Smooth trajectory generation for rotating extensible manipulators

In this study the generation of smooth trajectories of the end-effector of a rotating extensible manipulator arm is considered. Possible trajectories are modelled using Cartesian and polar piecewise cubic interpolants expressed as polynomial Hermite-type functions. The use of polar piecewise cubic interpolants devises continuous first and - in some cases - second order derivatives and allows easy calculation of kinematics variables such as velocity and acceleration. Moreover, the manipulator equations of motion can be easily handled, and the constrained trajectory of the non-active end of the manipulator derived directly from the position of the end-effector. To verify the proposed approach, numerical simulations are conducted for two different configurations

ExtrudeNet: Unsupervised Inverse Sketch-and-Extrude for Shape Parsing

Sketch-and-extrude is a common and intuitive modeling process in computer aided design. This paper studies the problem of learning the shape given in the form of point clouds by inverse sketch-and-extrude. We present ExtrudeNet, an unsupervised end-to-end network for discovering sketch and extrude from point clouds. Behind ExtrudeNet are two new technical components: 1) an effective representation for sketch and extrude, which can model extrusion with freeform sketches and conventional cylinder and box primitives as well; and 2) a numerical method for computing the signed distance field which is used in the network learning. This is the first attempt that uses machine learning to reverse engineer the sketch-and-extrude modeling process of a shape in an unsupervised fashion. ExtrudeNet not only outputs a compact, editable and interpretable representation of the shape that can be seamlessly integrated into modern CAD software, but also aligns with the standard CAD modeling process facilitating various editing applications, which distinguishes our work from existing shape parsing research. Code is released at https://github.com/kimren227/ExtrudeNet.Comment: Accepted to ECCV 202

A combined polar and Cartesian piecewise trajectory generation and analysis of a robotic arm

In this paper a combined polar-Cartesian approach to generate a smooth trajectory of a robotic arm along priori defined via-points is presented. Due to the characteristics/- geometry of the robotic arm, cylindrical coordinates are associated with the trajectory of motion. Possible trajectories representing the system dynamics are generated by mix matching higher order polar piecewise polynomials used to devise the radial trajectory and Cartesian piecewise polynomials used to calculate the related height in a normal plane unfolded along the radial trajectory of the motion. To describe the kinematic properties of the end-effector a moving non-inertial orthonormal Frenet frame is considered. Using the Frenet frame, the components of the velocity and acceleration along the frame unit vectors are calculated. Numerical simulations are performed for two different configurations in order to validate the approach

A path planning approach of 3D manipulators using zenithal gnomic projection and polar piecewise interpolation.

In this paper, the mathematical modeling and trajectory planning of a 3D rotating manipulator composed of a rotating-prismatic joint and multiple rigid links is considered. Possible trajectories of the end effector of the manipulator—following a sequence of 3D target points under the action of 2 external driving torques and an axial force—are modeled using zenithal gnomic projections and polar piecewise interpolants expressed as polynomial Hermite-type functions. Because of the geometry of the manipulator, the time-dependent generalized coordinates are associated with the spherical coordinates named the radial distance related to the manipulator length, and the polar and azimuthal angles describing the left and right and, respectively, up and down motion of the manipulator. The polar trajectories (left and right motion) of the end effector are generated using a inverse geometric transformation applied to the polar piecewise interpolants that approximate the gnomic projective trajectory of the 3D via-points. The gnomic via-points—located on a projective plane situated on the northern hemisphere—are seen from the manipulator base location, which represents the center of rotation of the extensible manipulator. The related azimuthal trajectory (up and down motion) is generated by polar piecewise interpolants on the azimuthal angles. Smoothness of the polygonal trajectory is obtained through the use of piecewise interpolants with continuous derivatives, while the kinematics and dynamics implementation of the model is well suited to computer implementation (easy calculation of kinematics variables) and simulation. To verify the approach and validate the model, a numerical example—implemented in Matlab—is presented, and the results are discussed

Active Manifold-Geodesics: A Riemannian View On Active Subspaces With Shape Sensitivity Applications

Aerospace designers routinely manipulate shapes in engineering systems toward design goals and study changes in the modeled system to facilitate new intuitions about the physical processes---e.g., shape optimization and parameter sensitivity analysis of an airfoil. The computational tools for such manipulation can include parameterized geometries, where the parameters provide a set of independent variables that control the geometry. Active subspaces provide an intuitive change of basis for studying differentiable functions with Euclidean domain of dimension greater than or equal to two. Recent work has developed and exploited active subspaces in the composition from geometry parameters to design quantities of interest (e.g., lift or drag of an airfoil); the active subspace is spanned by a set of directions in a parameter space which change the associated quantity of interest more, on average over the parameter design space, than orthogonal directions. Consequently, the active directions produce insight-rich geometry perturbations for a specific quantity of interest; however, these perturbations also depend on the chosen geometry parameterization. Several engineering applications explore this shape-parameterization dependency for optimization and sensitivity analysis. However, selection of a parameterization restricts any subsequent analysis to the class of chosen parameterization; including the approximation of an active subspace. Defining a precise calculus of shapes independent of engineering parameterizations requires a new interpretation of the domain of scalar-valued functions dependent on these shapes. The space of shapes admits a topological structure of a smooth manifold, a more general non-Euclidean domain for quantities of interest. This work extends the computation of active subspaces to differentiable functions defined on smooth manifolds M. We seek ordered geodesics defining submanifolds of a Riemannian manifold (M, g), endowed with a metric g, which change the differentiable function iv more, by an analogous globalizing notion of the average. These submanifolds representing analogous subspaces on a more general non-Euclidean domain are referred to as active manifold-geodesics. However, there are competing intrinsic and extrinsic perspectives regarding computations and approximations on Riemannian manifolds. Extrinsic perspectives rely on the existence of isometric embeddings of the manifold into an ambient Euclidean space while intrinsic perspectives work entirely with objects defined only on the manifold, i.e., not requiring an isometric embedding. The continuous form of an analogous average outer product of the gradient is presented from an intrinsic perspective. A discretization and approximation of the eigenspaces of the proposed intrinsic extension is applied to the sphere S2 &sub; R3 as an example which can be visualizedThese routines are then generalized to a matrix manifold of landmark-affine shapes to inform a global shape sensitivity analysis of transonic airfoils---independent of a shape-parameterization.</p

Active Manifold-Geodesics: A Riemannian View on Active Subspaces with Shape Sensitivity Applications

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