Finite Screw Research Articles

Kinematic equations of a rigid body in four-dimensional skew-symmetric operators and their application in inertial navigation

Abstract New dual matrix and biquaternion kinematic equations of motion of a free rigid body in dual four-dimensional matrix and biquaternion skew-symmetric operators are proposed. The equations are constructed using dual matrix and biquaternion analogues of Cayley's formulae, which are used to match a dual four-dimensional matrix operator and a biquaternion skew-symmetric operator to a classical biquaternion four-dimensional matrix and the biquaternion of a finite screw displacement of a free rigid body. New quaternion and biquaternion formulae for the summation of finite rotations and finite screw displacements of a free rigid body in four-dimensional skew-symmetric operators are also proposed. The proposed equations and formulae are constructed using the Kotelnikov–Study transference principle. The use of the proposed real and dual matrix kinematic equations of motion, as well as quaternion and biquaternion kinematic equations of motion, of a rigid body to construct new high-precision algorithms for operating strapdown inertial navigation systems is discussed.

Application of vector finite screw analysis to determine the roll centre from wheel points

In kinematic and compliance testing for suspensions, great importance is attached to the measurement of the instant centre and roll centre. As the suspension construction is truly three-dimensional, the screw axis theory can afford a much more accurate kinematic analysis. The purpose of this study was to present a vector finite displacement method (FSAv) for the kinematic analysis of instant centre and roll centre of suspensions. Two suspension simulation models, with and without approximately singular screw parameter, respectively, were employed to investigate the efficiency of the proposed method. It was found that the FSAv method prevailed over the method of general finite screw axis. In addition, it was not necessary to calculate the velocity matrix from the derivatives. This proposed method, therefore, is suitable for determining the roll centre height and the related parameters of instant centre. Moreover, it can be expanded for the measurement of the kingpin axis as well.

A Note on the Screw Triangle

This paper derives the expressions of an equivalent finite screw of two successive screw motions in a simplified form using purely vectorial analysis. This is achieved by tracing the trajectories of specific points on the moving body, which together with the known axis and angle of combined rotation, yield the expressions of the screw triangle. This paper also gives a short overview of different known expressions of the screw triangle and shows that the one given in this paper reduces the number of arithmetic operations by about a third compared with the most efficient algorithm in the literature.

Exponential and Cayley Maps for Dual Quaternions

In this work various maps between the space of twists and the space of finite screws are studied. Dual quaternions can be used to represent rigid-body motions, both finite screw motions and infinitesimal motions, called twists. The finite screws are elements of the group of rigid-body motions while the twists are elements of the Lie algebra of this group. The group of rigid-body displacements are represented by dual quaternions satisfying a simple relation in the algebra. The space of group elements can be though of as a six-dimensional quadric in seven-dimensional projective space, this quadric is known as the Study quadric. The twists are represented by pure dual quaternions which satisfy a degree 4 polynomial relation. This means that analytic maps between the Lie algebra and its Lie group can be written as a cubic polynomials. In order to find these polynomials a system of mutually annihilating idempotents and nilpotents is introduced. This system also helps find relations for the inverse maps. The geometry of these maps is also briefly studied. In particular, the image of a line of twists through the origin (a screw) is found. These turn out to be various rational curves in the Study quadric, a conic, twisted cubic and rational quartic for the maps under consideration.

Development of a method to compute the kingpin axis using screw axis theory based on suspension-parameter-measuring device data

Kingpin geometry is one of the most important design features in vehicle development, as it determines the steering feeling of drivers, the handling performance, and the steering stability of a vehicle. The idea of a geometric kingpin axis has been used to design a relatively simple suspension mechanism such as a double-wishbone type, and various techniques have been developed to design the kingpin axis of multi-link suspensions. However, a kingpin axis for a real vehicle has been unknown so far because there has been no method to measure or compute it. This paper presents the development of a method to compute the kingpin axis of a vehicle using screw theory and suspension-parameter-measuring device (SPMD) data. SPMD data, such as the wheel centre displacement, toe, camber, and side view angles, are used to derive the finite displacement matrix of the wheel. The finite screw axis is computed on the basis of the displacement matrix, and its compatibility with the kingpin axis is verified by an in-house kinematic analysis program and ADAMS/Car. The result of the finite screw axis method (FSAM) is also verified indirectly by comparing suspension parameters such as the caster trail with the SPMD test result. Both verification results show that the FSAM can successfully compute the kingpin axis of a real vehicle.

Experimental Analysis of Rigid Body Motion. A Vector Method to Determine Finite and Infinitesimal Displacements From Point Coordinates

This paper presents a vector method for measuring rigid body motion from marker coordinates, including both finite and infinitesimal displacement analyses. The common approach to solving the finite displacement problem involves the determination of a rotation matrix, which leads to a nonlinear problem. On the contrary, infinitesimal displacement analysis is a linear problem that can be easily solved. In this paper we take advantage of the linearity of infinitesimal displacement analysis to formulate the equations of finite displacements as a generalization of Rodrigues’ formula when more than three points are used. First, for solving the velocity problem, we propose a simple method based on a mechanical analogy that uses the equations that relate linear and angular momenta to linear and angular velocities, respectively. This approach leads to explicit linear expressions for infinitesimal displacement analysis. These linear equations can be generalized for the study of finite displacements by using an intermediate body whose points are the midpoint of each pair of homologous points at the initial and final positions. This kind of transformation turns the field of finite displacements into a skew-symmetric field that satisfies the same equations obtained for the velocity analysis. Then, simple closed-form expressions for the angular displacement, translation, and position of finite screw axis are presented. Finally, we analyze the relationship between finite and infinitesimal displacements, and propose vector closed-form expressions based on derivatives or integrals, respectively. These equations allow us to make one of both analyses and to obtain the other by means of integration or differentiation. An experiment is presented in order to demonstrate the usefulness of this method. The results show that the use of a set of markers with redundant information (n>3) allows a good accuracy of measurement of kinematic variables.

A Geometric Interpretation of Finite Screw Systems Using the Bisecting Linear Line Complex

This paper uses the concept of bisecting linear line complex of the two position theory in kinematics to present a geometric foundation for finite displacement screw systems, with an emphasis on incompletely specified displacement of points. It is shown that the bisecting linear line complex arising from the finite displacement of points is subject to a reciprocal condition if a specific definition of pitch of finite screws is used. The screw systems of finite displacements are then characterized in terms of intersections of bisecting linear line complexes. The line varieties corresponding to the two-system and four-system associated with finite displacements of two points and a point, respectively, are illustrated. This paper demonstrates that the bisecting linear line complex provides a geometric framework for studying finite and infinitesimal kinematics.

A method for predicting dynamic behaviour characteristics of a vehicle using the screw theory — part 1

This paper presents the development of a prediction method for vehicle dynamic behaviour characteristics using screw axis theory. A vehicle model is developed for quasi-static analysis which is applied to vehicles of three different suspension configurations under cornering conditions in which a specific lateral force acts. A finite screw axis is determined using quasi-static analysis based on a rigid body displacement of the vehicle model with respect to the ground. The fixed screw axis surface formed by the migration of the finite screw axis is also determined. The shape of the fixed screw surface shows that the fundamental characteristics of vehicle roll behaviour is determined not by suspension tuning element but by suspension geometry. The gradients of screw parameters with respect to lateral force when the vehicle model behaves in an initial cornering state are proposed as prediction parameters of vehicle dynamic motion. The dynamic analysis result shows that vehicle dynamic motion characteristics can be successfully predicted by the proposed parameters.

The correspondence between finite screw systems and projective spaces

This paper discusses the correspondence between vector subspaces and screw systems. The goal is to differentiate finite screw systems from screw systems arising in instantaneous kinematics and statics. This paper shows that a finite screw system corresponds to the projective space associated with the vector subspace spanned by the basis screws of the screw system. As a result, if finite displacement screws of a body constitute a screw system, the number of degrees of freedom of the motion of the body is one less than the order of the screw system.

Linear Property of the Screw Surface of the Spatial RPRP Linkage

This paper investigates the screw surface of the RPRP linkage, an overconstrained linkage with mobility one. The screw surface is a ruled surface containing screws for displacing the coupler of the RPRP linkage from one reference position to all reachable positions. In this paper, two types of RPRP linkages, in folded and unfolded forms, are constructed by using hexahedrons in accordance with Delassus’ criteria. It is shown that the screw surfaces of both types of RPRP linkages can be represented by screw systems of the second order. These novel finite screw systems are obtained by intersecting two 3-systems corresponding to the finite displacements of the RP and PR dyads. The intersection of finite screw systems is conducted by employing the intersection operation of vector subspaces of the six-dimensional vector space.

An historical review of the theoretical development of rigid body displacements from Rodrigues parameters to the finite twist

The development of the finite twist or the finite screw displacement has attracted much attention in the field of theoretical kinematics and the proposed q-pitch with the tangent of half the rotation angle has demonstrated an elegant use in the study of rigid body displacements. This development can be dated back to Rodrigues’ formulae derived in 1840 with Rodrigues parameters resulting from the tangent of half the rotation angle being integrated with the components of the rotation axis. This paper traces the work back to the time when Rodrigues parameters were discovered and follows the theoretical development of rigid body displacements from the early 19th century to the late 20th century. The paper reviews the work from Chasles motion to Cayley’s formula and then to Hamilton’s quaternions and Rodrigues parameterization and relates the work to Clifford biquaternions and to Study’s dual angle proposed in the late 19th century. The review of the work from these mathematicians concentrates on the description and the representation of the displacement and transformation of a rigid body, and on the mathematical formulation and its progress. The paper further relates this historic development to the contemporary development of the finite screw displacement and the finite twist representation in the late 20th century.

Robots and Screw Theory: Applications of Kinematics and Statics to Robotics

<i>Robots and Screw Theory: Applications of Kinematics and Statics to Robotics</i>

Moment arm of the patellar tendon in the human knee

The moment arm of the knee-extensor mechanism is described by the moment arm of the patellar tendon calculated with respect to the screw axis of the tibia relative to the femur. The moment arm may be found once the line of action of the patellar tendon and the position and orientation of the screw axis are known. In this study, the orientation of the patellar tendon and the position and orientation of the finite screw axis of the tibia relative to the femur were calculated from measurements of the three-dimensional positions of the bones obtained from fresh cadaver specimens. Peak values of the patellar tendon moment arm ranged from 4–6 cm for the six knees tested; the moment arm was maximum near 45° of knee flexion. The moment arm of the patellar tendon was nearly equal to the shortest (perpendicular) distance between the line of action of the patellar tendon and the axis of rotation of the knee at all flexion angles, except near full extension. Near full extension, the angle between the patellar tendon and the screw axis was significantly less than 90°, and the magnitude of the moment arm was then less than the perpendicular distance between these two lines. The patellar tendon moment arm remained roughly constant across individuals when normalized by femoral condyle width, suggesting that anatomical differences play a large role in determining the moment arm of the extensor mechanism.

The Finite Screw System Associated With the Displacement of a Line

In determining the screw systems associated with incompletely specified displacements, the displacement of a line was known to be an exceptional case. Recent research has concluded that all possible screws for the finite displacement of a line do not form a screw system. This paper utilizes Dimentberg’s definition of pitch to demonstrate that all possible screws for displacing a line from one position to another can indeed form a screw system of the third order. Two different approaches are taken: one uses the concept of a screw triangle, and the other is based on analytical geometry. A set of three linearly independent screws of the screw system is shown to be perpendicularly intersecting the external bisector of the initial and final positions of the line.

Decomposition of a Finite Rotation into Three Rotations about Given Axes

This paper consists of two parts. The subject of Part I is the following problem.Given are the axis and the angle of a finite rotation about a fixed point.Given are, furthermore, the axes of three consecutive rotations. To be determined are the angles of the three consecutive rotations such that theirresultant rotation is the given rotation. Explicit solutions to this problemare developed. Conditions for the existence of real solutions are given. In certain cases called critical the first and the third angleare not individually determinate. Only their sum or their difference is. Conditionsspecifying critical cases are developed. In Part II of the paper the more general problem of decomposition of a finite screw displacement into three consecutive screw displacements about given screw axes is solved.The solution is achieved by applying the transference principle of Kotelnikov and Study to the solution of the rotation problem.

On the third-order finite screw system: general displacement of a line

The problem of displacing a line, being a part of a rigid body, from one spatial position to another is studied by using the concept of screw matrix. It is known that there are ∞2 possible screws to realize such displacement; based on the definition of pitch given by Parkin, any one of the ∞2 screws cannot be expressed as the linear combination of three basis screws. In this paper an attempt is made towards a search of a new definition (expression) for pitch under which the set of ∞2 screws form a 3-system. It is shown that such expression for pitch exists and contains six arbitrary constants. By assigning some values to these constants two different but meaningful and valid expressions of pitch are obtained. These expressions are modified to include terms corresponding to Parkin's definition of pitches. Explicit analytical expression of the 3-system is also discussed.

On the Matrix Realization of the Theory of Biquaternions

This paper describes a matrix algebra realization of Clifford’s theory of biquaternions. By examining 4 × 4 skew-symmetric matrices, the paper shows the connection between infinitesimal screws in elliptic three-space and vector quaternions. By studying the matrix exponential of the skew-symmetric matrices, the paper also shows how finite screws in elliptic three-space lead to matrix realization of quaternions. Finally, it is shown that line transformations in elliptic three-space lead to double quaternions and that a dual quaternion is a limiting case of a double quaternion.

On the fourth order finite screw system: general displacement of a point in a rigid body

In this paper, the displacement of a point on a rigid body which moves from one spatial position to another is considered by the concept of screw matrix. Since there are ∞3 possible screws to achieve such displacement, using Parkin's definition (measure) for pitch, one can represent this infinite set of screws as a 4-system of screws. This has been demonstrated by Huang and Roth by the use of screw triangle [1]. In this paper, however, without any assumption for the expression of pitch we ask the question that whether there exist different appropriate measure of pitch other than Parkin's such that all the available finite screws also form a 4-system. In fact by the use of screw matrix concept, a general expression of the pitch is derived so that the screws realize the stated displacement form a 4-system. It is shown that Parkin's pitch, plus an arbitrary constant, is the only appropriate measure of pitch. To arrive at this conclusion, a special coordinate system is selected. This selection of the coordinate system also makes it possible to yield several new geometrical properties of the set of ∞3 screws.

On the Finite Screw Cylindroid Represented as a 2-system of Screws

The problem of displacing a line with a definite point on it from one spatial position to another is studied by utilizing the concept of screw matrix. It is known that all the available finite twists (screws) associated with this displacement form a ruled surface, the so-called finite screw cylindroid. If the definition of the pitch given by Parkin is used, then the finite screw cylindroid can be regarded as a 2-system of screws. This brings to one's mind the question as to whether there exist different appropriate measures for pitch other than Parkin's under which all the available finite twists form a 2-system. This question is answered in this paper. By deriving a general expression of the pitch for these available finite twists under the said condition, it is shown that Parkin's pitch plus an arbitrary constant is the only possible measure of pitch under which the finite screw cylindroid represents a 2-system of screws. However, since adding a constant to the pitches of all screws of any 2-system still gives a 2-system, constant term may be omitted. It is also shown that the determined 2-system of screws can be described as a linear combination of two special basis screws which are called in this paper the α = 0 and the α = π screws.

The Cylindroid Associated With Finite Motions of the Bennett Mechanism

The finite kinematic geometry of the Bennett mechanism is explored in this paper. The screw axes of all possible coupler displacements of the mechanism from any given configuration are shown to form a cylindroid. The screw system of the coupler displacement is obtained by using the intersection of finite screw systems associated with two corresponding revolute-revolute dyads. Inverse finite kinematic analysis is also performed to verify this new linear property of the Bennett mechanism.