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3- R RR spherical parallel mechanisms (SPMs) have been extensively studied due to their numerous applications. Substantial effort has been devoted to their forward kinematics (FK), which is essential for their calibration and feedback control. However, despite their simple architecture, rather few 3- R RR SPMs with closed-form FK solutions (CFFKS) have been reported; iterative procedures are thus required in most cases. This paper presents three classes of 3- R RR SPMs with CFFKS, with the univariate polynomials for their FK being linear, quadratic, or quartic. These classes include a large set of designs, thereby enhancing the flexibility in selecting their architecture parameters. Moreover, they cover the majority of 3- R RR SPMs with special geometries that have been reported, while encompassing 3- R RR SPMs with certain special geometries yielding exceptional features such as unlimited rotation capacity about certain directions. Notably, these formulations are also applicable to many SPMs with alternative topologies and certain parallel mechanisms of other types. This work expands the family of SPMs with CFFKS, highly desirable in many practical applications. • Three novel classes of analytic 3- R RR spherical parallel mechanisms are proposed. • Various formulations are proposed for their forward kinematics. • Their characteristic equations are derived, which can be linear, quadratic or quartic. • They cover the majority of 3- R RR SPMs with special geometries that have been reported. [ABSTRACT FROM AUTHOR]

• An algebraic synthesis method is proposed to construct novel single-loop 6R spatial mechanisms. • The mechanisms can achieve constant velocity transmission between designated input/output axes. • All the floating joints of the single-loop 6R spatial mechanisms rotate around fixed axes. • The transmission quality of the mechanisms is optimized within a specified range of motion. • Transmission characteristics of the single-loop mechanisms are verified by kinematics analysis. This paper addresses the algebraic synthesis of novel single-loop 6R spatial mechanisms, enabling constant -1:1 or 1:1 velocity ratio transmission between two adjacent axes, whether parallel, intersecting, or skew. Based on the motion polynomial over dual quaternions, an algebraic synthesis method including four steps is presented to construct and optimize single-loop 6R spatial mechanisms with a constant transmission ratio of -1:1 or 1:1 between arbitrarily designated input and output axes. Using this method, several novel single-loop 6R spatial mechanisms for constant velocity transmission are constructed by designating different poses and rotation directions of the input and output axes. Kinematics analysis of single-loop 6R spatial mechanisms is implemented to verify their transmission characteristics. The results reveal that the generated 1-DOF single-loop 6R novel mechanisms can indeed transmit motion with a constant transmission ratio of -1:1 or 1:1 between two adjacent parallel, intersecting, or skew axes. This work provides a framework for further investigation on single-loop mechanisms with special transmission characteristics. [ABSTRACT FROM AUTHOR]

• Constant-height deployable mechanisms are proposed with only revolute joints. • Deployable ring trusses for satellite mesh reflector antennas are constructed. • Mobility properties of deployable mechanisms are substantiated through screw theory. • The kinematic analysis is presented to investigate the deployable characteristics. • Prototypes of a deployable unit and a ring truss are built to validate the mechanisms. Deployable mechanisms play an increasingly crucial role in aerospace, yet their design still faces challenges: 1) The folded height increases while its deployed height decreases, which is not conducive to improving the folding performance; 2) Compared with revolute joints, prismatic joints have relatively weak reliability in complex space environments. To address the above challenges, novel constant-height deployable mechanisms with only revolute joints are proposed in this paper. The deployable mechanisms exhibit one degree of freedom, enabling them to attain modular expansion. Subsequently, the mobility properties of these deployable mechanisms are substantiated through screw theory. Assembly strategies for constant-height deployable mechanisms are proposed, and deployable ring trusses are further constructed to provide support structures for satellite mesh reflector antennas. The kinematics are analyzed to investigate the deployable characteristics of the proposed deployable mechanisms and ring trusses. The prototypes of a deployable mechanism and a ring truss are established to validate the designed deployable mechanisms. This work contributes novel types of deployable mechanisms with potential advantages, aiming to function as valuable references for the design and analysis of deployable trusses. [ABSTRACT FROM AUTHOR]

Cable-driven Continuum Robots (CCRs) represent a subset of flexible robotic systems with diverse applications encompassing medical, quality control, and search and rescue. Multi-segmented CCRs can independently actuate different CCR segments and deform into a series of connected circular arcs. While this method generates constant-curvature profiles, CCRs with varying radial cable offsets can generate shapes with varying curvatures. However, there are no kinematic models for such robots in the literature. This paper introduces optimization-based formulations to accurately estimate the deformed profile of CCRs, considering both variable cable offset and multi-segment configurations. The presented kinematic approach can predict the deformed profiles with an RMS error less than 5% of the total length of the robot. The application of the proposed method to determine the inverse kinematics of multi-segmented CCRs is demonstrated—even considering obstacles in the robot's workspace. Practical implementation of the formulation is illustrated using a 3-segmented CCR with a cable-driven soft gripper that imitates biological grippers such as an elephant's trunk or creeper vine to grasp objects. • Profile estimation for variably routed cable-driven continuum robot. • Forward and inverse kinematics of multi-segmented cable-driven continuum robot. • Experimental validation with and without an obstacle in robot workspace. • Example application of robot with soft gripper generating spiral profile for gripping. [ABSTRACT FROM AUTHOR]

• Bearing kinematics and friction are solved applying power minimization method. • Classical force equilibrium method and power minimization method are compared. • Friction calculation through power minimization presents a convex convergence. • Power minimization method has better performances than traditional method. • Power distribution within contact ellipses is represented. Two-point contact bearings are widely used in industry. In space applications, where the speed of the mechanisms is low, the main source of friction in the bearings is the one produced at the ball-raceway contact ellipses. This friction affects the motorization margin which is a key aspect of the operational envelope in space mechanisms. The traditional method to estimate this friction works by trying to find the force equilibrium for each ball. For that, an iterative sequence is arranged with six parameters defining the ball kinematics under the action of normal and friction forces assuming a constant friction coefficient. The force equilibrium model is reviewed analyzing the influence of the speed, and the implications of quasi-static assumption are assessed. For this last assumption, an energetic method is proposed established upon the minimization of the friction power generated within the contact patches. This model provides the same results as the force equilibrium model with a stronger convergence and a simple computational implementation. Finally, the two methods are compared obtaining better performances for the power minimization method. This work represents a first step in order to substitute force equilibrium methods with more stable energy based methods in bearing friction calculations. [ABSTRACT FROM AUTHOR]

• The coupling-constraint properties of single-loop kirigami are investigated. • A synthesis method of 1-degree-of-freedom multi-loop kirigami is proposed. • The forward & inverse design method of kirigami meshes are proposed. Kirigami has been utilized in designing deployable mechanisms for various engineering applications. This paper presents a novel construction method for one-degree-of-freedom (1-DOF) multi-loop kirigami mechanisms whose folding motions are kinematically equivalent to their original degree-4 rigid origami. Firstly, we introduce a family of single-loop kirigami (SLK) as the main loop of 1-DOF multi-loop kirigami. Secondly, the coupling-constraint properties of multi-DOF SLKs are clarified, and the structural characteristics of the 1-DOF multi-loop kirigami are defined, based on which, the construction method for the 1-DOF multi-loop kirigami is proposed. Thirdly, the representative models are selected and their kinematics are presented to demonstrate their capability to achieve folding motion equivalent to their origami prototypes, thus verifying the validity of our proposed methodology. Fourthly, the non-constraint condition of the 1-DOF multi-loop kirigami is presented. Finally, we propose extended configurations of 1-DOF multi-loop kirigami and provide the forward and inverse design methods for constructing kirigami meshes. The proposed methodology facilitates the design of large-scale deployable structures. [ABSTRACT FROM AUTHOR]

• A bistable kirigami topology decouples energy barrier and kinematic characteristics. • Parametrized kirigami dataset contains rich nonlinear responses is established. • Multi-objective inverse design in kirigami is achieved using ML-GA approach. • Programmable multistable kirigami chains realize target mechanical properties. Arraying bistable cells in a chain results in a multistable mechanical structure that is capable of sequential deformation paths and structural reconfigurability. However, to customize the multistability, for example, a prescribed energy barrier with variable snapping force/displacement, is constrained by the difficulty of decoupling energy from kinematic characteristics since they are always governed by the same geometric parameters. This paper proposes a bistable kirigami cut topology to realize decoupling. Cut parameters are classified into independent groups, corresponding to objective energy barrier or force/displacement, which are consequently independent also. On this basis, a machine learning (ML) and genetic algorithm (GA) based approach is presented to guide the inverse design of the nonlinear responses. Optimal individual kirigami cells that meet decoupled multi-objectives such as energy barrier, snapping force and stable displacement are obtained to construct kirigami chains. The conceivable deformation sequences, configurations and energy/force-displacement curves are validated by both the simulation and experimental results. The proposed mechanism decoupled strategy and inverse design framework open up new horizons for programming energy landscape and provide a general recipe for tailoring multistability in metamaterials. [Display omitted] [ABSTRACT FROM AUTHOR]

Cable-Driven Parallel Robots (CDPRs) displace the end-effector (EE) by means of cables configured in a parallel fashion. When a CDPR employs fewer cables than the degrees of freedom (DoFs) of its EE, it is underactuated and underconstrained. As a result, only a limited subset of the EE DoFs can be controlled assigning specific cable lengths, and determining the EE pose through direct kinematics (DK) is unfeasible, since it also depends on mechanical equilibrium. To estimate the EE pose, then, it may be convenient to acquire and combine redundant measurements. This paper introduces a sensor fusion technique that aims at achieving the optimal estimation of the pose of a generic UACDPR. Sensor fusion is achieved by an iterative nonlinear weighted least-square algorithm, which is solved by a Gauss–Newton-like method. A novel criterion for terminating iterations, which is physically sound and straightforward to implement, is put forward. Different redundant measurements are experimentally compared, so as to show the performance of the method with different sensors involved. • The pose of an underactuated cable-driven parallel robot is estimated. • Redundant measurements of lengths and angles are acquired and combined. • A sensor fusion technique for the optimal estimation of the pose is proposed. • An iterative nonlinear weighted least-square Gauss–Newton-like method is employed. [ABSTRACT FROM AUTHOR]

• LegUp robot enables the motion of the lower limb in the sagittal and coronal planes. • The mathematical models for legup are representative for the rehabilitation task. • Numerical optimizations lead to a singularity-free operational workspace. The paper presents the mathematical modeling and analysis of LegUp, a novel parallel robotic system for lower limb rehabilitation of bedridden patients. The operational workspace of the robotic system is defined based on a set of parameters that describe the motion of the lower limb joints, which is natural in the rehabilitation task. To comply with this representation of the operational workspace, the parallel robot kinematic models describe the dependency between the robotic system actuators and the lower limb joints. Furthermore, the singularity analysis is achieved in the joint space, which shows whether the operational workspace is singularity-free. To achieve a feasible mechanical design for the prescribed operational workspace and to ensure safe operation, the design parameters of the parallel robotic system are determined based on a multi-objective optimization problem. Numerical simulations show that the operational workspace is singularity-free for the selected design parameters, which are then used to construct the experimental model of LegUp. [ABSTRACT FROM AUTHOR]

Design of one-degree-of-freedom (1-DOF) mechanisms is of paramount interest. This work deals with the generation of exact regular polygonal paths by using two particular 1-DOF mechanisms. The design of two-bar and three-bar mechanisms including non-circular gears is presented and evaluated. Differences in the kinematics of both mechanisms are discussed. The three-bar mechanism allows derivable velocity and acceleration curves during the whole trajectory, including the polygon vertices, a feature that cannot be achieved with the simpler two-bar mechanism. Hence, the three-bar mechanism makes it possible to generate perfect vertices using non-circular gears. To the best of the authors' knowledge, this mechanism is the first 1-DOF mechanism with articulated links and a non-circular gear transmission that can generate exact regular polygonal paths. The proposed mechanisms can also be applied to generate two straight lines with a given angle, which is another novel contribution of this work. A prototype of the three-bar mechanism has been developed for experimental validation. • A 1-DOF mechanism that can generate exact regular polygonal paths is presented. • Solutions with two- and three-bar mechanisms are analyzed. • The mechanism can also generate two exact straight lines with a given angle. • Non-circular gears are used to generate the required non-linear transmission. [ABSTRACT FROM AUTHOR]

• A novel hybrid scara robot with large workspace and low inertia is proposed. • The robot can obtain a larger workspace based on a compact structure. • The robot has closed-form kinematics with convenience of control. • A physical prototype of the robot has been developed to verify the related design. This paper presents a novel four-degree-of-freedom (4-DOF) hybrid robot named Hybrid-Long-Scara robot, which has the advantages of large workspace and low inertia. First, the kinematic mechanism of the new robot is designed based on requirements of its DOFs and actuation joints. Second, the closed-form displacement and velocity models of the new robot are established. Third, the workspace of the Hybrid-Long-Scara robot is quantitatively analyzed and the advantages of the new robot are visually shown. At last, a physical prototype is developed to verify the effectiveness of the mechanical design and kinematic model of the Hybrid-Long-Scara robot. Comparing with a typical Scara robot, the new robot can obtain larger vertical workspace based on a compact structure. On the other hand, comparing with a typical palletizing robot, the new robot can obtain higher motion speed, has a more compact height dimension, and can applied to wider operation tasks. With those characteristics, the Hybrid-Long-Scara robot has a good application prospect in high-speed sorting, grabbing, unstacking and packing in large workspace. [ABSTRACT FROM AUTHOR]

• Instant centers and lines of centers for one-degree-of-freedom and two-degree-of-freedom planar linkages with up to 11 links. • Graphical methods to locate the secondary instant centers and the lines containing these instant centers. • An investigation of original methods to locate the secondary instant centers of 1-dof 8-bar linkages and 10-bar linkages and the lines of centers of 2-dof 9-bar linkages and 11-bar linkages. • Geometric insight into the first-order instantaneous kinematics of 1-dof and 2-dof planar linkages. This paper presents a detailed study of the instantaneous centers of velocity (henceforth referred to as instant centers) and the lines containing the instant centers for one-degree-of-freedom (1-dof) and two-degree-of-freedom (2-dof) planar linkages with up to 11 links. The focus is on graphical methods to locate secondary instant centers (that is, instant centers which are unknown from inspection and require graphical construction) or the lines containing secondary instant centers. The secondary instant center for a pair of links with one relative degree of freedom is a unique point, and for a pair of links with two relative degrees of freedom the secondary instant center must lie on a unique line. For a 2-dof linkage, the problem of finding the unique locations of the secondary instant centers on these lines is not addressed in this paper (this would correspond to knowing the ratio of the two input velocities). The paper reviews known graphical methods for the 4-bar, 5-bar, 6-bar, and 7-bar linkages and then presents original graphical methods to locate the secondary instant centers for the 1-dof 8-bar and 10-bar linkages, and the lines containing the secondary instant centers for the 2-dof 9-bar and 11-bar linkages. [ABSTRACT FROM AUTHOR]

The roots of the closure polynomial associated with a given mechanism determine its assembly modes. In the case of 6R closed-loop mechanisms, these polynomials are usually expressed in the half-angle tangent of one of its joints. In this paper, we derive closure polynomials of 6R robots in terms of distances, not angles. The use of a distance-based formulation provides, in general, a fundamental advantage since it leads to closure conditions without requiring neither variable eliminations nor variable substitutions. We restrict our attention, though, to robots with coplanar consecutive joint axes, i.e. , robots whose consecutive axes intersect at either proper or improper points. We show that this particular arrangement of joints does not result in a reduction in the maximum number of the inverse kinematic solutions with respect to the general case. Moreover, this family of robots include broadly used offset-wrist arms. For instance, in this paper, we obtain closure polynomials for robots such as the FANUC CRX-10iA/L, the UR10e, and the KUKA LBR iiwa R800 robot in generic form (i.e. , as a function of their end-effector locations). • The inverse kinematics of lobster arms is solved using distance geometry. • Closure polynomials are obtained in terms of distances. No angles are involved. • Neither variable eliminations nor variable substitutions are required. • The method is exemplified on a wide variety of cobots. [ABSTRACT FROM AUTHOR]

In the last decades, many efforts have been made to analyze and model small and large deflections of flexures, considering complex load cases and different solution techniques. However, few investigations focused on the kinematic aspects related to the deflection analysis of the flexible elements, and limited the study to the second-order kinematics. In this paper, an analytical formulation based on the instantaneous geometric invariants is developed to give deep kinematic insight, up to the fourth order, into the motion generated by the deflections. The problem is addressed from a geometrical point of view, defining the fundamental geometric entities that characterize the motion, that are inflection circle, cubic of stationary curvature and its derivative, Ball's point, and Burmester's points. Thanks to these special points on the plane, straight and circular paths can be approximated to the third and to the fourth order, respectively. The proposed formulation defines the geometric characteristics of flexures with different curvatures. An application regarding the definition of pseudo-rigid body models is discussed. Finite element simulations are performed to validate the results. • Geometric invariants are exploited to describe the motion of flexures. • Flexures with different curvatures are considered. • Geometric characteristics of motion are defined up to the forth order. • Ball's and Burmester's points have been determined. • High-accuracy PRBMs are defined. [ABSTRACT FROM AUTHOR]

The inverse kinematics (IK) problem of cable-driven manipulators with pure rolling joints (CDM-PRJs) presents unique challenges due to the equal angle constraints and strict joint limits. These factors render existing IK solvers ineffective or result in substantial degradation of their performance. To address these challenges, we propose a three-phase geometric iterative method to efficiently solve the IK problem for CDM-PRJs, which is a variant of the forward and backward reaching inverse kinematics method. Our method begins by establishing a surrogate model for CDM-PRJs to handle the equal angle constraints. Subsequently, we introduce a geometric iterative approach comprising a forward reaching phase, a backward reaching phase, and a state update phase. Additionally, we devise two novel measures, the random disturbance measure and the branch change measure, to effectively address deadlock situations and asymmetric joint limits, respectively. Simulation results demonstrate that our method surpasses the inverse Jacobian method and the sequential quadratic programming method in terms of success rate and computational efficiency. Furthermore, our method exhibits generality across various CDM-PRJs. • The kinematics of cable-driven manipulators with pure rolling joints is studied. • A three-phase geometric iterative method for inverse kinematics is proposed. • A random disturbance measure is presented to quickly detect and escape deadlocks. • A branch change measure is given to solve the problem of multi-branch solutions. • Results show the high efficiency and generality of the proposed method. [ABSTRACT FROM AUTHOR]

• New methods for planar and spatial jerk and jounce were developed and validated for multibody dynamics. • Precision for jerk and jounce is equivalent to velocity and acceleration. • These new equations are significantly more accurate than finite difference approximations. • Assembly tolerance of 1e-12 is recommended. • Kinematics developed for this paper are required to explore hyper-dynamic differential algebraic equations created by first and second derivatives of equations of motion for multibody systems using joint constraint methods. Planar and spatial joint constraint equations for jerk and jounce were derived for multibody dynamics. Exemplar derivations are provided. Results from kinematically driven numerical simulations of these new equations were compared to explicit geometric solutions for planar four-bar and inverted slider-crank mechanisms as well as spatial revolute-spherical-universal-revolute and revolute-spherical-prismatic-universal mechanisms. Root-mean-square-error between velocity, acceleration, jerk and jounce simulations versus explicit solutions were normalized by maximum absolute values for comparison. Relative precisions for new jerk and jounce computations were equivalent to relative precisions for extant velocity and acceleration computations. The new equations were significantly more accurate than finite difference approximations previously required for jerk and jounce. Third and fourth order kinematics developed for this paper are required to explore hyper-dynamic differential algebraic equations created by first and second derivatives of equations of motion for multibody systems using joint constraint methods. [ABSTRACT FROM AUTHOR]

• Instantaneous Kinematics (IK) of multi-DOF spatial mechanisms(SMs) is addressed • New geometric and analytic techniques for the analysis of this IK are presented • They relate multi-DOF SM's IK to the IK of the single-DOF SMs generated from it • They use Inst.-Screw-Axes (ISAs) and are implementable in a CAD system • The proposed methodology is relevant in mechanism design Multi-degree-of-freedom (multi-DOF) mechanisms generate single-DOF mechanisms by locking all their generalized coordinates but one. Here, the superposition principle is used to state a relationship between spatial multi-DOF mechanisms' instantaneous kinematics (IK) and the IK of the single-DOF mechanisms they generate. Firstly, the relationship between the instantaneous screw axes (ISAs) of a multi-DOF mechanism and the ISAs of the single-DOF mechanisms, it generates, is found; then, it is used for its singularity analysis. In particular, the IK model of a generic multi-DOF spatial mechanism is written through the ISA locations and, successively, it is studied to identify all the singular configurations of the multi-DOF mechanism through the analysis of the single-DOF mechanisms it generates. The results are a technique for the determination of ISAs' locations in multi-DOF spatial mechanisms and a singularity-analysis technique, for the same mechanism types, based on the singularity analysis of single-DOF spatial mechanisms. Eventually, the proposed techniques are applied to a case study. As far as this author is aware, both these results are presented for the first time in the literature. [ABSTRACT FROM AUTHOR]

The authors report on the kinematics and workspace analysis of a six-dof 3- C R(Pa)(Pa)R parallel mechanism (PM) with an orthogonal layout, which is found to bear various advantageous features. A compact forward-kinematics model is established, leading to a linear univariate characteristic polynomial, significantly reducing the computational cost. Moreover, the velocity-level kinematics indicates that the rotation and translation of the moving platform (MP) are decoupled, and the mechanism under study can achieve isotropy upon proper dimensioning; furthermore, the forward and inverse singularities are found to be solely dependent on the MP orientation, which can be characterized by six elliptical arcs in the orientation workspace; the study also reveals that the MP can theoretically rotate about any axis for at least ± 90 ° before encountering these singularities. Finally, its interference-free workspace is evaluated, indicating that the mechanism under study can offer a relatively large orientation workspace. The foregoing features make the proposed PM potentially attractive in a wide range of applications. • A 3- C R(Pa)(Pa)R PM with an "orthogonal layout" is studied in-depth. • A forward-kinematics model yielding a linear characteristic polynomial is established. • The robot bears decoupled motion, which can become isotropic with proper dimensions. • Six elliptical arcs fully characterize its forward and inverse singularities. • It offers a relatively large interference-free orientation workspace. [ABSTRACT FROM AUTHOR]

Designing and analyzing large cable-driven parallel robots (CDPRs) for precision tasks can be challenging, as the position kinematics are governed by kineto-statics and cable sag equations. Our aim is to find all equilibria for a given set of unstrained cable lengths using numerical continuation techniques. The Irvine sagging cable model contains both non-algebraic and multi-valued functions. The former removes the guarantee of finiteness on the number of isolated solutions, making homotopy start system construction less clear. The latter introduces branch cuts, which could lead to failures during path tracking. We reformulate the Irvine model to eliminate multi-valued functions and propose a heuristic numerical continuation method based on monodromy, removing the reliance on a start system. We demonstrate this method on an eight-cable spatial CDPR, resulting in a well-constrained non-algebraic system with 31 equations. The method is applied to four examples from literature that were previously solved in bounded regions. Our method computes the previously reported solutions along with new solutions outside those bounds much faster, showing that this numerical method enhances existing approaches for comprehensively analyzing CDPR kineto-statics. • We introduce a numerical continuation method for solving the kinetostatics of CDPRs. • The Irvine sagging cable model is reformulated to eliminate multi-valued functions. • We propose a new heuristic monodromy algorithm to solve non-algebraic systems. • We solve an eight-cable spatial CDPR, a well-constrained system with 31 equations. • The new approach finds solutions more quickly than previously published methods. [ABSTRACT FROM AUTHOR]

A modular approach to the kinematics of planar linkages is based on a structural analysis of the mechanisms. Kinematic chains that can be synthesized according to Assur principle are well classified by their structure. Such a classification allows the development of general methods of the kinematics of subchains that can be algorithmized and used for computer-aided design. The paper introduces a new method of the position analysis of Assur dyads based on vector algebra tools. Obtained in unified vector form formulas give the opportunity to calculate the radius-vectors of all points of the dyads in the standard basis for the three-dimensional space. The method can be applied to the position, complete kinematic and kinetostatic analyses of planar linkages of the 2nd class according to Artobolevsky. • The vector based approach for position analyses of Assur dyads is presented. • Chace vector equation is applicable for position analyses of Assur dyads. • The kinematics of planar linkages of the 2nd class by Artobolevsky is covered. [ABSTRACT FROM AUTHOR]

This paper aims at exploiting the unique characteristics of instantaneous invariants and pole locations in synthesizing motion generation four bar (4R) planar mechanisms with higher-order coupler point kinematics (i.e. specifying coupler point velocity and/or acceleration in addition to its position and coupler orientation). The proposed methodology targets to ease the complexity of such design problems by reducing the number of non-linear governing equations that are solved to generate the desired dyads of the final 4R mechanism design. The new approach integrates graphical and analytical techniques to synthesize 4R mechanisms with prescribed coupler point position, velocity, and acceleration. Two case studies are presented; the first presents the synthesis of a motion generation 4R with 3 prescribed coupler point positions and 2 prescribed coupler point velocities; the second presents the synthesis of a motion generation 4R with 2 prescribed coupler point positions, 2 prescribed coupler point velocities, and 1 prescribed coupler point acceleration. When compared to classical kinematic-based approaches, the proposed method prominently reduces the size and complexity of the set of governing non-linear equations. • New graphical/analytical technique for higher-order coupler point kinematics. • Exploits the unique characteristics of instantaneous geometric invariants. • Generates a prominently more concise set of governing non-linear equations. • Two case studies are presented and validated via classical kinematics techniques. [ABSTRACT FROM AUTHOR]

This study presents a disturbance torque observer-based variable impedance control method to mitigate the disturbance force acting on the 2-degree-of-freedom (DOF) transformable wheel mechanism. Compared to the previous study, the numerical solution of forward kinematic problem is calculated over the entire operating ranges of joint angles and the viability of dynamic model for the proposed transformable wheel is verified via the simulation study with the PD-based feedback. By tabularizing the numerical solution of kinematics, the complexity of forward kinematic problem of transformable wheel has been reduced to implement the disturbance torque observer in real time. In combination with the disturbance torque observer, the variable impedance control method is suggested, whose stiffness can be adjusted for mitigating the disturbance force acting on the transformable wheel and recovering its desired shape before the next step. The stability of proposed variable impedance control is proved with Lyapunov method without constraint on the stiffness of impedance model so that the adjustment of stiffness can be heuristically constructed. The extensive experiments using the proposed variable impedance control equipped with disturbance torque observer are carried out to demonstrate its excellent ability to mitigate the disturbance force acting on the transformable wheel during its stair-descending. • The solution of forward kinematics of transformable wheel is specifically calculated. • The disturbance torque observer is effectively implemented by streamlining the complexity of forward kinematics. • The stability of proposed variable impedance control in the Joint Space is proved under reasonable assumptions. • The performance of disturbance-mitigation of proposed method is experimentally validated. [ABSTRACT FROM AUTHOR]

Henry Léauté (1847–1916) was a French kinematician who taught at the École polytechnique. In 1878, Léauté derived the cubic of stationary curvature (CSC) equation in its modern form which is widely applied as a tool for synthesizing path generator mechanisms. The deduction of CSC equation by Léauté is based on an earlier work from the French school due to A. Mannheim in 1872. According to historical bibliographical evidence, Léauté and Mannheim are recognized as the true discoverers of CSC equation. However, the subsequent literature on CSC equation does not acknowledge the aforementioned French contributions, by not giving them credit. In the present work, early investigations on CSC is reviewed by quoting excerpts from original works and the precedence of discovery to Léauté is clarified. Then, the mathematical treatment of Léauté is elaborated, followed by the discussion of a graphical procedure for tracing the CSC through an example. A brief review of other scientific works of Léauté is also included. At the end, it is recommended to consider terming the CSC equation alternatively as "Mannheim-Léauté equation". • Path curvature analysis. • Cubic of stationary curvature equation. • History of kinematics. [ABSTRACT FROM AUTHOR]

• Constraint reduction of multi-loop overconstrained mechanisms is proposed. • Hamiltonian-path based method is for the first time applied in mechanism analysis. • Kinematic equivalence after mechanism reduction is formulated and verified. Most of the deployable polyhedral mechanisms (DPMs) are multi-loop overconstrained mechanisms that causes barriers for their applications due to the issues in assembly, operation and control. Yet, constraint reduction for these multi-loop overconstrained mechanisms is extremely challenging in kinematics. In this paper, by introducing the Hamiltonian path to investigate the 3D topological graphs of a group of Sarrus-inspired DPMs, we propose a systematic method for constraint reduction of multi-loop overconstrained DPMs. We demonstrate that through the removal of redundant joints with the assistant of tetrahedral Hamiltonian path, one equivalent simplest topological graph of tetrahedral mechanism is identified as a reduction basic unit. Subsequently, one simplest form of Sarrus-inspired cubic mechanism is obtained by investigating two Hamiltonian paths of its dual octahedron and sequentially arranging basic units. Furthermore, a total of nineteen simplest forms of Sarrus-inspired dodecahedral mechanisms are identified from seventeen Hamiltonian paths of its dual icosahedron. The overconstraints in each simplest Sarrus-inspired DPM are greatly reduced while preserving the original one-degree-of-freedom (DOF) motion behavior. The method proposed in this paper not only lays the groundwork for further research in wider deployable polyhedrons, but also inspires the reduction of other multi-loop overconstrained mechanisms, with potential applications in the fields of manufacturing, architecture and space exploration. [ABSTRACT FROM AUTHOR]

The dual quaternion plays an advantageous role in the field of robotic kinematics and dynamics due to its compact expression. It is well known that quaternion can be easily implemented as algebraic operations on vectors through matrix operators. Similarly, the dual matrix operators can convert dual quaternion operations to matrix operations. However, it has not received adequate attention, and thus there are very few applications at present. With the dual matrix operators of dual quaternion, this paper re-verifies the equivalency between the conjugate formula of unit dual quaternion and dual Euler–Rodrigues formula. To combine this equivalence with the homomorphic mapping of Lie groups, a theoretical correlation of the current hand-eye calibration methods is established. Motivated by the attempt to establish a more complete scheme for hand-eye calibration, a new simultaneous method based on the conjugate formula of dual quaternion is proposed. The method verifies that the rotation problem can be extended to the rigid-body transformation problem via the dual algebra. Finally, the simulation and experimental validations for the hand–eye calibration method are given. • The conjugate formula of unit dual quaternion is re-verified via the dual operations. • A new simultaneous method is defined for hand-eye calibration. • The theoretical correlation with the existing methods is established. [ABSTRACT FROM AUTHOR]

• A new 3T1R parallel manipulator with active and passive revolute joints. • Revolute joints are simpler and preloaded to raise stiffness and eliminate clearance. • Kinematic chains are placed symmetrically to achieve homogenous workspace. • The manipulator has low compliance and high workspace-to-footprint ratio. Pick-and-place automation requires manipulators to have at least three translational (3T) and one rotational (1R) degrees of freedom. Parallel manipulators usually include four limbs connected to a traveling plate to generate 3T1R motion. For the four limbs to be arranged with rotational symmetry, complicated types of passive joints and traveling plates are required, which make 3T1R manipulators structurally weaker and prone to wear and clearance issues. This paper presents a new 3T1R parallel manipulator that uses only revolute joints as the active and passive joints. Compared with other joint types, revolute joints are structurally simpler and stronger. They have an unlimited range of motion and can be preloaded to eliminate clearance. The proposed manipulator allows the four limbs to be connected to a simple H-shaped traveling plate with rotational symmetry. Kinematic analysis shows that the new 3T1R manipulator has a homogenous workspace with a high workspace-to-footprint ratio. Compliance analysis and experiments are given to verify that the manipulator can achieve low and homogenous compliance. The proposed manipulator is expected to provide an alternative in pick-and-place applications. [ABSTRACT FROM AUTHOR]

• A new method for constructing a family of Sarrus-inspired deployable polyhedral mechanisms is proposed. • Novel deployable tetrahedral, cubic and dodecahedral mechanisms with symmetric transformability are presented. • The equivalent analysis strategy for mobility of multiloop polyhedral mechanisms is proposed Deployable polyhedral mechanisms (DPMs) have witnessed flourishing growth in recent years because of their potential applications in robotics, space exploration, structure engineering, and so forth. This paper firstly presents the construction, mobility and kinematics of a family of Sarrus-inspired deployable polyhedral mechanisms. By carrying out expansion operation and implanting Sarrus linkages along the straight-line motion paths, deployable tetrahedral, cubic and dodecahedral mechanisms are identified and constructed following tetrahedral, octahedral, and icosahedral symmetry, respectively. Three paired transformations with synchronized radial motion between Platonic and Archimedean polyhedrons are revealed, and their significant symmetric properties are perfectly remained in each work configuration. Subsequently, with assistant of equivalent prismatic joints, the equivalent analysis strategy for mobility of multiloop polyhedral mechanisms is proposed to significantly simplify the calculation process. This paper hence presents the construction method and equivalent analysis of the Sarrus-inspired DPMs that are not only valuable in theoretical investigation, but also have great potential in practical applications such as mechanical metamaterials, deployable architectures and space exploration. [ABSTRACT FROM AUTHOR]

Cable-Driven Parallel Robots (CDPRs) move an end-effector (EE) using cables arranged in a parallel fashion. If a CDPR employs fewer cables than its EE degrees of freedom (DoFs), the robot is generally underactuated and underconstrained. Consequently, only a subset of the EE DoFs can be assigned for trajectory planning purposes, and the EE pose cannot be inferred by only relying on forward kinematics. Consequently, it is not trivial to assess the robot workspace (WS), even though WS computation is of paramount importance in analyzing the robot's performance. This paper introduces a novel algorithm for the computation of the reachable static WS of generic underactuated CDPRs , namely the set of EE positions that are statically attainable with at least one orientation and characterized by positive and bounded cable tensions. The algorithm leverages a novel geometrico-static problem, which, given a candidate EE position, seeks an orientation satisfying a stable static equilibrium characterized by a desired tension distribution. • Reachable static workspace of underactuated cable-driven robots is investigated. • A novel inverse geometrico-static model is proposed for workspace computation. • The inverse model Jacobian and the workspace computation methods are provided. • An application to trajectory planning shows the merits of the proposed approach. [ABSTRACT FROM AUTHOR]

• Two motion-decoupling methods based on cable length compensation are proposed. • A 7-DOF cable-driven spray-painting robotic system is developed. • An improved non-singular inverse kinematics approach is introduced. • Experiments are conducted to verify the decoupling performance of the robot. This paper presents the design and implementation of a 7-DOF cable-driven serial spray-painting robot (CDSSR) featuring motion-decoupling mechanisms and introduces an improved non-singular inverse kinematics method for redundant serial robots. In the study, first, the motion-coupling characteristics of cable-driven joints and motion-decoupling methods based on translational and rotational compensation were examined. Next, the mechanical structure of the CDSSR was designed, including motion-decoupling mechanisms, cable-driven differential wrists, and tension-amplification mechanisms. In addition, an improved non-singular inverse kinematics approach incorporating dual-arm angles and configuration control parameters was developed. Finally, an experimentally scaled prototype of the proposed robot was constructed, and motion decoupling and spray trajectory planning experiments were conducted. The experimental and simulation results demonstrate the effectiveness of the non-singular inverse kinematics method in achieving efficient inverse kinematics solutions under global configuration control. Furthermore, the motion decoupling mechanisms designed for the serial robot successfully accomplished motion decoupling. [ABSTRACT FROM AUTHOR]

This research aims to investigate the instantaneous kinematics of the terminal link of a planar two-link open chain using the canonical coordinate system of planar two-parameter motion. It is based on the complex-number technique. We review the higher-order instantaneous invariants of this motion of the terminal link with reference to the first and second-order instantaneous invariants. We apply these instantaneous invariants in complex number form to curvature theory. Finally, we give some examples by choice of different parameters and exhibit the comparisons. • Complex number form of instantaneous invariants of planar two-link open chain in curvature theory. • Complex number formulation of the inflection circle of two-parameter motions. • Complex number formulation of the cubic of stationary curvature of two-parameter motions. • Complex number formulation of the Ball points of two-parameter motions. [ABSTRACT FROM AUTHOR]

Kinematic singularities are configurations where the configuration space (c-space) V is not smooth or configurations where the rank of the constraint Jacobian drops while the c-space is locally smooth. Both can be analyzed by means of a higher-order analysis, for which the kinematic tangent cone provides a conceptual framework. It is desirable to complement such an analysis with more insight into the origin of specific singular phenomena. Such additional insight is obtained by investigating the kinematics from an algebraic viewpoint. Points of interest are then the singularities of a variety, referred to as algebraic singularities. Algebraic singularities are reflected as kinematic singularities or as shakiness. It is shown that these algebraic singularities can be attributed to properties of the ideal generated by the loop constraints — the constraint ideal. Particular situations arise depending on whether this constraint ideal is radical. If it is not radical, a singular branch is locally defined by a primary ideal. If it is radical, it must be distinguished whether real and complex dimensions of intersecting branches are different. Real intersections indicate kinematic singularities; in all other cases, the mechanism may be in a singularity or may be shaky. Various examples are presented. • Kinematics formulated algebraically with dual quaternions. • The ideal generated by the constraints is used for the analysis. • Singularities are distinguished when the constraint ideal is radical or not. • Shakiness is related to non-radical constrain ideal. • Shakiness is also related to different real and complex dimensions of the variety. [ABSTRACT FROM AUTHOR]

Kinematic redundancy offers robots many useful capabilities, such as an ability to fulfill several tasks simultaneously. However, the inverse kinematics (IK) problem for redundant manipulators is not trivial. Often, pseudoinverse-based methods are used, allowing for the projection of the desired joint space velocity vector on the null space of the manipulator's Jacobian. Unfortunately, it is not straightforward to include in the solution the joint position, velocity and acceleration bounds. On the other hand, the IK problem can be formulated as an optimization problem. A quadratic programming (QP) formulation is especially attractive for an online implementation, since it is computationally much lighter than nonlinear optimization methods. In a QP IK formulation, the joint constraints can be easily included. However, if the desired end effector trajectory becomes too demanding, then the joint space solution might not exist. To overcome this obstacle, we propose a QP IK formulation with trajectory scaling. The method is successfully tested on a model of KUKA LWR4+ 7-DOF manipulator; the joint constraints are satisfied and the motion is slowed down only when it is necessary. • A novel method for solving inverse kinematics for redundant robots is presented. • Trajectory scaling is introduced into the redundancy resolution method. • The scaling is non-uniform, allowing for slowing down the motion only when necessary. • Online capabilities of the QP-based method are preserved. [ABSTRACT FROM AUTHOR]

• Design a six-legged robot prototype using coupled parallelogram legs. • The maximum velocity distribution of the foot-tip in body frame is calculated. • Propose a design method of trajectory parameters based on wading scenarios. • This method is based on the Jacobian matrix and has good transferability. This paper proposes a novel design method for trajectory parameters based on the velocity index and wading height, which is implemented on a six-legged robot with a coupled parallelogram leg (CPL) for wading scenarios such as shallow waters, harbors, and coastal areas. Design of trajectory parameters includes motion planning and initialization of the joint space angles. The parameters of motion planning are obtained from pre-tests, considering the characteristics of CPL, motor performance, and legged motion. The initial angles of the joint space are obtained using an objective function that considered the velocity index and wading height. By adjusting the coefficients of the objective function, the optimal initial angles of the joint space can be determined for different scenarios, leading to improved overall performance. The proposed approach was validated through experiments using a prototype and provided a reference for setting trajectory parameters in other legged robots. [ABSTRACT FROM AUTHOR]

• A DBCM is proposed for geometrically nonlinear analysis of compliant mechanisms. • The nonlinearities of load-stiffening and kinematics-arching effects are considered. • The nonlinear kinetostatics and large-amplitude vibration analyses are achieved. • The nonlinear amplitude-dependent resonance frequency is revealed. Compliant mechanisms with intermediate-deformation ranges are increasingly applied in large-stroke precision manipulators and other actuators. The nonlinearities of axially-loaded stiffening and kinematics-arching effects (P-Δ effects) have a pronounced influence on the kinetostatic and dynamic behaviors of such a class of compliant mechanisms. A dynamic beam constraint model (DBCM) with the pseudo-static characteristic is reported for the nonlinear kinetostatics and large-amplitude vibration analyses of intermediate-deformation compliant mechanisms. Firstly, the DBCM of flexure beams is derived in a closed-form equation on the frequency domain, including the P-Δ effects. A matrix-based modeling methodology with a step-by-step iteration procedure is then introduced to calculate the nonlinear performance of general planar compliant mechanisms with serial-parallel configurations avoiding inner force analysis and kinematics calculation. Except for the static constraint behaviors in the BCM, the amplitude-dependent resonance frequency is captured with the DBCM. Theoretical, numerical and experimental comparisons for a double parallel guiding flexure pivot and Scott-Russell compliant amplifying mechanism verify the feasibility of the DBCM. In summary, the DBCM enables both the nonlinear kinetostatics and large-amplitude vibration analyses of moderately large-stroke compliant mechanisms in a straightforward way. [ABSTRACT FROM AUTHOR]

• The mobility of mechanisms comprising spatial scissor units are analyzed. • The translational assembly has 4 DOFs, while the 4th DOF is infinitesimal. • The mirrored assembly has a single DOF and thus can be deployed in one mode. Mechanisms consisting of spatial scissor units have different kinematic behaviors than those of planar scissors. However, their kinematics, especially the mobility analysis, has not received enough attention. Two types of deployable assemblies are analyzed in this paper, namely the translational and mirrored assemblies. Both the assemblies are made of tripod scissor units, and their instantaneous mobility are examined using screw theory. The study starts on the configuration where all the members have the identical deployment angle. Firstly, the geometric property of each assembly was studied. Then, screw-loop equations were developed based on graph theory and closure equations. Finally, the mobility of each assembly was computed using linear algebra. Following the analysis, physical prototypes were constructed to validate the results, and several different motion modes were obtained for the translational assembly. The analysis reveals different kinematic behaviors of the two assemblies. In the given configuration, the translational assemblies have four instantaneous degrees of freedom, while the mirrored assemblies have only a single instantaneous degree of freedom. [ABSTRACT FROM AUTHOR]