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
[1.] In literature, for planar and spatial arms with a few segments, this has often been achieved by explicit trigonometric formulae developed from the geometry of the specific arm @cite_10 @cite_18 . However, very often such formulations are limited to the specific arm that the IK is designed for, becomes increasingly complex with the increase in the number of segments, and does suffer from isolated singularities and or discontinuities. For more complex arms, traditionally numerical gradient-decent type algorithms have been used @cite_3 @cite_7 @cite_21 @cite_9 . However, the problem very often being nonlinear and non-convex, guarantees of completeness or even continuity are difficult to achieve. Furthermore, numerical techniques are often computationally expensive. A mixed numerical and analytic technique has been used in @cite_13 for computing inverse kinematics in context of path planning for the end effector of an arm with @math segments and point obstacles in the environment. A closed form solutions to the inverse kinematics problem has recently been hinted in @cite_11 using a approach. But the problem being addressed for a spatial arm, some numerical techniques were also adopted.
@cite_13 @cite_18 @cite_7 @cite_9 @cite_21 @cite_3 @cite_10 @cite_11
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A new and efficient kinematic position and velocity solution scheme for spatial hyper-redundant manipulators is presented. The manipulator's arm has discrete links and universal joints. Backbone curve concepts and a modal approach are used to resolve the manipulator's redundancy. The effects of the mode shapes and the slope of the backbone curve at the starting point on the workspace are studied. It is shown that the usage of conventional mode shapes limits the workspace of the hyper-redundant arm. By introducing new mode shapes, an improved workspace is obtained. A simple and efficient recursive fitting method is introduced to avoid complications involved with solving systems of nonlinear algebraic equations. This method also guarantees the existence of solutions for the inverse kinematic problem at the velocity level. Velocity properties of the backbone curve are investigated and the inverse velocity propagation is solved for the spatial hyper-redundant arm. The velocity propagation scheme is recursive and is efficiently applicable to any number of links. Few computational models have addressed the spatiotemporal features of unconstrained three-dimensional (3D) arm motion. Empirical observations made on hand paths, speed profiles, and arm postures during point-to-point movements led to the assumption that hand path and arm posture are independent of movement speed, suggesting that the geometric and temporal properties of movements are decoupled. In this study, we present a computational model of 3D movements for an arm with four degrees of freedom based on the assumption that optimization principles are separately applied at the geometric and temporal levels of control. Geometric properties (path and posture) are defined in terms of geodesic paths with respect to the kinetic energy metric in the Riemannian configuration space. Accordingly, a geodesic path can be generated with less muscular effort than on any other, nongeodesic path, because the sum of all configuration-speed-dependent torques vanishes. The temporal properties of the movement (speed) are determined in task space by minimizing the squared jerk along the selected end-effector path. The integration of both planning levels into a single spatiotemporal representation simplifies the control of arm dynamics along geodesic paths and results in movements with near minimal torque change and minimal peak value of kinetic energy. Thus, the application of Riemannian geometry allows for a reconciliation of computational models previously proposed for the description of arm movements. We suggest that geodesics are an emergent property of the motor system through the exploration of dynamical space. Our data validated the predictions for joint trajectories, hand paths, final postures, speed profiles, and driving torques. 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. Trajectory and kinematics of drawing movements are mutually constrained by functional relationships that reduce the degrees of freedom of the hand-arm system. Previous investigations of these relationships are extended here by considering their development in children between 5 and 12 years of age. Performances in a simple motor task—the continuous tracing of elliptic trajectories—demonstrate that both the phenomenon of isochrony (increase of the average movement velocity with the linear extent of the trajectory) and the so-called two-thirds power law (relation between tangential velocity and curvature) are qualitatively present already at the age of 5. The quantitative aspects of these regularities evolve with age, however, and steady-state adult performance is not attained even by the oldest children. The power-law formalism developed in previous reports is generalized to encompass these developmental aspects of the control of movement. Two general frameworks are currently available to conceptualize the motor-control problem. Broadly, the two frameworks differ in the answer that they give to the question "Where do form and structure come from?" According to the motor-program view (e.g., Schmidt, 1988), form and structure in a movement come from a central abstract representation of the intended result. Thus, making a movement entails a detailed mapping of this representation into an appropriate motor plan. The competing task-dynamic view (e.g., Kugler, 1989; Saltzman & Kelso, 1987) emphasizes instead the global morphogenetic power of the nonlinearities of the motor system. These two views need not be mutually exclusive. In fact, different classes of movements may require different conceptual frameworks for their analysis. At any rate (at least in the case of drawing movements considered here), an internal representation of the intended trajectory and kinematics must be available before execution. Consequently, much effort is being devoted to understanding how this internal model is translated into a set of motor commands. In this respect, one basic conceptual difficulty is because the geometry of the arm-forearm-hand system affords a large number of degrees of freedom. Provided that there are no constraints in the workspace, the only trajectory that matters in many manual