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
2113407491 1564897360 2081897478 2146097584 2159806973 2139691007 1495118766 2159911618
In this paper we develop a systematic topological approach to motion planning for a planar 2-R manipulator with point obstacles. By considering components in the free space for the second joint as the first joint varies, we build a two-dimensional array representing the cells of the free space and an associated graph representing the boundaries of those cells. Using this graph, we derive a closed formula for the number of components of the free space. At the same time we solve the motion existence problem, namely, when are two arbitrary configurations in the same component? If so, we develop two explicit algorithms for constructing the path - a middle path method and a linear interpolation method. These algorithms give complete solutions to the path planning problem. Extensive examples are worked out which verify the correctness and efficiency of the resulting program. Then we briefly discuss how these methods generalize to a 3-R planar manipulator. INTRODUCTION: Brief History. Multifingered Hands and Dextrous Manipulation. Outline of the Book. Bibliography. RIGID BODY MOTION: Rigid Body Transformations. Rotational Motion in R3. Rigid Motion in R3. Velocity of a Rigid Body. Wrenches and Reciprocal Screws. MANIPULATOR KINEMATICS: Introduction. Forward Kinematics. Inverse Kinematics. The Manipulator Jacobian. Redundant and Parallel Manipulators. ROBOT DYNAMICS AND CONTROL: Introduction. Lagrange's Equations. Dynamics of Open-Chain Manipulators. Lyapunov Stability Theory. Position Control and Trajectory Tracking. Control of Constrained Manipulators. MULTIFINGERED HAND KINEMATICS: Introduction to Grasping. Grasp Statics. Force-Closure. Grasp Planning. Grasp Constraints. Rolling Contact Kinematics. HAND DYNAMICS AND CONTROL: Lagrange's Equations with Constraints. Robot Hand Dynamics. Redundant and Nonmanipulable Robot Systems. Kinematics and Statics of Tendon Actuation. Control of Robot Hands. NONHOLONOMIC BEHAVIOR IN ROBOTIC SYSTEMS: Introduction. Controllability and Frobenius' Theorem. Examples of Nonholonomic Systems. Structure of Nonholonomic Systems. NONHOLONOMIC MOTION PLANNING: Introduction. Steering Model Control Systems Using Sinusoids. General Methods for Steering. Dynamic Finger Repositioning. FUTURE PROSPECTS: Robots in Hazardous Environments. Medical Applications for Multifingered Hands. Robots on a Small Scale: Microrobotics. APPENDICES: Lie Groups and Robot Kinematics. A Mathematica Package for Screw Calculus. Bibliography. Index Each chapter also includes a Summary, Bibliography, and Exercises A necessary feature of robot systems is a capability to process robot paths in terms of Cartesian coordinates. However, the so-called articulated robot arms which represent the more advanced systems operate in terms of their revolute and sliding joint coordinates, and these do not lend themselves easily to translation into Cartesian equivalents. As a rule, the direct (joint-to-Cartesian space) coordinate transformation can be solved for in closed form. As for the inverse (Cartesian-to-joint) transformation, for some rather common arm designs practical solutions cannot be found in closed form. An iterative procedure for one such class of articulated robots is presented. The procedure produces an exact solution for the wrist tip position coordinates and an approximate solution for the wrist orientation coordinates. The convergence characteristics of the procedure are discussed. A new method for computing numerical solutions to the inverse kinematics problem of robotic manipulators is developed. The method is based on a combination of two nonlinear programming techniques and the forward recursion formulas, with the joint limitations of the robot being handled implicitly as simple boundary constraints. This method is numerically stable since it converges to the correct answer with virtually any initial approximation, and it is not sensitive to the singular configuration of the manipulator. In addition, this method is computationally efficient and can be applied to serial manipulators having any number of degrees of freedom. > The kinematic transformation between task space and joint configuration coordinates is nonlinear and configuration dependent. A solution to the inverse kinematics is a vector of joint configuration coordinates that corresponds to a set of task space coordinates. For a class of robots closed form solutions always exist, but constraints on joint displacements cannot be systematically incorporated in the process of obtaining a solution. An iterative solution is presented that is suitable for any class of robots having rotary or prismatic joints, with any arbitrary number of degrees of freedom, including both standard and kinematically redundant robots. The solution can be obtained subject to specified constraints and based on certain performance criteria. The solution is based on a new rapidly convergent constrained nonlinear optimization algorithm which uses a modified Newton-Raphson technique for solving a system nonlinear equations. The algorithm is illustrated using as an example a kinematically redundant robot. Numerical solution of the inverse kinematics of robots using modified Newton-Raphson (MNR) and modified predictor-corrector (MPC) algorithms is discussed. These modified algorithms are highly reliable and stable. Both algorithms always find a solution if a physically realizable robot configuration exists. They are also capable of approaching singular configurations of the robot as E = C em, where E is the error between the theoretical and numerically computed singular joint values, the convergence criteria, and C and m are appropriate constants. It is also shown that the modified predictor-corector (MPC) algorithm is 5-15 times faster than the modified Newton-Raphson (MNR) algorithm. "Richard Paul is perhaps the world's leading authority on the science of robot manipulation. He has contributed to almost every aspect of the field. His impressive publication record includes important articles on the kinematics of robot arms, their dynamics, and their control. He has developed a succession of interesting ideas concerning representation, specifically the use of homogeneous matrices.... Paul's book is written in his usual clear style, and it contains numerous interesting examples."--Patrick H. Winston and Mike Brady, editors, The MIT Press Artificial Intelligence Series"Robot Manipulators" is firmly grounded on the theoretical principles of the subject and makes considerable use of vector and matrix methods in its development. It is the first full treatment to be published, and it is designed for graduate courses in robotics as well as for practicing engineers. Following an introduction, the book's ten chapters cover homogeneous transformations, defining transformation equations, solving transformation equations, differential transformation relationships, motion trajectories, dynamics, digital servo systems, force transformations, compliance, and manipulation languages.Paul writes that the impact of robot manipulators on the workplace and the economy over the coming decade could be profound: "While currently available industrial robots will probably not have a major impact on manufacturing, a low-cost, mass-produced, sensor-controlled robot could have a revolutionary effect.... Such robots would represent the conclusion of the industrial revolution, replacing the type of labor required at its outset to perform the repetitive machine-linked tasks whose ideal performance is characterized by our conception of a robot, not a human. Based on current research work, laboratory demonstrations, and the general level of technology in this country, we believe that it is possible to achieve such a robot within the coming decade." Inverse Kinematics is used in computer graphics and robotics to control the posture of an articulated body. We introduce a new method utilising the law of cosines, which quickly determines the joint angles of a kinematic chain when given a target. Our method incurs a lower computational and rotational cost than Cyclic-Coordinate Descent (CCD), yet is also guaranteed to find a solution if one is available. Additionally it moves the upper joints of any kinematic chain first, thus ensuring a closer simulation of natural movement than CCD, which tends to overemphasise the movement of joints closer to the end-eector of the kinematic chain.