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In the recent paper (Wang et al. in Differ Geom Appl 80:101840, 2022), the authors and Xu have established a Simons-type integral inequality for holomorphic curves in a complex Grassmann manifold G(k, N). In this paper, we completely classify holomorphic immersions from the two-sphere of constant curvature into G(3, N) with the norm of the second fundamental form satisfying the equality case of the inequality and prove that any such immersion can be decomposed as the “direct sum” of some “foundation stones” up to congruence. [ABSTRACT FROM AUTHOR]

Deployable structures based on origami are widely used in the application of actuators. In this paper, we present a novel family of origami-based deployable structures with constant curvature. Two categories of non-flat-foldable and non-developable degree-4 vertices (NFND degree-4 vertices) are introduced. Pyramid structures are constructed based on the NFND degree-4 vertices. Doubly symmetric and singly symmetric spherical origami tubular cells (SOTCs) are established based on pyramid structures. To construct deployable origami modules using SOTCs, linking units are introduced. The mobility of the SOTCs and origami modules is analyzed using constraint screws. To realize the construction and simulation of deployable structures, kinematic and geometric analyses of the doubly symmetric and singly symmetric SOTCs are performed. Finally, we introduce four cases for deployable structures in spherical actuators based on the combination of multiple origami modules. These case studies demonstrate the potential of these deployable origami structures in the design of spherical actuators. [ABSTRACT FROM AUTHOR]

Continuum robots stand out due to their high dexterity, which allows them to effectively navigate in confined spaces and dynamic environments. At present, motor‐controlled tendons represent the most prominent actuation method used in continuum robots which require large drive units, increasing the overall system size and weight. A potential alternative technology to overcome those limitations is represented by shape memory alloy (SMA) wire actuators, which are characterized by extremely high energy density and flexibility, leading to a reduction of the size, weight, and design complexity of continuum robots. The complex thermomechanical behavior of SMA wires, however, makes the design of SMA‐based applications a challenging task, and systematic approaches to design SMA‐driven continuum robots are poorly understood. To overcome this issue, this article presents a novel systematic methodology for designing SMA‐driven continuum robots capable of motion in a three‐dimensional environment. First, the kinematic relationship between SMA wires and continuum robot deformation as well as the required actuator force in quasi‐static conditions, is mathematically described based on the assumption of a constant curvature deformation. Subsequently, the model is validated by in‐plane experiments for different design parameters. Based on the results, a fully integrated, antagonistic SMA continuum robot is built and validated. [ABSTRACT FROM AUTHOR]

In this paper, we give a characterization of homogeneous totally real minimal two-spheres in a complex hyperquadric Q n . Let f be a totally real minimal immersion from two-sphere in Q n , and τ X Y ,   τ X c (see Sec. 2) are globally defined invariants relative to the first and second fundamental forms. We prove that if its Gauss curvature K and τ X Y are constants, and τ X c vanishes identically, then f is congruent to F 2 k , 2 l constructed by the Boruvka spheres with n = 2 (k + l). [ABSTRACT FROM AUTHOR]

In this paper, firstly we study the geometry of conformal minimal two-spheres immersed in the complex hyperquadric Q n - 2 . Then we classify the linearly full irreducible conformal minimal immersions with constant curvature from S 2 to Q n - 2 ( n ⩾ 7 ) of isotropy order r = n - 6 under some conditions, which shows that all such immersions can be expressed by Veronese surfaces in C P n - 1 only under some conditions. [ABSTRACT FROM AUTHOR]

Soft crawling robots have attracted considerable attentions due to their merits of flexibility, safety, cost, and their unique applications that are not available for the rigid robots. However, poor precision resulting from the strongly nonlinear reconfiguration prevents such soft robots from wider applications. This paper reported a novel SMA spring driven soft crawling robot with feet of constant curvature, and the constant height of the feet during crawling enabled effective alleviation of the nonlinearity of the robotic reconstruction. An analytical static model for the step length of the robot's crawling gait was built based on 11 independent design parameters, and the influential significance of each parameter was parametrically studied based on the static model. These parameters were then qualitatively classified as strong, medium, and weak factors based on their influences on the theoretical step length of the crawling robot, among which the minimum and the maximum bending angles and the length of the silicone body functioned as the dominant factors. This work provides an efficient approach to the design, prediction, evaluation, and optimization of such soft crawling robots for diverse application surroundings. [ABSTRACT FROM AUTHOR]

Consider a geodesic triangle on a surface of constant curvature and subdivide it recursively into four triangles by joining the midpoints of its edges. We show the existence of a uniform δ > 0 such that, at any step of the subdivision, all the triangle angles lie in the interval (δ , π - δ) . Additionally, we exhibit stabilising behaviours for both angles and lengths as this subdivision progresses. [ABSTRACT FROM AUTHOR]

In design matters, mechanisms with deformable elements are a step behind those with rigid bars, particularly if dimensional synthesis is considered a fundamental part of mechanism design. For the purposes of this work, a hybrid rigid–flexible four-bar mechanism has been chosen, the input bar being a continuum tendon of constant curvature. The coupler curves are noticeably more complex but offer more possibilities than the classical rigid four-bar counterpart. One of the objectives of this work is to completely characterize the coupler curves of this hybrid rigid–flexible mechanism, determining the number and type of circuits as well as constituent branches. Another important aim is to apply optimization techniques to the dimensional synthesis of path generation. Considerable progress in finding the best design solutions can be obtained if all the acquired knowledge about the coupler curves of this hybrid mechanism is integrated into the optimization algorithm. [ABSTRACT FROM AUTHOR]

Reinforcement learning (RL) is explored for motor control of a novel pneumatic-driven soft robot modeled after continuum media with a varying density. This model complies with closed-form Lagrangian dynamics, which fulfills the fundamental structural property of passivity, among others. Then, the question arises of how to synthesize a passivity-based RL model to control the unknown continuum soft robot dynamics to exploit its input–output energy properties advantageously throughout a reward-based neural network controller. Thus, we propose a continuous-time Actor–Critic scheme for tracking tasks of the continuum 3D soft robot subject to Lipschitz disturbances. A reward-based temporal difference leads to learning with a novel discontinuous adaptive mechanism of Critic neural weights. Finally, the reward and integral of the Bellman error approximation reinforce the adaptive mechanism of Actor neural weights. Closed-loop stability is guaranteed in the sense of Lyapunov, which leads to local exponential convergence of tracking errors based on integral sliding modes. Notably, it is assumed that dynamics are unknown, yet the control is continuous and robust. A representative simulation study shows the effectiveness of our proposal for tracking tasks. [ABSTRACT FROM AUTHOR]

Deployable structures based on origami are widely used in the application of actuators. In this paper, we present a novel family of origami-based deployable structures with constant curvature. Two categories of non-flat-foldable and non-developable degree-4 vertices (NFND degree-4 vertices) are introduced. Pyramid structures are constructed based on the NFND degree-4 vertices. Doubly symmetric and singly symmetric spherical origami tubular cells (SOTCs) are established based on pyramid structures. To construct deployable origami modules using SOTCs, linking units are introduced. The mobility of the SOTCs and origami modules is analyzed using constraint screws. To realize the construction and simulation of deployable structures, kinematic and geometric analyses of the doubly symmetric and singly symmetric SOTCs are performed. Finally, we introduce four cases for deployable structures in spherical actuators based on the combination of multiple origami modules. These case studies demonstrate the potential of these deployable origami structures in the design of spherical actuators.

We construct globally hyperbolic spacetimes such that each slice {t D t0} of the universal time t is a model space of constant curvature k.t0/which may not only vary with ... somewhere, a slight extension of the class shows how the topology of the t -slices changes. This makes at least one comoving observer to disappear in finite time t showing some similarities with an inflationary expansion. Anyway, the spacetime is foliated by Cauchy hypersurfaces homeomorphic to spheres, not all of them t-slices. [ABSTRACT FROM AUTHOR]

If both the arc length and the intrinsic curvature of a curve or surface are preserved, then the flow of the curve or surface is said to be inextensible. The absence of motion-induced strain energy is the physical characteristic of inextensible curve and surface flows. In this paper, we study inextensible tangential, normal and binormal ruled surfaces generated by a curve with constant torsion, which is also called a Salkowski curve. We investigate whether or not these surfaces are minimal or can be developed. In addition, we prove some theorems which are related to inextensible ruled surfaces within three-dimensional Euclidean space.

If both the arc length and the intrinsic curvature of a curve or surface are preserved, then the flow of the curve or surface is said to be inextensible. The absence of motion-induced strain energy is the physical characteristic of inextensible curve and surface flows. In this paper, we study inextensible tangential, normal and binormal ruled surfaces generated by a curve with constant torsion, which is also called a Salkowski curve. We investigate whether or not these surfaces are minimal or can be developed. In addition, we prove some theorems which are related to inextensible ruled surfaces within three-dimensional Euclidean space. [ABSTRACT FROM AUTHOR]

Reinforcement learning (RL) is explored for motor control of a novel pneumatic-driven soft robot modeled after continuum media with a varying density. This model complies with closed-form Lagrangian dynamics, which fulfills the fundamental structural property of passivity, among others. Then, the question arises of how to synthesize a passivity-based RL model to control the unknown continuum soft robot dynamics to exploit its input–output energy properties advantageously throughout a reward-based neural network controller. Thus, we propose a continuous-time Actor–Critic scheme for tracking tasks of the continuum 3D soft robot subject to Lipschitz disturbances. A reward-based temporal difference leads to learning with a novel discontinuous adaptive mechanism of Critic neural weights. Finally, the reward and integral of the Bellman error approximation reinforce the adaptive mechanism of Actor neural weights. Closed-loop stability is guaranteed in the sense of Lyapunov, which leads to local exponential convergence of tracking errors based on integral sliding modes. Notably, it is assumed that dynamics are unknown, yet the control is continuous and robust. A representative simulation study shows the effectiveness of our proposal for tracking tasks.

In design matters, mechanisms with deformable elements are a step behind those with rigid bars, particularly if dimensional synthesis is considered a fundamental part of mechanism design. For the purposes of this work, a hybrid rigid–flexible four-bar mechanism has been chosen, the input bar being a continuum tendon of constant curvature. The coupler curves are noticeably more complex but offer more possibilities than the classical rigid four-bar counterpart. One of the objectives of this work is to completely characterize the coupler curves of this hybrid rigid–flexible mechanism, determining the number and type of circuits as well as constituent branches. Another important aim is to apply optimization techniques to the dimensional synthesis of path generation. Considerable progress in finding the best design solutions can be obtained if all the acquired knowledge about the coupler curves of this hybrid mechanism is integrated into the optimization algorithm.

General (α,β)-metrics form a rich and important class of Finsler metrics. Many well-known Finsler metrics of constant flag curvature can be lo-cally expressed as a general (α,β) metrics. In this paper, we study the general (α,β)-metrics with constant Ricci curvature (tensor) and constant flag curvature. Moreover, we study general (α; β) metrics with vanishing X-curvature. [ABSTRACT FROM AUTHOR]

We provide an extension of Obata's theorem to Finsler geometry and establish some rigidity results based on a second-order differential equation. Mainly, we prove that every complete simply connected Finsler manifold of positive constant flag curvature is isometrically homeomorphic to a Euclidean sphere endowed with a certain Finsler metric and vice versa. Based on these results, we present a classification of Finsler manifolds that admit a transnormal function. Specifically, we show that if a complete Finsler manifold admits a transnormal function with exactly two critical points, then it is homeomorphic to a sphere. [ABSTRACT FROM AUTHOR]

In this paper, a full-kinematic model and a visual-servo control system for recently introduced cooperative continuum robots (CCRs) are proposed. The focus will be on operative-supportive configuration which consists of two continuum robots (CRs), coupled to perform a positioning task. First, a hybrid kinematic modeling approach based on Denavit-Hartenberg method and Cosserat rod theory is developed. An efficient control system is also introduced within which the issues related to numerical Jacobian of Cosserat-based approach are addressed. A switching image-based visual servoing (IBVS) control strategy is proposed. The switching controller enables switching between the standard and proposed scaled-and non-dimensional (SND) IBVS sub-control systems. The stability of the proposed system in the sense of Lyapunov is also proved. Extensive simulations verified that SND IBVS is well capable of reducing large error signals at the beginning of the motion. In addition, the performance of the proposed switching approach, standard IBVS, and SND IBVS are compared to highlight the effectiveness of the proposed switching controller. Robustness analysis is also included here to show that even in the presence of imaging and actuation noise, the controlled system smoothly accomplishes the positioning task with an acceptable convergence speed. [ABSTRACT FROM AUTHOR]

Twisted and coiled polymer (TCP) is increasingly used for soft robots due to its good compaction. However, the lack of kinematics model of elastomer soft robots limits precise control. For this purpose, a kinematic model is proposed in this work for TCP-based elastomer soft robots. This paper has two main contributions. First, constant curvature assumption (CCA), the foundation in soft robots is conceptualized and the pioneering proof is constituted. Second, an innovative but foundational soft robot kinematic model is established based on CCA called constant curvature elastomer model (CCE). Experiments are conducted to verify the proposed kinematic model, showing the validity of the model. A soft endoscope for flaw detection in pipelines is shown as a practical case of TCP-based soft robots.

The purpose of this study is to obtain a characterization of unit speed spacelike curve with constant curvature and torsion in the Minkowski 3-space. According to this purpose, the position vector of a spacelike curve is expressed by a linear combination of its Serret Frenet Frame with differentiable functions. Since a spacelike curve has different kinds of frames, then we investigate the curve with respect to the Lorentzian casual characterizations of the frame. Hence we examine the results in three different cases including different subcases. Moreover, we illustrate some examples for each case. [ABSTRACT FROM AUTHOR]

We give sharp upper bounds on the injectivity radii of complete hyperbolic surfaces of finite area with some geodesic boundary components. The given bounds are over all such surfaces with any fixed topology; in particular, boundary lengths are not fixed. This extends the first author's earlier result to the with-boundary setting. In the second part of the paper we comment on another direction for extending this result, via the systole of loops function. [ABSTRACT FROM AUTHOR]

Aiming at the problem that the kinematic analytical solution cannot be obtained in the existing continuum robotic arm and its kinematic model, a tendon-driven continuum arm was presented with two bending degrees of freedom (DOFs), and its forward kinematics and inverse kinematics were studied. Based on the constant curvature hypothesis and combined with the modified Denavit-Hartenberg ( D-H) method, the mapping from the tendons' lengths to the arm end position was derived and the forward kinematic analytical solution was obtained. By extracting the arm backbone curve, the inverse kinematics was established using the geometric analysis method and the mapping from the arm end position to the tendons' lengths was derived. What's more, the partial inverse kinematics closed-form solution was obtained. Finally, the proposed model was validated by an experiment. The results indicate that the average position error of the proposed kinematics model is no more than 2. 56 mm, which accounts for less than 2.57% of the total arm length, and therefore the model can be well used for the end position control of continuum arm. [ABSTRACT FROM AUTHOR]

In this paper, we study Eta-Ricci soliton (η-Ricci soliton) on three dimensional N(k)-paracontact metric manifolds. We prove that the scalar curvature of an N(k)-paracontact metric manifold admitting η-Ricci solitons is constant and the manifold is of constant curvature k. Also, we prove that such manifolds are Einstein. Moreover, we show the condition of that the η-Ricci soliton to be expanding, steady or shrinking. In such a case we prove that the potential vector field is Killing vector field. Also, we show that the potential vector field is an infinitesimal automorphism or it leaves the structure tensor in the direction perpendicular to the Reeb vector field ξ. Finally, we illustrate an example of a three dimensional N(k)-paracontact metric manifold admitting an η-Ricci soliton. [ABSTRACT FROM AUTHOR]

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For any given k ∈ ℕ , this paper gives upper bounds on the radius of a packing of a complete hyperbolic surface of finite area by k equal-radius disks in terms of the surface's topology. We show that these bounds are sharp in some cases and not sharp in others. [ABSTRACT FROM AUTHOR]

The aim of the present paper is to provide an intrinsic investigation of special Finsler spaces of Hp-scalar curvature and of Hp- constant curvature. Characterizations of such spaces are given. Sufficient condition for Finsler space of Hp-scalar curvature to be of perpendicular scalar curvature is investigated. Necessary and sufficient condition under which a Finsler space of scalar curvature turns into a Finsler space of Hp-scalar curvature is shown. Further, certain conditions under which a Finsler manifolds of Hp-scalar curvature and of scalar curvature reduce to a Finsler space of Hp-constant curvature are obtained. Finally, various examples are studied and constructed. [ABSTRACT FROM AUTHOR]

In this paper, we explore the energy-based swing-up control for a soft robot equipped with an actuated constant curvature soft pendulum and an unactuated rotational base joint. The aim is to swing the robot up towards its upright equilibrium point (UEP) with the pendulum at an upright position. We establish a necessary and sufficient condition to protect the control law from singularities and demonstrate the robot's potential motion towards a homoclinic orbit or a closed-loop equilibrium point. After examining the robot's closed-loop equilibrium points, we generate formulae to compute all such points and introduce two conditions concerning control parameters, eliminating all but the UEP and downward equilibrium point (DEP) with the robot in a downward position. We prove the instability of the equilibrium points with negative gravitational potential energy. This obviates the need for one of the two conditions concerning such equilibrium points. We also prove the robot's linear controllability at the UEP. Our obtained results reveal that, regardless of its initial state, the soft robot can be swung-up towards its UEP using the energy-based controller, provided that it meets the proposed control parameter conditions, and can subsequently be balanced around the UEP using a locally stabilizing controller. Our simulation investigations validate the presented theoretical results. [ABSTRACT FROM AUTHOR]

We prove that the classical Lambert theorem about the elapsed time on an arc of Keplerian orbit extends without change to the Kepler problem on a space of constant curvature. We prove that the Hooke problem has a property similar to Lambert's theorem, which also extends to the spaces of constant curvature. [ABSTRACT FROM AUTHOR]

We prove Runge-type theorems and universality results for locally univalent holomorphic and meromorphic functions. Refining a result of M. Heins, we also show that there is a universal bounded locally univalent function on the unit disk. These results are used to prove that on any hyperbolic simply connected plane domain there exist universal conformal metrics with prescribed constant curvature. [ABSTRACT FROM AUTHOR]

Einstein–Hilbert action with a determinantal invariant has been considered. The obtained field equation contains the inverse Ricci tensor, ℜ α β . The linearized solution of invariant has been examined, and it is concluded that the constant curvature spacetime metric solution of the field equation gives different curvature constant for each values of σ. The σ = 0 gives a trivial solution for constant curvature, R 0 . [ABSTRACT FROM AUTHOR]

We study the geometry of conformal minimal two spheres immersed in G(2; 7; ℝ): Then we classify the linearly full irreducible conformal minimal immersions with constant curvature from S2 to G(2; 7; ℝ); or equivalently, a complex hyperquadric Q5 under some conditions. We also completely determine the Gaussian curvature of all linearly full totally unramified irreducible and all linearly full reducible conformal minimal immersions from S2 to G(2; 7; ℝ) with constant curvature. For reducible case, we give some examples, up to SO(7) equivalence, in which none of the spheres are congruent, with the same Gaussian curvature. [ABSTRACT FROM AUTHOR]