Empirical design of non-circular centrodes

TO THE INSTRUCTOR / TO THE STUDENT 13. INTRODUCTION TO DYNAMICS, KINEMATICS OF PARTICLES Introduction to Dynamics / Motion of a Particle on an Axis / Motion of a Particle in Three Dimensions / Rotation of a Line in a Plane, Angular Velocity, and Angular Acceleration / Simple Harmonic Motion / Curvature of a Plane Curve / Tangential and Normal Components of Acceleration, Plane Motion / Differentiation of Vectors / Vector Interpretation of Tangential and Centripetal / Polar Coordinates / Kinematics Frames of Reference / Problems 14. KINEMATICS OF PARTICLES Newton's Law of Universal Gravitation / Newton's Laws of Motion / Newtonian Reference Frames / Applications of Newton's Second Law / Motion of a Particle under the Action of Gravity / The Inertial Force: Particle Motion in an Accelerated Frame / Centrally Directed Force: Planetary Motion / Problems 15. WORK AND ENERGY PRINCIPLES FOR PARTICLES Introduction / Work-Force Relationships / Power-Force Relationship / Conservative and Nonconservative Systems / Potential Energy of External and Internal Forces / The General Concept of Energy / Problems 16. MOMENTUM PRINCIPLES FOR PARTICLES Introduction / Laws of Momentum and Conservation of Momentum / Center of Mass of a System of Particles / Collisions, or Impacts / Inelastic Collisions / Law of Moment of Momentum Conservation of Moment of Momentum / Moment of Momentum with Respect to a Moving Reference Frame / Variable-Mass Systems / The Principle of Momentum in Fluid Mechanics / Chapter Summary / Problems 17. KINEMATICS OF RIGID BODIES Part I: Plane Kinematics of Rigid Bodies / Types of Plane Displacements / Plane Motion of a Rigid Body / Velocity of a Rigid Body Slice That Executes Plane Motion / The Instantaneous Center of Velocity / Accelerations in a Rigid Body Slice That Executes Plane Motion / Part II: Three-Dimensional Kinematics of Rigid Bodies / Vector Character of Angular Velocity / Rotational Couples / General Motion of a Rigid Body / Velocities and Accelerations in a Rigid Body / Relative Motion and Rotating Reference Frames / Problems 18. PLANE KINETICS OF RIGID BODIES Translation of a Rigid Body / Inertial-Force Method for Translation of a Rigid Body / Mass Moment of Inertia of a Body with Respect to an Axis / Rotation of a Rigid Body about a Fixed Axis / The Pendulum, a Body That Rotates about a Fixed Axis / Plane Motion of a Rigid Body / Use of the Inertial-Force Method and Inertial Couples to Analyze. (Part contents).

This article discusses a direct analytical method for calculating the instantaneous center of rotation and the instantaneous axis of rotation for the two-dimensional and three-dimensional motion, respectively, of rigid bodies. In the case of planar motion, this method produces a closed-form expression for the instantaneous center of rotation based on a single point located on the rigid body. It can also be used to derive closed-form expressions for the body and space centrodes. For three-dimensional, rigid body motion, an extension of the technique used for planar motion locates a point on the instantaneous axis of rotation, which is parallel to the body angular velocity vector. In addition, methods are demonstrated that can be used to map the body and space cones for general rigid body motion, and locate the fixed point for the body.

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For a general J wheeled mobile platform capable of up to three-degrees-of-freedom planar motion, there are up to two J independent input parameters yet the output of the platform is completely represented by three independent variables. This leads to an input parameter resolution problem based on operational criteria, which are in development just as they have been developed for n input manipulator systems. To resolve these inputs into a meaningful decision structure means that all motions at the wheel attachment points must have clear physical meaning. To this effect, we propose a methodology for kinematic modeling of multiwheeled mobile platforms using instant centers to efficiently describe the motion of all system points up to the nth order using a generalized algebraic formulation. This is achieved by using a series of instant centers (velocity, acceleration, jerk, and jerk derivative), where each point in the system has a motion property with its magnitude proportional to the radial distance of the point from the associated instant center and at a constant angle relative to that radius. The method of instant center provides a straightforward and physically intuitive way to synthesize a general order planar motion of mobile platforms. It is shown that a general order motion property of any point on a rigid body follows two properties, namely, directionality and proportionality, with respect to the corresponding instant center. The formulation presents a concise expression for a general order motion property of a general point on the rigid body with the magnitude and direction separated and identified. The results are summarized for up to the fifth order motion in the summary table. Based on the initial formulation, we propose the development of operational criteria using higher order properties to efficiently synthesize the motion of a J wheeled mobile platform.

Bradley (1932) showed that if two rigid spheres of radii R 1 and R 2 are placed in contact, they will adhere with a force 2πΔ R γ, where R is the equivalent radius R 1 R 1 /( R 1 + R 2 ) and Δγ is the surface energy or ‘work of adhesion’ (equal to γ1+γ2-γ12). Subsequently Johnson et al. (1971) (JKR theory) showed by a Griffith energy argument (assuming that contact over a circle of radius a introduces a surface energy -π a 2 Δγ) how the Hertz equations for the contact of elastic spheres are modifed by surface energy, and showed that the force needed to separate the spheres is equal to (3/2)πΔ R γ, which is independent of the elastic modulus and so appears to be universally applicable and therefore to conflict with Bradley9s answer. The discrepancy was explained by Tabor (1977), who identified a parameter 3 Δγ 2 / 3 / E * 2 / 3 \e governing the transition from the Bradley pull-off force 2π R Δ|γ to the JKR value (3/2)π R Δγ. Subsequently Muller et al. (1980) performed a complete numerical solution in terms of surface forces rather than surface energy, (combining the Lennard–Jones law of force between surfaces with the elastic equations for a half-space), and confirmed that Tabor9s parameter does indeed govern the transition. The numerical solution is repeated more accurately and in greater detail, confirming the results, but showing also that the load–;approach curves become S-shaped for values of μ greater than one, leading to jumps into and out of contact. The JKR equations describe the behaviour well for values of μ of 3 or more, but for low values of μ the simple Bradley equation better describes the behaviour under negative loads.

SUMMARY This paper extends an earlier investigation by Willis & Young (1987), in which an exact equation was derived for the magnetic field lines of the general axisymmetric magnetic multipole of arbitrary degree (n). Exact equations are derived for the magnetic field lines of two special non-axisymmetric magnetic multipoles of arbitrary degree, which may be classified as either symmetric or antisymmetric sectorial multipoles. These results have applications in studies of the possible nature of solar-terrestrial physics during geomagnetic polarity reversals. The magnetic-field-line configurations for these two non-axisymmetric multipoles are discussed in detail and are illustrated for low-degree (2 ≤ n ≤ 4) multipoles. It is shown that the existence of magnetic neutral lines is a characteristic feature of such non-axisymmetric multipoles. These neutral lines define localized regions of space (‘magnetospheric cusps’) through which charged particles can gain access to the inner magnetosphere. Moreover, both non-axisymmetric magnetic fields are locally perpendicular to a set of meridional planes and one is locally perpendicular to the equatorial plane. Therefore, there exist classes of charged-particle trajectories for which motion is confined completely to these meridional planes, or the equatorial plane, and every particle executing such planar motion is scattered non-adiabatically when it encounters a neutral line. This result emphasizes the need for a more detailed study of the motion of charged particles in both axisymmetric and non-axisymmetric multipole magnetic fields before firm conclusions can be reached on possible changes in geomagnetically trapped radiation during polarity reversals.

Camera motion is said to be planar if the direction of translation is perpendicular to the axis of rotation. A parabolic catadioptric camera is a camera realizing the orthogonal projection of rays reflected on a parabolic mirror. We consider the planar motion of a parabolic catadioptric camera, especially the motion restricted to a plane perpendicular to the optical axis, a common case in mobile robots working in urban environments. We begin by deriving the catadioptric fundamental matrix for such a motion and the intrinsic degrees of freedom in this matrix, which turn out to be 8. We show that the camera intrinsics and the 3D motion can be recovered from the fundamental matrix. We derive the necessary and sufficient condition for a fundamental matrix to be induced by a planar motion. Based on the additional constraint for a planar motion, we present an algorithm to compute epipolar geometry and recover the camera parameters and motion.

Camera motion is said to be planar if the direction of translation is perpendicular to the axis of rotation. A parabolic catadioptric camera is a camera realizing the orthogonal projection of rays reflected on a parabolic mirror. We consider the planar motion of a parabolic catadioptric camera, especially the motion restricted to a plane perpendicular to the optical axis, a common case in mobile robots working in urban environments. We begin by deriving the catadioptric fundamental matrix for such a motion and the intrinsic degrees of freedom in this matrix, which turn out to be 8. We show that the camera intrinsics and the 3D motion can be recovered from the fundamental matrix. We derive the necessary and sufficient condition for a fundamental matrix to be induced by a planar motion. Based on the additional constraint for a planar motion, we present an algorithm to compute epipolar geometry and recover the camera parameters and motion.

Constrained planar motion analysis by decomposition

Vision-based relative pose estimation serves as a core technology for high-precision localization in autonomous vehicles and mobile platforms. To overcome the limitations of conventional three-view pose estimation methods that rely heavily on dense feature matching and incur high computational costs, this paper proposes an efficient three-point correspondence algorithm based on planar motion constraints. The method constructs trifocal tensor constraint equations and develops a linearized three-point solution framework, enabling rapid relative pose estimation using merely three corresponding points in three views. In simulation experiments, we systematically analyzed the robustness of the algorithm under complex conditions that included image noise, angular deviation, and vibration. The method was further validated in real-world scenarios using the KITTI public dataset. Experimental results demonstrate that under the condition of satisfying the planar motion assumption, the proposed method achieves significantly improved computational efficiency compared with traditional methods (including general three-view methods, two-view planar motion estimation methods, and classical two-view methods), with the single-solution time reduced by more than 80% compared to general three-view methods. In the public dataset, our algorithm achieves a median rotation estimation error of less than 0.0545 degrees and maintains a translation estimation error of less than 2.1319 degrees. The proposed method exhibits higher computational efficiency and better numerical stability compared to conventional algorithms. This research provides an effective pose estimation solution with real-time performance and high accuracy for planar motion platforms such as autonomous vehicles and indoor mobile robots, demonstrating substantial engineering application value.

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Shoulder motions consist of a composite movement of three joints and one pseudo-joint, which together dictate the humerothoracic motion. The purpose of this work was to quantify the location of the centre of rotation (CoR) of the shoulder complex as a whole. Dynamic motion of 12 participants was recorded using optical motion tracking during coronal, scapular and sagittal plane elevation. The instantaneous CoR was found for each angle of elevation using helical axes projected onto the three planes of motion. The location of an average CoR for each plane was evaluated using digitised and anthropometric measures for normalisation. When conducting motion in the coronal, scapular, and sagittal planes, respectively, the coefficients for locating the CoRs of the shoulder complex are −61%, −61%, and −65% of the anterior–posterior dimension – the vector between the midpoint of the incisura jugularis and the xiphoid process and the midpoint of the seventh cervical vertebra and the eighth thoracic vertebra; 0%, −1%, and −2% of the superior–inferior dimension – the vector between the midpoint of the acromioclavicular joints and the midpoint of the anterior superior iliac spines; and 57%, 57%, and 78% of the medial–lateral dimension −0.129 times the height of the participant. Knowing the location of the CoR of the shoulder complex as a whole enables improved participant positioning for evaluation and rehabilitation activities that involve movement of the hand with a fixed radius, such as those that employ isokinetic dynamometers.

The goal of this study was to investigate the effect of ligament failure on the instantaneous center of rotation (ICR) in the lower lumbar spine. A 3D finite element model of the L4-5 segment was obtained and validated. Ligament failure was simulated by reducing ligaments in a stepwise manner from posterior to anterior. A pure bending moment of 7.5 Nm was applied to the model in 3 anatomical planes for the purpose of validation, and a 6-Nm moment was applied to analyze the effect of ligament failure. For each loading case, ligament reduction step, and load increment, the range of motion of the segment and the ICR of the mobile (L-4) vertebra were calculated and characterized. The present model showed a consistent increase in the range of motion as the ligaments were removed, which was in agreement with the literature reporting the kinematics of the L4-5 segment. The shift in the location of the ICR was below 5 mm in the sagittal plane and 3 mm in both the axial and coronal planes. The location of the ICR changed in all planes of motion with the simulation of multiple ligament injury. The removal of the ligaments also changed the load sharing within the motion segment. The change in the center of rotation of the spine together with the change in the range of motion could have a diagnostic value, revealing more detailed information on the type of injury, the state of the ligaments, and load transfer and sharing characteristics of the segment.

Analyzing center of rotation during opening and closing movements of the mandible using computer simulations

In this paper we study combined translational and rotational (general) motion of planar rigid bodies in the presence of dry coulomb friction contact. Despite the cases where the body has pure translational/ rotational motion or can be assumed as a point mass, during the general motion, distinct points of the rigid body move in different directions which cause the friction force vector at each point to be different. Therefore, the direction and the magnitude of the overall friction force cannot be intuitively defined. Here the concept of instantaneous center of rotation is used as an effective method to determine the resultant friction force and moment. The main contribution of this paper is to propose novel stick-slip switching conditions for the general in-plane motion of rigid bodies. Simulation results for some combination of external forces are provided and some experimental tests are designed and conducted for practical verification of the proposed stick-slip conditions.