An Explicit Formulation of Singularity-Free Dynamic Equations of Mechanical Systems in Lagrangian Form---Part one: Single Rigid Bodies

This paper presents the explicit dynamic equations of a mechanical system. The equations are presented so that they can easily be implemented in a simulation software or controller environment and are also well suited for system and controller analysis. The dynamics of a general mechanical system consisting of one or more rigid bodies can be derived from the Lagrangian. We can then use several well known properties of Lie groups to guarantee that these equations are well defined. This will, however, often lead to rather abstract formulation of the dynamic equations that cannot be implemented in a simulation software directly. In this paper we close this gap and show what the explicit dynamic equations look like. These equations can then be implemented directly in a simulation software and no background knowledge on Lie theory and differential geometry on the practitioner's side is required. This is the first of two papers on this topic. In this paper we derive the dynamics for single rigid bodies, while in the second part we study multibody systems. In addition to making the equations more accessible to practitioners, a motivation behind the papers is to correct a few errors commonly found in literature. For the first time, we show the detailed derivations and how to arrive at the correct set of equations. We also show through some simple examples that these correspond with the classical formulations found from Lagrange's equations. The dynamics is derived from the Boltzmann--Hamel equations of motion in terms of local position and velocity variables and the mapping to the corresponding quasi-velocities. Finally we present a new theorem which states that the Boltzmann--Hamel formulation of the dynamics is valid for all transformations with a Lie group topology. This has previously only been indicated through examples, but here we also present the formal proof. The main motivation of these papers is to allow practitioners not familiar with differential geometry to implement the dynamic equations of rigid bodies without the presence of singularities. Presenting the explicit dynamic equations also allows for more insight into the dynamic structure of the system. Another motivation is to correct some errors commonly found in the literature. Unfortunately, the formulation of the Boltzmann-Hamel equations used here are presented incorrectly. This has been corrected by the authors, but we present here, for the first time, the detailed mathematical details on how to arrive at the correct equations. We also show through examples that using the equations presented here, the dynamics of a single rigid body is reduced to the standard equations on a Lagrangian form, for example Euler's equations for rotational motion and Euler--Lagrange equations for free motion

An Explicit Formulation of Singularity-Free Dynamic Equations of Mechanical Systems in Lagrangian Form---Part Two: Multibody Systems

This paper presents the explicit dynamic equations of multibody mechanical systems. This is the second paper on this topic. In the first paper the dynamics of a single rigid body from the Boltzmann--Hamel equations were derived. In this paper these results are extended to also include multibody systems. We show that when quasi-velocities are used, the part of the dynamic equations that appear from the partial derivatives of the system kinematics are identical to the single rigid body case, but in addition we get terms that come from the partial derivatives of the inertia matrix, which are not present in the single rigid body case. We present for the first time the complete and correct derivation of multibody systems based on the Boltzmann--Hamel formulation of the dynamics in Lagrangian form where local position and velocity variables are used in the derivation to obtain the singularity-free dynamic equations. The final equations are written in global variables for both position and velocity. The main motivation of these papers is to allow practitioners not familiar with differential geometry to implement the dynamic equations of rigid bodies without the presence of singularities. Presenting the explicit dynamic equations also allows for more insight into the dynamic structure of the system. Another motivation is to correct some errors commonly found in the literature. Unfortunately, the formulation of the Boltzmann-Hamel equations used here are presented incorrectly. This has been corrected by the authors, but we present here, for the first time, the detailed mathematical details on how to arrive at the correct equations. We also show through examples that using the equations presented here, the dynamics of a single rigid body is reduced to the standard equations on a Lagrangian form, for example Euler's equations for rotational motion and Euler--Lagrange equations for free motion

General Solutions to Functional and Kinematic Redundancy

A systematic and general approach to represent functional redundancy is presented. It is shown how this approach allows the freedom provided by functional redundancy to be integrated into the inverse geometric problem for real-time applications and how it can be utilised to improve performance. A set of new iterative solutions to the inverse geometric problem, well suited for kinematically redundant manipulators, is also presented

Iterative Solutions to the Inverse Geometric Problem for Manipulators with no Closed Form Solution

A set of new iterative solutions to the inverse geometric problem is presented. The approach is general and does not depend on intersecting axes or calculation of the Jacobian. The solution can be applied to any manipulator and is well suited for manipulators for which convergence is poor for conventional Jacobian-based iterative algorithms. For kinematically redundant manipulators, weights can be applied to each joint to introduce stiffness and for collision avoidance. The algorithm uses the unit quaternion to represent the position of each joint and calculates analytically the optimal position of the joint when only the respective joint is considered. This sub-problem is computationally very efficient due to the analytical solution. Several algorithms based on the solution of this sub-problem are presented. For difficult problems, for which the initial condition is far from a solution or the geometry of the manipulator makes the solution hard to reach, it is shown that the algorithm finds a solution fairly close to the solution in only a few iterations

Software Components of the Thorvald II Modular Robot

In this paper, we present the key software components of the Thorvald II mobile robotic platform. Thorvald~II is a modular system developed by the authors for creating robots of arbitrary shapes and sizes, primarily for the agricultural domain. Several robots have been built and are currently operating on farms and universities at various locations in Europe. Robots may take many different forms, and may be configured for differential drive, Ackermann steering, all-wheel drive, all-wheel steering with any number of wheels etc. The software therefore needs to be configuration agnostic. In this paper we present an architecture that allows for simple setup of never-seen-before robot configurations. The presented software is organized in a collection of ROS packages, made available to the reader. These packages allow a user to create her or his own robot configurations and simulate these robots in Gazebo using a provided plugin. Although the presented packages were created to be used with Thorvald robots, they may also be useful for people who are looking to develop their own robot and are interested in testing various robot configurations in simulation before settling on a specific design. To create a robot, the user lists modules with key parameters in one single configuration file and gives this as an input to the robot at startup. Example configuration files are provided within the packages. In this paper, we discuss key aspects of the ROS packages and provide directions on where to find updated information on how to install and use these

Towards Safe Robotic Agricultural Applications: Safe Navigation System Design for a Robotic Grass-Mowing Application through the Risk Management Method

Safe navigation is a key objective for autonomous applications, particularly those involving mobile tasks, to avoid dangerous situations and prevent harm to humans. However, the integration of a risk management process is not yet mandatory in robotics development. Ensuring safety using mobile robots is critical for many real-world applications, especially those in which contact with the robot could result in fatal consequences, such as agricultural environments where a mobile device with an industrial cutter is used for grass-mowing. In this paper, we propose an explicit integration of a risk management process into the design of the software for an autonomous grass mower, with the aim of enhancing safety. Our approach is tested and validated in simulated scenarios that assess the effectiveness of different custom safety functionalities in terms of collision prevention, execution time, and the number of required human interventions