A tetrahedron approach for a unique closed‐form solution of the forward kinematics of six‐dof parallel mechanisms with multiconnected joints

This article presents a new formulation approach that uses tetrahedral geometry to determine a unique closed‐form solution of the forward kinematics of six‐dof parallel mechanisms with multiconnected joints. For six‐dof parallel mechanisms that have been known to have eight solutions, the proposed formulation, called the Tetrahedron Approach, can find a unique closed‐form solution of the forward kinematics using the three proposed Tetrahedron properties. While previous methods to solve the forward kinematics involve complicated algebraic manipulation of the matrix elements of the orientation of the moving platform, or closed‐loop constraint equations between the moving and the base platforms, the Tetrahedron Approach piles up tetrahedrons sequentially to directly solve the forward kinematics. Hence, it allows significant abbreviation in the formulation and provides an easier systematic way of obtaining a unique closed‐form solution. © 2002 Wiley Periodicals, Inc.     

Wireframe Projections: Physical Realisability of Curved Objects and Unambiguous Reconstruction of Simple Polyhedra

The reconstruction of an object from a single 2D projection of a 3D wireframe model is a vision problem with applications in CAD/CAM and computer graphics.We propose an algorithm for the interpretation of wireframe projections based on assigning semantic and numerical depth labels to lines. This method allows us to state necessary and sufficient conditions for the physical realisability of a wireframe projection of a curved object. The presence of linear features provides further constraints on the positions of object vertices. For example, each straight line gives rise to a coplanarity constraint between a set of object vertices.We show that extra information, such as vanishing points, parallel lines or user-entered depth-parity information, is sufficient to uniquely determine the face-circuits in wireframe projections of polyhedra with simple trihedral vertices. In fact, a polyhedron with simple trihedral vertices can be unambiguously reconstructed from its 3D wireframe model.     

Perspective angle transform: Principle of shape from angles

This paper introduces a new relation, called the perspective angle transform (PAT), to deal with shape-from-angle problems, together with its application to 3D configuration recovery from an image. Three main aspects of PAT are presented in this paper. The first is the derivation of PAT which holds between the apparent and real angles under perspective projection. A concept is proposed of a virtual image plane and a new coordinate system, named the first perspective moving coordinate (FPMC) system, for analysis of the shape-from-range problems. Characteristics of FPMC are discussed briefly. The second point is the analysis of PAT properties, for which the general PAT form is introduced on another new coordinate system, named the second perspective moving coordinate (SPMC) system. Using this general form, the gradient of the plane including the real angle is constrained on a curve (PAT curve) of the fourth degree in the virtual image plane. The characteristics of PAT curves and relations between the general PAT form and skewed symmetry are summarized briefly. The last point concerns an application of PAT. As an example, we treat 3D configuration recovery from three arbitrary line segments in the image plane. This recovery corresponds to a generalization of the right-angled interpretation problem proposed and discussed by S.T. Barnard. A solution to the problem is shown using the general PAT form in conjunction with the concept of a virtual crossing point. The right-angled interpretation problem is ascribed to a quadratic equation. A simplified solution is also provided for the case where three line segments have a common crossing point in the image plane. This solution is based on the primary PAT form and is applicable to interpretation of a trihedral vertex with at least two right angles.This paper is an enlarged and polished version of a previous paper in Proc. IEEE Conf CVPR'86 [11]. A part of this paper summarizes the contents of the previous paper. However, perspective projection is treated here in more rigorous and more compact manner than in the previous paper.     

An efficient local approach for the path generation of robot manipulators

In this article an efficient local approach for the path generation of robot manipulators is presented. The approach is based on formulating a simple nonlinear programming problem. This problem is considered as a minimization of energy with given robot kinematics and subject to the robot requirements and a singularities avoidance constraint. From this formulation a closed form solution is derived which has the properties that allows to pursue both singularities and obstacle avoidance simultaneously; and that it can incorporate global information. These properties enable the accomplishment of the important task that while a specified trajectory in the operational space can be closely followed, also a desired joint configuration can be attained accurately at a given time. Although the proposed approach is primarily developed for redundant manipulators, its application to nonredundant manipulators is examplified by considering a particular commercial manipulator.     

Tracking and Motion Estimation of the Articulated Object: a Hierarchical Kalman Filter Approach

Real-time motion capture plays a very important role in various applications, such as 3D interface for virtual reality systems, digital puppetry, and real-time character animation. In this paper we challenge the problem of estimating and recognizing the motion of articulated objects using theoptical motion capturetechnique. In addition, we present an effective method to control the articulated human figure in realtime.The heart of this problem is the estimation of 3D motion and posture of an articulated, volumetric object using feature points from a sequence of multiple perspective views. Under some moderate assumptions such as smooth motion and known initial posture, we develop a model-based technique for the recovery of the 3D location and motion of a rigid object using a variation of Kalman filter. The posture of the 3D volumatric model is updated by the 2D image flow of the feature points for all views. Two novel concepts – the hierarchical Kalman filter (KHF) and the adaptive hierarchical structure (AHS) incorporating the kinematic properties of the articulated object – are proposed to extend our formulation for the rigid object to the articulated one. Our formulation also allows us to avoid two classic problems in 3D tracking: the multi-view correspondence problem, and the occlusion problem. By adding more cameras and placing them appropriately, our approach can deal with the motion of the object in a very wide area. Furthermore, multiple objects can be handled by managing multiple AHSs and processing multiple HKFs.We show the validity of our approach using the synthetic data acquired simultaneously from the multiple virtual camera in a virtual environment (VE) and real data derived from a moving light display with walking motion. The results confirm that the model-based algorithm works well on the tracking of multiple rigid objects.     

Optimum design of composite shells subject to natural frequency constraints

A technique for the optimum (minimum weight) design of a composite shell subject to constraints on its natural frequencies is presented. The optimization problem is posed as a general mathematical programming problem in which one or more of the inequality constraints involves the shell natural frequencies, which must be evaluated numerically during the optimization. For this reason, a method for numerically evaluating the natural frequencies of composite shells is also presented. The method is based upon the finite element method of structural analysis and Rayleigh's principle. Because the element used is applicable to anisotropic shells of arbitrary shape, the method is very general. By using Rayleigh's principle, the necessity of assembling overall mass and stiffness matrices for the shell is eliminated. The optimization is performed by nondimensionalizing the mathematical programming problem and using the penalty function method of Fiacco and McCormick to transform the problem to a sequence of unconstrained minimizations having solutions which converge to the solution of the original (constrained) problem. The unconstrained minimizations are performed using the variable metric method of Fletcher and Powell. Derivatives of the nondimensional frequency constraints are evaluated numerically using difference equations. The frequency calculation method is demonstrated by calculating the fundamental frequency for the transverse vibration mode of a multilayered cylindrical shell with fixed overall geometry and variable composite geometry. Results indicate that the frequency increases with increasing fiber orientation angle, fiber volume fraction, or lamina thickness. The optimization technique is demonstrated by minimizing the weight of the shell discussed above subject to a constraint on its fundamental transverse frequency. The design variables are the fiber orientation angle, the fiber volume fraction, and the lamina thickness. Results are presented and explained in terms of the physical aspects of the problem.     

A hierarchical multiple-view approach to three-dimensional objectrecognition

A hierarchical approach is proposed for solving the surface and vertex correspondence problems in multiple-view-based 3D object-recognition systems. The proposed scheme is a coarse-to-fine search process, and a Hopfield network is used at each stage. Compared with conventional object-matching schemes, the proposed technique provides a more general and compact formulation of the problem and a solution more suitable for parallel implementation. At the coarse search stage, the surface matching scores between the input image and each object model in the database are computed through a Hopfield network and are used to select the candidates for further consideration. At the fine search stage, the object models selected from the previous stage are fed into another Hopfield network for vertex matching. The object model that has the best surface and vertex correspondences with the input image is finally singled out as the best matched model. Experimental results are reported using both synthetic and real range images to corroborate the proposed theory.     

A simple method for fitting of bounding rectangle to closed regions

In this paper, we introduce a new approach for fitting of a bounding rectangle to closed regions. In this approach the coordinates of the vertices are computed directly using a closed-form solution. This approach is based on simple coordinate geometry and uses the boundary points of regions. Using a least-square approach we determine the directions of major and minor axes of the object, which gives the orientation of the object. The four vertexes of the bounding rectangle are computed by pair wise solving the four straight lines. Examples from synthetic data and some real-life data show that the approach is both accurate and efficient.     

Recovering the position and orientation of free-form objects fromimage contours using 3D distance maps

The accurate matching of 3D anatomical surfaces with sensory data such as 2D X-ray projections is a basic problem in computer and robot assisted surgery, In model-based vision, this problem can be formulated as the estimation of the spatial pose (position and orientation) of a 3D smooth object from 2D video images. The authors present a new method for determining the rigid body transformation that describes this match. The authors' method performs a least squares minimization of the energy necessary to bring the set of the camera-contour projection lines tangent to the surface. To correctly deal with projection lines that penetrate the surface, the authors consider the minimum signed distance to the surface along each line (i.e., distances inside the object are negative). To quickly and accurately compute distances to the surface, the authors introduce a precomputed distance map represented using an octree spline whose resolution increases near the surface. This octree structure allows the authors to quickly find the minimum distance along each line using best-first search. Experimental results for 3D surface to 2D projection matching are presented for both simulated and real data. The combination of the authors' problem formulation in 3D, their computation of line to surface distances with the octree-spline distance map, and their simple minimization technique based on the Levenberg-Marquardt algorithm results in a method that solves the 3D/2D matching problem for arbitrary smooth shapes accurately and quickly     

Improving object orientation estimates by considering multiple viewpoints

This article describes a probabilistic approach for improving the accuracy of general object pose estimation algorithms. We propose a histogram filter variant that uses the exploration capabilities of robots, and supports active perception through a next-best-view proposal algorithm. For the histogram-based fusion method we focus on the orientation of the 6 degrees of freedom (DoF) pose, since the position can be processed with common filtering techniques. The detected orientations of the object, estimated with a pose estimator, are used to update the hypothesis of its actual orientation. We discuss the design of experiments to estimate the error model of a detection method, and describe a suitable representation of the orientation histograms. This allows us to consider priors about likely object poses or symmetries, and use information gain measures for view selection. The method is validated and compared to alternatives, based on the outputs of different 6 DoF pose estimators, using real-world depth images acquired using different sensors, and on a large synthetic dataset.     

Analytic minimum-norm solution for rate coordination in redundant manipulators

In this article, we present a novel method that results in efficient minimum norm solution for the rate coordination problem in redundant manipulators. The theory is developed based upon a geometric interpretation that, for minimum norm criterion, vectors orthogonal to constraint space should pass through the origin of the solution space. It is shown that, for any spatial manipulator with 1, 2, or 3 degrees of redundancy, the minimum norm rate solution can be derived in analytic closed form. An example of the analytic formulation is given for a 3R planar case, substantiated with simulation results. The behavior of this algorithm in nonredundant and near singular situations is also discussed. The method offers an equivalent but much more efficient alternative to using the pseudoinverse in redundancy resolution and, in fact, is applicable to any underdetermined linear system. An alternative formulation of pseudoinverse arrived at in the course of the development is also presented. © 1992 John Wiley & Sons, Inc.     

基于变结构控制的固定前置角制导律设计

弹载合成孔径雷达（SAR）的方位成像能力受飞行导弹的观测角影响,其末制导段采用前侧视工作方式。因此,必需合理设计制导律使前侧视条件成立。为了保证末制导段弹目间具有合适的夹角,采用变结构控制方法设计了一种固定前置角制导律,该制导律能够解决合成孔径雷达导引头对于前置角约束的问题。在制导律设计过程中,首先建立了弹目相对运动关系模型和具有终端角度约束时的视线角变化模型;在此基础上采用变结构控制的方法设计了固定前置角制导律;进而,对该固定前置角导引律进行了性能分析,得出了其一般攻击特性;最后,通过弹道仿真论证了其正确性与有效性。     

约束可满足方法对单幅图片的三维猜测与个体识别

从二维图像提取到的特征点相对位置信息,利用对于目标物体的先验知识如边缘比例、夹角特征等,形成基于投影成像规则的变量约束方程,从而将物体姿态计算问题转化为一个非线性优化问题,通过多个局部约束线索形成全局约束,能够迅速对物体的姿态做出比较准确的估计,从而推算出视角等信息.实现这个转化的一个前提假设是,对于我们理解图像中物体的姿态,特征位置点的相对位置而不是绝对位置,起着关键作用.因此计算是在一个假设的深度上进行的,从效果来看这样的假设并不影响物体的住姿计算.本文的方法计算量小,利用几何特征来识别稳定、可靠、泛化能力好,实践证明使用几何特征的约束可满足方法能获得关于姿态的极少量可能解,识别出的姿态在各条边之间的比例关系上具有不变性,继而可以将其应用于不变性识别的实际问题之中.     

Shock Isolation with Anticipating Control for External Disturbances of Various Shapes

Limiting performance analysis is performed for isolation of an object on a movable base from short-duration impact excitations by means of an active shock isolator with anticipating control. The external disturbance (excitation) is modeled by the time history of the absolute acceleration of the base. The control is performed by a force that acts between the base and the object to be protected. The absolute value of the control force is subject to a constraint. A procedure is proposed for constructing optimal anticipating controls that minimize the peak magnitude of the displacement of the object relative to the base for external disturbances from a certain class. For a number of types of external disturbances, the solution of the optimal control problem is obtained in closed form. The controls that are constructed in closed form are modified to be applicable for the disturbances for which closedform solutions are absent.     

Iterative Pose Estimation Using Coplanar Feature Points

This paper presents a new method for the computation of the position and orientation of a camera with respect to a known object, using four or morecoplanarfeature points. Starting with the scaled orthographic projection approximation, this method iteratively refines up to two different pose estimates, and provides an associated quality measure for each pose. When the camera distance is large compared with the object depth, or when the accuracy of feature point extraction is low because of image noise, the quality measures for the two poses are similar, and the two pose estimates are plausible interpretations of the available information. In contrast, known methods using a closed form pose solution for four coplanar points are not robust for distant objects in the presence of image noise because they provide only one of the two possible poses and may choose the wrong pose.     

Task-oriented approaches to the inverse kinematics problem for a reclaimer excavating and transporting raw material

The inverse kinematics problem of the reclaimer that excavates and transports raw materials in a raw yard is investigated. Link coordinates are introduced by the Denavit-Hartenberg representation. The middle bucket, among the buckets which are in contact with the raw material pile, is treated as the end-effector of a reclaimer. Two task-oriented approaches are investigated. The first approach assumes the complete removal of a pile via a level plan on the pile. In this case, the end-effector is assumed to be a particle moving in the three-dimensional (3D) space. A closedform solution is provided. The second approach assumes the reclamation of an arbitrary pile. The end-effector is regarded as a rigid body which requires both position and orientation information. Because there is no solution for the second approach in general, an approximate solution is provided by exploiting a geometric constraint. The 3D information near the excavation point is approximated as a plane and the orientation of the end-effector is given in the normal direction of the plane.     

On the constraints violation in forward dynamics of multibody systems

It is known that the dynamic equations of motion for constrained mechanical multibody systems are frequently formulated using the Newton–Euler’s approach, which is augmented with the acceleration constraint equations. This formulation results in the establishment of a mixed set of partial differential and algebraic equations, which are solved in order to predict the dynamic behavior of general multibody systems. The classical solution of the equations of motion is highly prone to constraints violation because the position and velocity constraint equations are not fulfilled. In this work, a general and comprehensive methodology to eliminate the constraints violation at the position and velocity levels is offered. The basic idea of the described approach is to add corrective terms to the position and velocity vectors with the intent to satisfy the corresponding kinematic constraint equations. These corrective terms are evaluated as a function of the Moore–Penrose generalized inverse of the Jacobian matrix and of the kinematic constraint equations. The described methodology is embedded in the standard method to solve the equations of motion based on the technique of Lagrange multipliers. Finally, the effectiveness of the described methodology is demonstrated through the dynamic modeling and simulation of different planar and spatial multibody systems. The outcomes in terms of constraints violation at the position and velocity levels, conservation of the total energy and computational efficiency are analyzed and compared with those obtained with the standard Lagrange multipliers method, the Baumgarte stabilization method, the augmented Lagrangian formulation, the index-1 augmented Lagrangian, and the coordinate partitioning method.     

On simultaneous shape and material layout optimization of shell structures

This work presents a computational method for integrated shape and topology optimization of shell structures. Most research in the last decades considered both optimization techniques separately, seeking an initial optimal topology and refining the shape of the solution later. The method implemented in this work uses a combined approach, were the shape of the shell structure and material distribution are optimized simultaneously. This formulation involves a variable ground structure for topology optimization, since the shape of the shell mid-plane is modified in the course of the process. It was considered a simple type of design problem, where the optimization goal is to minimize the compliance with respect to the variables that control the shape, material fraction and orientation, subjected to a constraint on the total volume of material. The topology design problem has been formulated introducing a second rank layered microestructure, where material properties are computed by a “smear-out” procedure. The method has been implemented into a general optimization software called ODESSY, developed at the Institute of Mechanical Engineering in Aalborg. The computational model was tested in several numerical applications to illustrate and validate the approach.     

A finite element method for the analysis of rubber parts, experimental and analytical assessment

A finite element method for the analysis of non linear rubber type parts is developed. Although using a penalty function approach, it is not based on reduced integration techniques but on a reduced constraint concept.

We compare computations to a closed form solution for a Mooney material. Finally computations of a test rubber part using a 9 coefficient Rivlin Law are compared to measurements.     

Non-rigid metric reconstruction from perspective cameras

The metric reconstruction of a non-rigid object viewed by a generic camera poses new challenges since current approaches for Structure from Motion assume the rigidity constraint of a shape as an essential condition. In this work, we focus on the estimation of the 3-D Euclidean shape and motion of a non-rigid shape observed by a perspective camera. In such case deformation and perspective effects are difficult to decouple – the parametrization of the 3-D non-rigid body may mistakenly account for the perspective distortion. Our method relies on the fact that it is often a reasonable assumption that some of the points on the object’s surface are deforming throughout the sequence while others remain rigid. Thus, relying on the rigidity constraints of a subset of rigid points, we estimate the perspective to metric upgrade transformation. First, we use an automatic segmentation algorithm to identify the set of rigid points. These are then used to estimate the internal camera calibration parameters and the overall rigid motion. Finally, we formulate the problem of non-rigid shape and motion estimation as a non-linear optimization where the objective function to be minimized is the image reprojection error. The prior information that some of the points in the object are rigid can also be added as a constraint to the non-linear minimization scheme in order to avoid ambiguous configurations. We perform experiments on different synthetic and real data sets which show that even when using a minimal set of rigid points and when varying the intrinsic camera parameters it is possible to obtain reliable metric information.