A Lie group is an old mathematical abstract object dating back to the XIX century, when mathematician Sophus Lie laid the foundations of the theory of continuous transformation groups. Its influence has spread over diverse areas of science and technology many years later. In robotics, we are recently experiencing an important trend in its usage, at least in the fields of estimation, and particularly in motion estimation for navigation. Yet for a vast majority of roboticians, Lie groups are highly abstract constructions and therefore difficult to understand and to use.In estimation for robotics it is often not necessary to exploit the full capacity of the theory, and therefore an effort of selection of materials is required. In this paper, we will walk through the most basic principles of the Lie theory, with the aim of conveying clear and useful ideas, and leave a significant corpus of the Lie theory behind. Even with this mutilation, the material included here has proven to be extremely useful in modern estimation algorithms for robotics, especially in the fields of SLAM, visual odometry, and the like.Alongside this micro Lie theory, we provide a chapter with a few application examples, and a vast reference of formulas for the major Lie groups used in robotics, including most jacobian matrices and the way to easily manipulate them. We also present a new C++ template-only library implementing all the functionality described here.
Motion planning is one of the main problems studied in the field of robotics. However, it is still challenging for the state-of-the-art methods to handle multiple conditions that allow better paths to be found. For example, considering joint limits, path smoothness and a mixture of Cartesian and joint-space constraints at the same time, pose a significant challenge for many of them. This work proposes to use Timed-Elastic Bands for representing the manipulation motion planning problem, allowing to apply continuously optimized constraints to the problem during the search for a solution. Due to the nature of our method, it is highly extensible with new constraints or optimization objectives. The proposed approach is compared against state-of-the-art methods in various manipulation scenarios. Results show that it is more consistent and less variant while performing in a comparable manner to the state-of-the-art. This behaviour allows the proposed method to set a lower bound performance guarantee for other methods to build upon.
This paper describes a self-calibration procedure that jointly estimates the extrinsic parameters of an exteroceptive sensor able to observe ego-motion, and the intrinsic parameters of an odometry motion model, consisting of wheel radii and wheel separation. We use iterative nonlinear onmanifold optimization with a graphical representation of the state, and resort to an adaptation of the pre-integration theory, initially developed for the IMU motion sensor, to be applied to the differential drive motion model. For this, we describe the construction of a pre-integrated factor for the differential drive motion model, which includes the motion increment, its covariance, and a first-order approximation of its dependence with the calibration parameters. As the calibration parameters change at each solver iteration, this allows a posteriori factor correction without the need of re-integrating the motion data. We validate our proposal in simulations and on a real robot and show the convergence of the calibration towards the true values of the parameters. It is then tested online in simulation and is shown to accommodate to variations in the calibration parameters when the vehicle is subject to physical changes such as loading and unloading a freight.
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