Using results on the topology of moduli space of polygons [Jaggi, 92; Kapovich and Millson, 94], it can be shown that for a planar robot arm with @math segments there are some values of the base-length, @math , at which the configuration space of the constrained arm (arm with its end effector fixed) has two disconnected components, while at other values the constrained configuration space has one connected component. We first review some of these known results. Then the main design problem addressed in this paper is the construction of pairs of continuous inverse kinematics for arbitrary robot arms, with the property that the two inverse kinematics agree when the constrained configuration space has a single connected component, but they give distinct configurations (one in each connected component) when the configuration space of the constrained arm has two components. This design is made possible by a fundamental theoretical contribution in this paper -- a classification of configuration spaces of robot arms such that the type of path that the system (robot arm) takes through certain critical values of the forward kinematics function is completely determined by the class to which the configuration space of the arm belongs. This classification result makes the aforesaid design problem tractable, making it sufficient to design a pair of inverse kinematics for each class of configuration spaces (three of them in total). We discuss the motivation for this work, which comes from a more extensive problem of motion planning for the end effector of a robot arm requiring us to continuously sample one configuration from each connected component of the constrained configuration spaces. We demonstrate the low complexity of the presented algorithm through a Javascript + HTML5 based implementation available at this http URL
The IK algorithm that we propose and use in this paper is an analytically computable one of @math complexity ( @math being the number of segments of the arm), akin to the triangulation approach of @cite_11 . However our additional requirement of identifying and passing through certain critical points in the configuration space, as will be described in the next point, has necessitated a more careful and formal construction of the IK algorithm. In particular, our algorithm has a recursive or incremental flavor to it, wherein we construct the inverse kinematics by breaking up the arm into smaller components.
@cite_11
2139795879 2107275435 2962743311 2132558143
In this paper we develop a set of inverse kinematics algorithms suitable for an anthropomorphic arm or leg. We use a combination of analytical and numerical methods to solve generalized inverse kinematics problems including position, orientation, and aiming constraints. Our combination of analytical and numerical methods results in faster and more reliable algorithms than conventional inverse Jacobian and optimization-based techniques. Additionally, unlike conventional numerical algorithms, our methods allow the user to interactively explore all possible solutions using an intuitive set of parameters that define the redundancy of the system. In this paper, a feature extraction method for online classification problems is presented by extending Kernel principal component analysis (KPCA). The proposed incremental KPCA (IKPCA) constructs a nonlinear high-dimensional feature space incrementally by not only updating eigen-axes but also adding new eigen-axes. The augmentation of a new eigen-axis is carried out when the accumulation ratio falls below a threshold value. We mathematically derive the incremental update equations of eigen-axes and the accumulation ratio without keeping all training samples. From the experimental results, we conclude that the proposed IKPCA works well as an incremental learning algorithm of a feature space in the sense that a minimum number of axes are augmented to maintain a designated accumulation ratio, and that the eigenvectors with major eigenvalues can converge closely to those of the batch type of KPCA. In addition, the recognition accuracy of IKPCA is similar to or slightly better than that of KPCA Motivated by applications in computational advertising and systems biology, we consider the problem of identifying the best out of several possible soft interventions at a source node @math in an acyclic causal directed graph, to maximize the expected value of a target node @math (located downstream of @math ). Our setting imposes a fixed total budget for sampling under various interventions, along with cost constraints on different types of interventions. We pose this as a best arm identification bandit problem with @math arms where each arm is a soft intervention at @math and leverage the information leakage among the arms to provide the first gap dependent error and simple regret bounds for this problem. Our results are a significant improvement over the traditional best arm identification results. We empirically show that our algorithms outperform the state of the art in the Flow Cytometry data-set, and also apply our algorithm for model interpretation of the Inception-v3 deep net that classifies images. Real-time control of the end-effector of a humanoid robot in external coordinates requires computationally efficient solutions of the inverse kinematics problem. In this context, this paper investigates inverse kinematics learning for resolved motion rate control (RMRC) employing an optimization criterion to resolve kinematic redundancies. Our learning approach is based on the key observations that learning an inverse of a nonuniquely invertible function can be accomplished by augmenting the input representation to the inverse model and by using a spatially localized learning approach. We apply this strategy to inverse kinematics learning and demonstrate how a recently developed statistical learning algorithm, locally weighted projection regression, allows efficient learning of inverse kinematic mappings in an incremental fashion even when input spaces become rather high dimensional. Our results are illustrated with a 30-DOF humanoid robot.