Screw Theory

This paper presents the theoretical analysis and the experimental validation of the force sensing capabilities of continuum robots. These robots employ super-elastic NiTi backbones and actuation redundancy. The paper uses screw theory to analyze the limitations and to provide geometric interpretation to the sensible wrenches. The analysis is based on the singular value decomposition of the Jacobian mapping between the configuration space and the twist space of the end effector. The results show that the sensible wrenches belong to a two-dimensional screw system and the insensible wrenches belong to a four-dimensional screw system. The theory presented in this paper is validated through simulations and experiments. It is shown that the force sensing errors have an average of 0.34 grams with a standard deviation of 0.83 grams. Another experiment of generating the stiffness map of a silicone strip suggests possible medical application of palpation for tumor detection. The presented study allows force sensing in challenging environments where placing force sensors at the distal end of a robot is not possible due to limitations such as size and MRI compatibility. x x x x J J J J T T J J I J E J W APPENDIX B Since x J is formulated as p J J , p J can be found by taking time derivative of p b , as in , while J is found by formulating the angular velocity of GCS, (43). p

A robot kinematic calibration method based on the local frame representation of the product-of-exponentials (Local POE) formula is introduced. In this method, the twist coordinates of the joint axes are expressed in their respective local (body) frames. The advantages of this new approach are threefolds: (1) revolute and prismatic joints can be uniformly expressed in the twist coordinates based on the line geometry; (2) the twist coordinates of the joint axes can be set up with simple values because the local frames can be arbitrarily de®ned on the links; (3) the kinematic parameters described by the twist coordinates vary smoothly that makes the method robust and singularity-free. By assuming that the kinematic errors exist only in the relative initial poses of the consecutive link frames, the kinematic calibration models can be formulated in a simple and elegant way. The calibration process then becomes to re-de®ne a set of new local link frames that are able to re¯ect the actual kinematics of the robot. This method can be applied to robot manipulators with generic open chain structures (serial or tree-typed). The simulation and experiment results on a 4-DOF SCARA type robot and a 5-DOF tree-typed modular robot have shown that the average positioning accuracy of the end-eector increases signi®cantly after calibration.

In this paper, we present a novel kinematic-based framework for collision detection and estimation of contact location along multisegment continuum robots. Screw theory is used to define a screw motion deviation (SMD) as the distance between the expected and the actual instantaneous screw axis (ISA) of motion. The expected ISA is computed based on the unconstrained kinematics model of the robot, while the actual ISA is computed based on sensory information. Collisions with rigid environments at any point along the robot are detected by monitoring the SMD. Contact locations are estimated by the minimization of the SMD between the ISA that is obtained from a constrained kinematic model of the continuum robot and the one that is obtained from sensor data. The proposed contact detection and localization methods only require the relative motion of each continuum segment with respect to its own base. This strategy allows the straightforward generalization of these algorithms for an n-segment continuum robot. The framework is evaluated via simulations and experimentally on a three-segment multibackbone continuum robot. Results show that the collision-detection algorithm is capable of detecting a single collision at any segment, multiple collisions occurring at multiple segments, and total-arm constraint. It is also shown that the estimation of contact location is possible at any location along the continuum robot with an accuracy better than 20% of the segment nominal length. We believe this study will enhance manipulation safety in unstructured environments and confined spaces. since the Fall semester of 2010.

A new, systematic approach to the synthesis of Cartesian sti®ness by springs is presented using screw (spatial vector) algebra. The space of solutions is fully characterized for all sti®nesses realizable by springs. The main result shows that a rank r sti®ness can always be synthesized by r springs. Further, it can also be synthesized by an arbitrarily large number of springs greater than r: Synthesis algorithms are presented and numerical results support the theory.

Just as the 3-D Euclidean space can be inverted through any of its points, the special Euclidean group SE(3) admits an inversion symmetry through any of its elements and is known to be a symmetric space. In this paper, we show that the symmetric submanifolds of SE(3) can be systematically exploited to study the kinematics of a variety of kinesiological and mechanical systems and, therefore, have many potential applications in robot kinematics. Unlike Lie subgroups of SE(3), symmetric submanifolds inherit distinct geometric properties from inversion symmetry. They can be generated by kinematic chains with symmetric joint twists. The main contribution of this paper is: 1) to give a complete classification of symmetric submanifolds of SE(3); 2) to investigate their geometric properties for robotics applications; and 3) to develop a generic method for synthesizing their kinematic chains.

The twist space of a plunging constant-velocity (CV) coupling with intersecting shafts consists, in all configurations, of a planar field of zero-pitch screws. Recently, we reported an important discovery about this screw system: it is closed under two consecutive Lie bracket operations, thus being referred to as a Lie triple system; taking the exponential of all its twists generates the motion manifold of the coupling. In this paper, we first give a geometric characterization of the Lie product and the Lie triple product of a generic screw system. Then, we present a systematic identification of all Lie triple screw systems of se(3), by an approach based on both algebraic Lie group theory and descriptive screw theory. We also derive the exponential motion manifolds of the Lie triple screw systems in dual quaternion representation. Finally, several important applications of Lie triple screw systems in mechanism and machine theory are highlighted in the conclusions.

This paper deals with the singular configurations of symmetric 5-DOF parallel mechanisms performing three translational and two independent rotational DOFs. The screw theory approach is adopted in order to obtain the Jacobian matrices. The regularity of these matrices is examined using Grassmann-Cayley algebra and Grassmann geometry. More emphasis is placed on the geometric investigation of singular configurations by means of Grassmann-Cayley Algebra for a class of simplified designs whereas Grassmann geometry is used for a matter of comparison. The results provide algebraic expressions for the singularity conditions, in terms of some bracket monomials obtained from the superbracket decomposition. Accordingly, all the singularity conditions can be enumerated.

This study addresses the kinematics, including position, velocity and acceleration analyses, of a modular spatial hyper-redundant manipulator built with a variable number of serially connected identical mechanical modules with autonomous motions. First, the kinematics of the base module, a three-legged in-parallel manipulator, is formulated using the theory of screws. After that, the results thus obtained, are applied recursively for accomplishing

Rigid body displacement and motion parameterization is a research subject with continuous development in many theoretical and applied fields. A very important objective for the design of parameterization methods is to obtain a reduced number of algebraic equations and fewer variables for a more compact notation. This criterion increased the interest into tensorial based approaches for rigid motion parameterization. The present research analyzes the performance of tensorial parameterization and proposes new methods to compute the rigid-body motion parameters. Our rigid body motion parameterization solutions are developed using rigid bases of dual vectors, a new notion defined in this paper. Considering a rigid body motion, computational techniques are proposed for the orthogonal dual tensor, the screw parameters and the instantaneous screw parameters. The parameterization techniques proposes in this paper are described both algebraic and geometric. Our solutions are sustained by the symbolic and numerical results gathered from Matlab implementations.

When a rigid body is axially reflected through a moving line, its image undergoes a so-called line-symmetric motion. The space comprising all possible line-symmetric motions that share a common initial line is a four-dimensional submanifold, denoted M4 , in the special Euclidean group SE(3). Recently, we showed that M4 may be used to characterize motions of a line-symmetric body that are free of self-spin and sliding, thus lending itself to applications such as remote center of motion devices for minimal invasive surgery and haptic interfaces. Aiming at designing robot mechanisms for these applications, we present in this paper a systematic enumeration of line-symmetric motion generators (LSMGs), i.e., robot mechanisms that generate the line-symmetric motion manifold M 4 , following a procedure based on symmetric space theory. LSMGs present a ubiquitous line symmetry of their joint axes, thus offering a new understanding of the line-symmetric motions.

In recent times, there has been considerable interest in creating and deploying modular cooperating collectives of robots. Interest in such cooperative systems typically arises when certain tasks are either too complex to be performed by a single agent or when there are distinct benefits that accrue by cooperation of many simple robotic modules. However, the nature of both the individual modules as well as their interactions can affect the overall system performance. In this paper, we examine this aspect in the context of cooperative payload transport by robot collectives wherein the physical nature of the interactions between the various modules creates a tight coupling within the system. We leverage the rich theoretical background of analysis of constrained mechanical systems to provide a systematic framework for formulation and evaluation of system-level performance on the basis of the individual-module characteristics. The composite multi-degree-offreedom (DOF) wheeled vehicle, formed by supporting a common payload on the end-effectors of multiple individual mobile manipulator modules, is treated as an in-parallel system with articulated serial-chain arms. The system-level model, constructed from the twist-and wrench-based models of the attached serial chains, can then be systematically analyzed for performance (in terms of mobility and disturbance rejection). A two-module composite system example is used throughout the paper to highlight various aspects of methodical system model formulation, effects of selection of active, passive or locked articulations on system performance, and experimental validation on a hardware prototype test bed.

An efficient approach for determining the force-unconstrained poses (singularity loci) of planar parallel manipulators (PPMs) is presented. The approach is implemented for the 3-RPR, 3-PRR, and 3-RRR PPMs, where the underline indicates the actuated joint. The force-unconstrained poses are identified based upon concepts of reciprocal screws, kinematics, and elimination methods. The use of screw quantities provides a clear insight as to how the forces are acting on the mechanism. This allows finding conditions of singular configurations that are related to displacements of passive joints. Based on these conditions, the force-unconstrained poses are obtained by solving the forward kinematic analysis of manipulators with the same architecture but different actuation layouts. It is demonstrated that the singularity loci of the 3-RPR, 3-PRR, and 3-RRR PPMs can be described with single-variable polynomials of degrees 2, 2, and 6, respectively. j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / m e c h m t

In this work we show that the concept of a one-parameter persistent rigid-body motion is a slight generalisation of a class of motions called Ribaucour motions by Study. This allows a simple description of these motions in terms of their axode surfaces. We then investigate other special rigid-body motions, and ask if these can be persistent. The special motions studied are line-symmetric motions and motions generated by the moving frame adapted to a smooth curve. We are able to find geometric conditions for the special motions to be persistent and, in most cases, we can describe the axode surfaces in some detail. In particular, this work reveals some subtle connections between persistent rigid-body motions and the classical differential geometry of curves and ruled surfaces.

In order to build an linear, high computational efficiency geometric error model for machine tools, a new error modeling method based on screw theory is presented. Geometrical error sources are expressed as error twists, and the integrated model is calculated as the sum of error twists. The method is demonstrated on a 2-DOF mechanism and a simulation error modeling procedure at 1000 random points for a 5-axis machine tool is conducted. The simulation results show that the computational efficiency increase by 5 times. The feasibility of the proposed method is also verified and the physical influences of the error sources are also analyzed. This modeling method has high computational efficiency and clear physical meanings. It is useful to guide the design of new machine tools.

Mechanisms and robots often share the following fundamental property: the instantaneous twist space generated by the end-effector at a generic pose is a rigidly-displaced copy of the one generated at the home configuration, i.e., the tangent spaces at all points of its motion manifold (a manifold of the Lie group of rigid displacements SE(3)) are mutually congruent. A manifold of this kind, hereafter denoted as persistent, can be seen as the envelope of a persistent twist subspace rigidly moving in SE(3). In this paper, we shall summarize three important classes of persistent manifolds that have so far been discovered and systematically investigated in the literature, namely the Lie subgroups, the persistent product-of-exponential (POE) manifolds, and the symmetric subspaces. In each case, the persistence property arises from a distinct manifold structure, which dictates the ensuing classification and underlies the framework for the synthesis of mechanical devices that are capable of generating such manifolds. In this regard, we attempt to offer a guideline to classification and mechanism synthesis of persistent manifolds for a general audience.

The determination of principal twists of the end-effector of a multi-degree-of-freedom manipulator plays a central role in their analysis, design, motion planning and determination of singularities. Most approaches to obtain principal twists and the distributions of twists, such as the well-known classical results of cylindroid and hyperboloid, are based on geometric reasoning and involve intuitive choice of coordinate systems. In this paper, we present a formal algebraic approach to obtain the principal twists of any multidegree-of-freedom serial, parallel or hybrid manipulator, by making use of the algebra of dual numbers, vectors and matrices. We present analytical expressions for the principal twists and the pitches for any manipulator of arbitrary degree-of-freedom. A consequence of our approach is that we can obtain analytical expressions for the screws along which a manipulator can lose or gain degrees-of-freedom at a singularity. The theoretical results are illustrated with the help of examples of parallel and hybrid manipulators.

In this paper we present a singularity free trajectory tracking method for the cooperative working of multi-arm robot manipulators. It is based on an inverse kinematic transformation which determines the manipulator's joint angles corresponding to the end-effector trajectory given in the task space. The kinematic problem of multi-arm robot system is solved by using screw theory and quaternion algebra. Screw theory is an effective way to establish a global description of rigid body and avoids singularities due to the use of the local coordinates. Dualquaternion is the most compact and efficient dual operator to express screw displacement. Inverse kinematic solutions are obtained by using Paden-Kahan sub-problems. The proposed method is implemented using two Stäubli TX60 robot arms and simulation results are given.

This study derives the equation of parasitic motion of 3-DoFs parallel manipulator from the velocity-level analytic-constraint equation and compares it with a well-known positionlevel geometric method. The velocity-level constraint is formulated based on the extended Jacobian, derived from the instantaneous motion space (IMS) and the instantaneous restriction space (IRS) for free motion and constraint. In contrast, the position-level constraint, adopted in previous studies, is geometrically obtained by analyzing the moving platform and limb motions. The velocity-level analytic-constraint matrix is used to further analyze the task-space motion. In this paper, the procedure of detecting and identifying the parasitic terms from the independent terms is introduced utilizing the property that comes from the virtue of analytic constraint and inverse rate kinematics algorithm. Then, an equation for the constraint-compatible task motion and a coupling relation between the parasitic and independent motions are derived during further analysis. The paper derived the parasitic motion from position constraint, and an algebraic equivalency with the velocity constraint is shown by taking the time derivative of a point-plane position level constraint for comparison. Finally, numerical simulations are provided to validate the proposed approach and demonstrate the effect of the constraints on the given input velocity within the entire rotational workspace.

The subject of this paper is a diagram due to Sir Robert Stawell Ball and Martin Disteli. The diagram, illustrating the relations among the velocities of a pair of gears with skew axes and the geometric parameters of the relative layout of the axes, is intended for the analysis and design of the gear train. A brief historical account on the contributions of both Ball and Disteli is included, which shows that, very likely, these two geometers found the diagram independently. While this diagram appears to be an invaluable tool for the design of gears with skew axes, its significance has remained virtually unknown to the community of gear-geometry researchers and designers. One aim of this paper is to provide a succinct derivation of the diagram, which we do with the aid of dual algebra. The significance of the geometric and kinematic relations summarized in the diagram are thus made more transparent to the reader.

Screw theory is well known to do kinematic computations. Recently it has been used to create kinematic models of assembly features so allowing assembly analysis. Motion limit analysis uses the mathematics of screw theory to model the ability of mechanical assembly features to allow or constrain rigid body motions in six degrees of freedom. A user of this theory is able to determine the directions and quantitative amounts of possible finite rigid body motions of a part that is being added to an assembly via calculation applied to a defined set of assembly features. The ability to calculate rigid body motions of a part is important for enabling in-process adjustment during assembly to precisely establish key assembly dimensions. MLA software is a part of a suite of software programs used to do assembly analysis

A new inverse kinematic solution for serial robot manipulators is represented in this paper. Major aims of this paper are to obtain singularity avoiding inverse kinematic solutions and formulize kinematic problems in a compact closed form. Our solution method is based on screw theory and it uses quaternions as a screw motion operator. Screw theory methods based on line transformation. All screw motions are represented as a rotation about a line together with a translation along the line with respect to base frame. Thus screw theory methods do not suffer from singularities. Two quaterninos are used to represent screw motion. First one is for orientation and second one is for translation. Thus we formulize kinematic problems in a compact closed form. 6R-DOF industrial robot manipulators forward and inverse kinematic equations are derived using this new formulation and also it compared with D-H convention that is the most common method in robot kinematic.

A clear and rigorous definition of muscle moment-arms in the context of musculoskeletal 9 systems modelling is presented, using classical mechanics and screw theory. The definition 10 provides an alternative to the tendon excursion method, which can lead to incorrect moment 11 arms if used inappropriately due to its dependency on the choice of joint coordinates. The 12 definition of moment-arms, and the presented construction method, apply to musculoskeletal 13 models where the bones are modelled as rigid bodies, the joints are modelled as ideal me-14 chanical joints, and the muscles are modelled as massless, frictionless cables wrapping over 15 the bony protrusions, approximated using geometric surfaces. In this context, the definition 16 is independent of any coordinate choice. It is then used to solve a muscle-force estimation 17 problem for a simple two-dimensional conceptual model and compared with an incorrect ap-18 plication of the tendon excursion method. The relative errors between the two solutions vary 19 between 0% and 100%. 20

This paper presents a 3D extension of our previous work on the synthesis of assemblies whose dimensional integrity is insensitive to the dimensional variations of individual parts. Assuming that assemblies can be built in the reverse sequence of decomposition, the method recursively decomposes a given product geometry into two subassemblies until parts become manufacturable. At each recursion, joints are assigned to the interfaces between two subassemblies to ensure the two criteria for robust dimensional integrity, in-process dimensional adjustability and proper part constraints. Screw Theory is utilized as a unified 3D representation of the two criteria. A case study on an automotive space frame is presented to demonstrate the method.

This article presents a systematic graph theory-based method for the topological synthesis of planar metamorphic mechanisms including metamorphic transformations of
links and changes in the degrees-of-freedom. The parts to move, with input and output motion defined, and the topological design space, are represented by graphs of simplejointed
mechanisms. The topological requirements involving link transformations are expressed in terms of subgraphs with a given degrees-of-freedom containing prescribed
input and output parts. The algorithm executes two subgraph searches inside atlases of mechanisms with different degrees-of-freedom. An application to the design of a family
of low-voltage circuit-breaker mechanisms is shown.

The location of the instantaneous screw axis (ISA) is essential in order to obtain useful kinematic models of the human body for applications such as prosthesis and orthoses design or even to help in disease diagnosis techniques. In this paper, dual vectors will be used to represent and operate with kinematic screws with the purpose of locating the instantaneous screw axes which characterize this instantaneous motion. A photogrammetry system based on markers will be used to obtain the experimental data from which the kinematic magnitudes will be obtained. A comprehensive analysis of the errors in the measurement of kinematic parameters has been developed, obtaining explicit expressions for them based on the number of markers and their distribution. Finally, the developed methodology has been applied to the experimental determination of the ISA during an alternative motion of flexion and extension of the back.

In 1978, Hunt found a set of vector subspaces of screws that guarantee `full-cycle mobility' of mechanisms.
They are subalgebras of the Lie algebra se(3) of the Euclidean group and they are at the basis of most families of mechanisms with special motion capabilities.
Recently, a more general concept was presented.
Persistent screw systems (PSSs) are not subalgebras of se(3), but they still exhibit relevant properties for full-cycle motions, namely the invariance of both the space dimension and the pitch of the principal screws.
For this reason, they are believed to play an important role in both mobility analysis and mechanism synthesis.
This paper provides the detailed derivation and the comprehensive classification of PSSs of dimension three.

The Lie product involves two screws, namely, a driver screw (so-called screwer in this paper) and a driven screw (termed screwen in this paper). Thus, in a spatial motion, the screwer performs a displacement of the screwen. Under this context, the Lie product represents the change of the screwen due to the infinitesimal motion performed by the screwer. To this end, this paper sets forth an integrated formulation that brings together a general geometric interpretation of the Lie product and a novel and simple derivation of the Lie product in terms of the involved screws. Moreover, since the formulation is purely geometric, a great deal of physical insight into the description of the motion problem is gained. As a result, the screw partial derivatives, in terms of the involved Lie products, are obtained for three frequently encountered types of kinematic joints, namely, helical, revolute and prismatic. Since the derivation of the Lie product is completely general, this derivation should provide a sound foundation for further applications of the Lie product.

The problem of determining the tensors and the vector invariants that describe the vector field of the order accelerations is generally avoided in rigid body kinematics. This paper extends the discussion from velocities and accelerations to order accelerations. Using the tensor calculus and the dual numbers algebra, a computing method for studying the order acceleration field properties is proposed for the case of the general motion of the rigid body. This approach uses the isomorphism between the Lie group of the rigid displacements and the Lie group of the orthogonal dual tensors.

The 4×4 homogeneous transformation matrix is extensively used for representing rigid body displacement in 3D space and has been extensively used in the analysis of mechanisms, serial and parallel manipulators, and in the field of geometric modeling and CAD. The properties of the transformation matrix are very well known. One of the well known properties is that a general 4 × 4 homogeneous transformation matrix cannot be diagonalized, and at best can be reduced to a Jordan form . In this note, we show that the 4 × 4 homogeneous transformation matrix can be diagonalized if and only if displacement along the screw axis is zero. For the general transformation with non-zero displacement along the axis, we present an explicit expression for the fourth basis vector of the Jordan basis. We also present a variant of the Jordan form which contains the motion variables along and about the screw axis and the corresponding basis vectors which contains the information only about the screw axis and its location. We present a novel expression for a point on the screw axis closest to the origin, which is then used to form a simple choice of basis for different forms. Finally the theoretical results are illustrated with a numerical example. * Graduate Student, 1 Unless stated otherwise, we assume [R] to be nontrivial, i.e [R] = [I3].

Flexible robot arms have been developed for various medical and industrial applications because of their compliant structures enabling safe environmental interactions. This paper introduces a novel flexible robot arm comprising a number of elastically deformable planar spring elements arranged in series. The effects of flexure design variations on their layer compliance properties are investigated. Numerical studies of the different layer configurations are presented and finite Element Analysis (FEA) simulation is conducted. Based on the suspended platform’s motion of each planar spring, this paper then provides a new method for kinematic modeling of the proposed robot arm. The approach is based on the concept of simultaneous rotation and the use of Rodrigues' rotation formula and is applicable to a wide class of continuum-style robot arms. At last, the flexible robot arms respectively integrated with two different types of compliance layers are prototyped. Preliminary test results are reported.

Ortho-planar spring is a compact spring that generates motion based on the deformation of flexure elements, and it has wide applications in compliant robotic designs. Previous studies only investigate the out-of-plane compliance of ortho-planar spring and fewer work pays attention to its angular compliance. To address this issue, this paper provides an analytical method to study both the linear and angular compliance characteristics of ortho-planar spring for the first time. In the frame work of screw theory, the symbolic formula of platform's compliance matrix was obtained based on the hybrid integration of beam elements. Subsequently non-dimensional geometric parameters were introduced to compare the compliance characteristics of planar spring, which revealed the ortho-planar spring also demonstrates large bending compliance. Numerical studies of angular compliance of planar spring were provided, and they showed good agreements with the finite element simulations in a large working range. Based on the numerical study, a physical prototype of one continuum structure assembled with pla-nar springs was provided and it demonstrated a large flexibility compared to other previous designs, which suggests ortho-planar spring has potential value in developing continuum manipulators in related medical surgery and bio-robotic research fields.

for their friendship, encouragement and help with various issues. On a personal note, I thank my parents and brothers for their inexhaustible financial and emotional support during the last three years. CONTENTS 4.3 The first fundamental theorem of invariant theory for SO(3, R) 4.4 The second fundamental theorem of invariant theory for

This paper presents a novel compliant parallel mechanism
that utilizes shape-memory-alloy (SMA) spring based actuators. By employing SMA coil springs, the traditional line constraint that resists translation along its axis but no other forms of motions is transformed into a linear actuator that can generate deflection along its axis, which leads to the design of SMA-spring linear actuators. In accordance with this SMA actuator, an constraint-based approach in the framework of screw theory is utilized to synthesize the constraint and actuation space of parallel mechanisms, and a novel 4 DOF parallel platform is developed based on this analytical approach. A physical prototype is manufactured by employing the SMA-spring actuators, and its mobility and workspace are verified with both finite element simulation
and experiment observations. The results illustrate this parallel mechanism has a large workspace in all desired mobility configurations. The presented work on the parallel platform demonstrates the efficiency of the constraint-based approach in determining the layout of actuation systems, also the developed SMA actuators pave a new way for applying the SMA technique in the future development of compliant parallel mechanisms and robotics.

When a mechanism moves, the twist system S of the end-effector generally varies.
In significant special cases, S is a subalgebra of the Lie algebra of the special Euclidean group, and it remains constant.
In more general cases, S remains invariant up to a proper isometry, thus preserving its class.
A mechanism of this kind is said to generate a persistent screw system (PSS) of the end-effector.
PSSs play an important role in mobility analysis and mechanism design.
This paper presents the serial generators of 4-dimensional PSSs with a constant class of the general type.