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Geometric interpretation of the general POE model for a serial-link robot via conversion into D-H parameterization
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Abstract and Figures
While the Product of Exponentials (POE) formula has been gaining increasing popularity in modeling the kinematics of a serial-link robot, the Denavit-Hartenberg (D-H) notation is still the most widely used due to its intuitive and concise geometric interpretation of the robot. This paper outlines the development of an analytical solution to automatically convert a POE model into a D-H model for a robot with revolute, prismatic, and helical joints, which are the complete set of three basic one degree of freedom lower pair joints for constructing a serial-link robot. The conversion algorithm developed can be used in applications such as calibration where it is necessary to convert the D-H model to the POE model for identification and then back to the D-H model for compensation. The equivalence of the two models proved in this paper also benefits the analysis of the identifiability of the kinematic parameters. It is found that the maximum number of identifiable parameters in a general POE model is 5h + 4r + 2t + n + 6 where h, r, t, and n stand for the number of helical, revolute, prismatic, and general joints, respectively. It is also suggested that the identifiability of the base frame and the tool frame in the D-H model is restricted rather than the arbitrary six parameters as assumed previously.
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The Denavit-Hartenberg (D-H) model and the product of exponentials (POE) model have been two popular methods for modeling the kinematics of a serial-link robot. While these two models are equivalent in essence, no study has revealed how to convert from the POE model to the D-H model. The conversion enables direct utilization of established algorithms formulated with D-H parameters or compensation of the D-H model after calibration with the POE parameters. It also provides a simpler method to determine the D-H parameters of a robot. For these reasons, this paper proposes an analytic approach to automatically convert a group of POE parameters into the associated D-H parameters. Three lemmas are put forward and proved to derive the final algorithm. An implementation of the algorithm in MATLAB is provided as well.
The product of exponential model based robot calibration approach eliminates parameter discontinuity and simplifies coordinate frame setup, but demands extra effort to normalize twist coordinates and differentiate parameter-varying exponential maps. In this paper, we show that such an endeavor can be exempted by respecting the nonlinear geometry of the joint axis configuration space (ACS), the set of all possible axis locations. We analyze the geometry of the ACS models for prismatic and revolute joints, and treat the errors as Adjoint transformations on joint twists. We propose a novel robot kinematic calibration algorithm based on the ACS and Adjoint error model. It is geometrically intuitive, computationally efficient, and can easily handle additional assumptions on joint axes relations. We present a comparative study with simulations and experiments to show that our algorithm outperforms the existing ones in various aspects.
This paper presents an analytical approach to deter-mine and eliminate the redundant model parameters in serial-robot kinematic calibration based on the product of exponentials formula. According to the transformation principle of the Lie algebra se(3) between different frames, the connection between the joints' twist errors and the links' geometric ones is established. Identifiability analysis shows that the redundant errors are sim-ply equivalent to the commutative elements of the robot's joint twists. Using the Lie bracket operation of se(3), a linear parti-tioning operator can be constructed to analytically separate the identifiable parameters from the system error vector. Then, error models satisfying the completeness, minimality, and model conti-nuity requirements can be obtained for any serial robot with all combinations and configurations of revolute and prismatic joints. The conventional conclusion that the maximum number of inde-pendent parameters is 4r + 2p + 6 in a generic serial robot with r revolute and p prismatic joints is verified. Using the quotient manifold of the Lie group SE(3), the links' geometric errors and the joints' offset errors can be integrated as a whole, such that all these errors can be identified simultaneously. To verify the ef-fectiveness of the proposed method, calibration simulations and experiments are conducted on an industrial six-degree-of-freedom (DoF) serial robot. Index Terms—Calibration and identification, kinematics, lie bracket, product of exponentials (POE) formula.
This paper proposes an algorithm for robotic kinematic calibration based on a minimal product of exponentials (POE)-based model for the applications where only position measurements are required. Both joint zero-offset errors and initial frame twist error can be involved in this model. Analysis of the identifiability of these errors shows that at most six elements of these parameters can be identified. It also suggests that at least three noncollinear points on the end-effector should be measured to maximize the identifiability. Compared with the traditional POE-based model with full pose (position and orientation) measurements, the minimal model with only position measurements outperforms in terms of convenience, efficiency, and accuracy.
A Mathematical Introduction to Robotic Manipulation presents a mathematical formulation of the kinematics, dynamics, and control of robot manipulators. It uses an elegant set of mathematical tools that emphasizes the geometry of robot motion and allows a large class of robotic manipulation problems to be analyzed within a unified framework. The foundation of the book is a derivation of robot kinematics using the product of the exponentials formula. The authors explore the kinematics of open-chain manipulators and multifingered robot hands, present an analysis of the dynamics and control of robot systems, discuss the specification and control of internal forces and internal motions, and address the implications of the nonholonomic nature of rolling contact are addressed, as well. The wealth of information, numerous examples, and exercises make A Mathematical Introduction to Robotic Manipulation valuable as both a reference for robotics researchers and a text for students in advanced robotics courses.
A symbolic notation devised by Reuleaux to describe mechanisms did not recognize the necessary number of variables needed for complete description. A reconsideration of the problem leads to a symbolic notation which permits the complete description of the kinematic properties of all lower-pair mechanisms by means of equations. The symbolic notation also yields a method for studying lower-pair mechanisms by means of matrix algebra; two examples of application to space mechanisms are given.
Safety issue is a great concern for rehabilitation robots that are expected to contribute to future aging society. Appropriate compliance is required for their joints. However, combination of servomotors and high-ratio gears, such as harmonic gears, makes the joint of robots nonbackdrivable. The nonbackdrivability causes lack of adaptability and safety. On the other hand, conventional direct-drive systems, including linear motors, are relatively big for such application. This paper presents the development and analysis of compact high-backdrivable direct-drive linear actuator. The motor consists of a helical structure stator and mover. The mover does not contact the stator and moves helically in the stator under a proper magnetic levitation control. Thus, the motor realizes direct-drive motion without mechanical gears. Decoupling control is proposed and integrated with disturbance observer to achieve robustness against model uncertainties and input disturbance. The main contribution of this paper is to experimentally realize the direct-drive feature of the helical motor.