Displacement Subgroups Research Articles

CT mapping for complex tibial pilon fractures: Understanding the injury pattern and its relation to the approach choice

ObjectiveTo map OTA/AO type 43C3 tibial pilon fractures by means of computed tomography and analyze the difference between varus and valgus fractures. We hypothesized that valgus fractures are less frequent than varus, and the affected zones of the tibial pilon are different among the patterns. Material and methodsRetrospective, cross-sectional, observational study, using images of 73 computed tomographies of patients who had presented OTA/AO type 43C3 tibial pilon fractures. The radiographs and computed tomography were subdivided into two groups: varus and valgus fractures. Also, the presence or absence of fibula fractures. Both groups were subdivided into eight zones: four lateral to the pilon and four medial. ResultsThe distribution of total affected zones is significantly different in the male and female subgroups (p = 0.027). The incidence of cases in zone 1 is significantly different in varus and valgus displacement subgroups (p = 0.002). In the patients with valgus displacements, 61.9% of the fractures affect zone 1; in the patients with varus displacements, only 25.8% of the fractures affect zone 1. In the patients without fibula fracture, 58.3% of the fractures affect zone 2; in the patients with fibula fracture, 24.6% of the fractures affect zone 2. ConclusionThe typical profile of the patient with a pilon fracture is age ranging from 27 to 57 years, male, with fibular fracture, and the injury affecting two or three pilon zones, with zones 6, 1, and 3 being the most affected ones. There is no typicality regarding the valgus or varus fracture displacement, although a difference was found in zone 1.

Type synthesis of 2-DoF rotational parallel mechanisms actuating the inter-satellite link antenna

This paper focuses on the type synthesis of two degree-of-freedom (2-DoF) rotational parallel mechanisms (RPMs) that would be applied as mechanisms actuating the inter-satellite link antenna. Based upon Lie group theory, two steps are necessary to synthesize 2-DoF RPMs except describing the continuous desired motions of the moving platform. They are respectively generation of required open-loop limbs between the fixed base and the moving platform and definition of assembly principles for these limbs. Firstly, all available displacement subgroups or submanifolds are obtained readily according to that the continuous motion of the moving platform is the intersection of those of all open-loop limbs. These subgroups or submanifolds are used to generate all the topology structures of limbs. By describing the characteristics of the displacement subgroups and submanifolds intuitively through employing simple geometrical symbols, their intersection and union operations can be carried out easily. Based on this, the assembly principles of two types are defined to synthesize all 2-DoF RPMs using obtained limbs. Finally, two novel categories of 2-DoF RPMs are provided by introducing a circular track and an articulated rotating platform, respectively. This work can lay the foundations for analysis and optimal design of 2-DoF RPMs that actuate the inter-satellite link antenna.

Kinematics and dynamics of planar mechanisms reinterpreted in rigid-body’s configuration space

Mechanisms with lower mobility can be studied by using tools that are directly deduced from those of spatial kinematics as screw theory. Nevertheless, ad-hoc tools that fully exploit the peculiarities of the displacement subgroups these mechanisms move in are usually more efficient both in showing mechanisms’ features and when used to conceive numerical algorithms. Planar displacements constitute a three-dimensional subgroup with many peculiarities that allow the use of simplified tools (e.g., complex numbers) for studying planar mechanisms. Here, the systematic use of three-dimensional vector spaces to represent link poses and velocities in planar motion and planar system of forces is investigated. The result is a novel coherent set of tools that make it possible to geometrically describe kinematics and dynamics of planar mechanisms in the three-dimensional configuration space of links’ planar poses. The effectiveness of this novel approach is shown through a case study.

Symplectic reduction of holonomic open-chain multi-body systems with constant momentum

This paper presents a two-step symplectic geometric approach to the reduction of Hamilton’s equation for open-chain, multi-body systems with multi-degree-of-freedom holonomic joints and constant momentum. First, symplectic reduction theorem is revisited for Hamiltonian systems on cotangent bundles. Then, we recall the notion of displacement subgroups, which is the class of multi-degree-of-freedom joints considered in this paper. We briefly study the kinematics of open-chain multi-body systems consisting of such joints. And, we show that the relative configuration manifold corresponding to the first joint is indeed a symmetry group for an open-chain multi-body system with multi-degree-of-freedom holonomic joints. Subsequently using symplectic reduction theorem at a non-zero momentum, we express Hamilton’s equation of such a system in the symplectic reduced manifold, which is identified by the cotangent bundle of a quotient manifold. The kinetic energy metric of multi-body systems is further studied, and some sufficient conditions are introduced, under which the kinetic energy metric is invariant under the action of a subgroup of the configuration manifold. As a result, the symplectic reduction procedure for open-chain, multi-body systems is extended to a two-step reduction process for the dynamical equations of such systems. Finally, we explicitly derive the reduced dynamical equations in the local coordinates for an example of a six-degree-of-freedom manipulator mounted on a spacecraft, to demonstrate the results of this paper.

Nonholonomic dynamical reduction of open-chain multi-body systems: A geometric approach

This paper studies the geometry behind nonholonomic Hamilton's equation to present a two-stage reduction procedure for the dynamical equations of nonholonomic open-chain multi-body systems with multi-degree-of-freedom joints. In this process, we use the Chaplygin reduction and an almost symplectic reduction theorem. We first restate the Chaplygin reduction theorem on cotangent bundle for nonholonomic Hamiltonian mechanical systems with symmetry. Then, under some conditions we extend this theorem to include a second reduction stage using an extended version of the symplectic reduction theorem for almost symplectic manifolds. We briefly introduce the displacement subgroups and accordingly open-chain multi-body systems consisting of such joints. For a holonomic open-chain multi-body system, the relative configuration manifold corresponding to the first joint is a symmetry group. Hence, we focus on a class of nonholonomic distributions on the configuration manifold of an open-chain multi-body system that is invariant under the action of this group. As the first stage of reduction procedure, we perform the Chaplygin reduction for such systems. We then introduce a number of sufficient conditions for a reduced system to admit more symmetry due to the action of the relative configuration manifolds of other joints. Under these conditions, we present the second stage of the reduction process for nonholonomic open-chain multi-body systems with multi-degree-of-freedom joints. Finally, we explicitly derive the reduced dynamical equations in the local coordinates for an example of a two degree-of-freedom crane mounted on a four-wheel car to illustrate the results of this paper.

Topology synthesis of three-legged spherical parallel manipulators employing Lie group theory

Driven by the requirements of the bionic joint or tracking equipment for the spherical parallel manipulators (SPMs) with three rotational degrees-of-freedom (DoFs), this paper carries out the topology synthesis of a class of three-legged SPMs employing Lie group theory. In order to achieve the intersection of the displacement subgroups, the subgroup characteristics and operation principles are defined in this paper. Mainly drawing on the Lie group theory, the topology synthesis procedure of three-legged SPMs including four stages and two functional blocks is proposed, in which the assembly principles of three legs are defined. By introducing the circular track, a novel class of three-legged SPMs is synthesized, which is the important complement to the existing SPMs. Finally, four typical examples are given to demonstrate the finite displacements of the synthesized three-legged SPMs.

A generalized exponential formula for forward and differential kinematics of open-chain multi-body systems

This paper presents a generalized exponential formula for Forward and Differential Kinematics of open-chain multi-body systems with multi-degree-of-freedom, holonomic and nonholonomic joints. The notion of lower kinematic pair is revisited, and it is shown that the relative configuration manifolds of such joints are indeed Lie groups. Displacement subgroups, which correspond to different types of joints, are categorized accordingly, and it is proven that except for one class of displacement subgroups the exponential map is surjective. Screw joint parameters are defined to parameterize the relative configuration manifolds of displacement subgroups using the exponential map of Lie groups. For nonholonomic constraints the admissible screw joint speeds are introduced, and the Jacobian of the open-chain multi-body system is modified accordingly. Computational aspects of the developed formulation for Forward and Differential Kinematics of open-chain multi-body systems are explored by assigning coordinate frames to the initial configuration of the multi-body system, employing the matrix representation of SE(3) and choosing a basis for se(3). Finally, an example of a mobile manipulator mounted on a spacecraft, i.e., a six-degree-of-freedom moving base, elaborates the computational aspects.

Synthesis of a Rear Wheel Suspension Mechanism With Pure Rectilinear Motion

This paper focuses on the synthesis of an independent suspension that can guide the wheel to track a straight line when moving up (jounce) and down (rebound). With displacement subgroups, it first synthesizes a rigid body guidance mechanism and verifies the result through screw theory. To simplify and optimize the loads of each kinematic chain of the knuckle, it investigates the static equations and ultimately synthesizes a symmetric redundant-constraint suspension structure, which could not only eliminate the shambling shocks induced by the jumping of wheels but also decrease the abrasion of tires. Theoretically, only one pair of noncoplanar kinematic chains is necessary to realize straight line guidance. However, a second pair of noncoplanar kinematic chains is particularly utilized to improve the load status of the links. Because of the redundant constraints induced by the suspension structures, the whole weight can be significantly reduced compared with the initial one. ADAMS simulations with a set of real parameters indicate that the rear suspension mechanism proposed in this paper can guide the wheel to follow a rectilinear locus during jounce and rebound. Therefore, this kind of independent suspension can improve the ride and handling properties of advanced vehicles.

CONJUGATION IN THE DISPLACEMENT GROUP AND MOBILITY IN MECHANISMS

The paper deals with the Lie group algebraic structure of the set of Euclidean displacements, which represent rigid-body motions. We begin by looking for a representation of a displacement, which is independent of the choice of a frame of reference. Then, it is a simple matter to prove that displacement subgroups may be invariant by conjugation. This mathematical tool is suitable for solving special problems of mobility in mechanisms.

The mathematical group structure of the set of displacements

With the aim of representing physical rigid body motions, this article deals with the Euclidean displacement operators. Remaining as elementary as possible, an intrinsic mathematical formalism is presented with the use of affine geometry together with Lie's theory of groups. We start by defining a differential operator associated with the velocity field. Then, thanks to the Lie acceptance of the exponential function, we derive the formal Lie expression of finite displacements which are shown to be equivalent to affine direct orthonormal transformations. Lie sub-algebras of screw velocity fields lead to the description of the displacement sub-groups. The geometric modelling of small displacements is also evoked. This paper unifies elements previously published in [2] and [5].