver, respectively. To illustrate the validity and applicability of the proposed approach, three design examples are provided. The examples demonstrate the advantages of the SOS-based approaches for the existing LMI approaches to T-S fuzzy observer designs.

In this paper, we discuss an optimal shape of the grounded part of a snake robot with passive wheels for jumping. First, we derive a relationship between constraint forces for the wheels and necessary friction forces, and propose a constraint force ellipse. Next, we introduce an evaluation index for the optimal shape of the grounded part (base part) of a snake robot using the constraint force ellipse. Finally, in order to demonstrate the validity of the optimal shape that minimizes the evolution index, simulations have been carried out.

In this paper, we consider “sinus-lifting motion” which is the effective motion of a snake for rapid movement consisting of lifting some parts of its body off the ground, and switching the lifted and grounded parts dynamically. This motion is analyzed based on control theory. We focus on the constraint force of passive wheels and switching of grounded parts of the snake robot to minimize the constraint force. Simulations show that the motion that minimizes the constraint force of the snake robot is similar to the sinus-lifting motion in appropriate amplitudes and periods of the body shape.

In this paper, we consider the trajectory tracking control of a 3-dimensional (3D) snake robot by using redundancy. We discuss the motion control which accomplishes the trajectory tracking and other tasks, for example singular configuration avoidance, obstacle avoidance, etc., for the 3D snake robot. Features of the internal motion caused by the kinematic redundancy are considered and the dynamic model of the 3D snake robot is derived by introducing shape controllable points. By utilizing redundancy, we propose the controller which accomplishes the trajectory tracking as the main-task and several sub-tasks. Simulation results show the effectiveness of the proposed controller.

This paper presents a path-tracking control method for a snake-like robot using screw drive mechanism. The operators are required to command only one unit in the head, then commands for the rest of the units are automatically calculated to track the path of the preceding units. Although the velocity commands for exact path tracking can be calculated using past command signals due to the omni-directional property of the robot, a simpler control law without using past signals is adopted from the implementation viewpoint. Asymptotic tracking error is investigated based on a Lyapunov approach for the case of a constant curvature. Furthermore, the effectiveness of the proposed method is investigated by computer simulations and laboratory experiments.

In this paper we consider modeling and control of a redundant snake robot with wheeled link mechanism based on kinematic model. We derive a kinematic model of a snake robot with introducing links without wheels and shape controllable points in the snake robot's body in order to make the system redundancy controllable. By using redundancy, it becomes possible to accomplish the main objective of controlling the position and the attitude of the snake robot head and the shape of the snake robot, and the sub-objective of the singular configuration avoidance. Simulations and experimental results show the validity of the control law and how snake robots move at the neighborhood of the singular configuration.

In this paper, we derive a dynamic model and a control law for cooperative task of two snake robots which have wheeled links and demonstrate the validity of proposed controller by simulation and experiment. First, we derive a dynamic model of single snake robot with redundancy by introducing the shape controllable points. Next, we extend it to the model for the cooperative task of two redundant snake robots and propose a control law to accomplish this task. Finally, in order to demonstrate the validity of the derived model and effectiveness of the proposed control law, simulation and experiment are carried out

In this paper, we derive a dynamic model and a control law for cooperative task of three snake robots which have wheeled links and demonstrate the validity of proposed controller by simulation. First, we derive a dynamic model of single snake robot with redundancy by introducing the shape controllable points. Next, we extend it to the model for the cooperative task of three redundant snake robots and propose a control law to accomplish this task. Finally, in order to demonstrate the validity of the derived model and effectiveness of the proposed control law, simulation is carried out