Numerical Methods to Compute the Coriolis Matrix and Christoffel Symbols for Rigid-Body Systems

The growth of model-based control strategies for robotics platforms has led to the need for additional rigid-body-dynamics algorithms to support their operation. Toward addressing this need, this article summarizes efficient numerical methods to compute the Coriolis matrix and underlying Christoffel Symbols (of the first kind) for tree-structure rigid-body systems. The resulting algorithms can be executed purely numerically, without requiring any partial derivatives that would be required in symbolic techniques that do not scale. Properties of the presented algorithms share recursive structure in common with classical methods such as the Composite-Rigid-Body Algorithm. The algorithms presented are of the lowest possible order: $O(Nd)$ for the Coriolis Matrix and $O(Nd^2)$ for the Christoffel symbols, where $N$ is the number of bodies and $d$ is the depth of the kinematic tree. A method of order $O(Nd)$ is also provided to compute the time derivative of the mass matrix. A numerical implementation of these algorithms in C/C++ is benchmarked showing computation times on the order of 10-20 $\mu$s for the computation of the Coriolis matrix and $40-120$ $\mu$s for the computation of the Christoffel symbols for systems with $20$ degrees of freedom. These results demonstrate feasibility for the adoption of these numerical methods within control loops that need to operate at $1$kHz rates or higher, as is commonly required for model-based control applications

Computed torque control of a free-flying cooperat ing-arm robot

The unified approach to solving free-floating space robot manipulator end-point control problems is presented using a control formulation based on an extension of computed torque. Once the desired end-point accelerations have been specified, the kinematic equations are used with momentum conservation equations to solve for the joint accelerations in any of the robot's possible configurations: fixed base or free-flying with open/closed chain grasp. The joint accelerations can then be used to calculate the arm control torques and internal forces using a recursive order N algorithm. Initial experimental verification of these techniques has been performed using a laboratory model of a two-armed space robot. This fully autonomous spacecraft system experiences the drag-free, zero G characteristics of space in two dimensions through the use of an air cushion support system. Results of these initial experiments are included which validate the correctness of the proposed methodology. The further problem of control in the large where not only the manipulator tip positions but the entire system consisting of base and arms must be controlled is also presented. The availability of a physical testbed has brought a keener insight into the subtleties of the problem at hand

ROBOTRAN: a powerful symbolic gnerator of multibody models

The computational efficiency of symbolic generation was at the root of the emergence of symbolic multibody programs in the eighties. At present, it remains an attractive feature of it since the exponential increase in modern computer performances naturally provides the opportunity to investigate larger systems and more sophisticated models for which real-time computation is a real asset. <br><br> Nowadays, in the context of mechatronic multibody systems, another interesting feature of the symbolic approach appears when dealing with enlarged multibody models, i.e. including electrical actuators, hydraulic devices, pneumatic suspensions, etc. and requiring specific analyses like control and optimization. Indeed, since symbolic multibody programs clearly distinguish the modeling phase from the analysis process, extracting the symbolic model, as well as some precious ingredients like analytical sensitivities, in order to export it towards any suitable environment (for control or optimization purposes) is quite straightforward. Symbolic multibody model portability is thus very attractive for the analysis of mechatronic applications. <br><br> In this context, the main features and recent developments of the ROBOTRAN software developed at the Université catholique de Louvain (Belgium) are reviewed in this paper and illustrated via three multibody applications which highlight its capabilities for dealing with very large systems and coping with multiphysics issues

Proceedings of the 3rd Annual Conference on Aerospace Computational Control, volume 1

Conference topics included definition of tool requirements, advanced multibody component representation descriptions, model reduction, parallel computation, real time simulation, control design and analysis software, user interface issues, testing and verification, and applications to spacecraft, robotics, and aircraft

Dynamical formulations and control of an automatic retargeting system

The Poincare equations, also known as Lagrange's equations in quasi coordinates, are revisited with special attention focused on a diagonal form. The diagonal form stems from a special choice of quasi velocities that were first introduced by Georg Hamel nearly a century ago. The form has been largely ignored because the quasi velocities create so-called Hamel coefficients that appear in the governing equations and are based on the partial derivative of the mass matrix factorization. Consequently, closed-form expressions for the Hamel coefficients can be difficult to obtain and relying on finite-dimensional, numerical methods are unattractive. In this thesis we use a newly developed operator overloading technique to automatically generate the Hamel coefficients through exact partial differentiation together with numerical evaluation. The equations can then be numerically integrated for system simulation. These special Poincare equations are called the Hamel Form and their usefulness in dynamic modeling and control is investigated. Coordinated control algorithms for an automatic retargeting system are developed in an attempt to protect an area against direct assaults. The scenario is for a few weapon systems to suddenly be faced with many hostile targets appearing together. The weapon systems must decide which weapon system will attack which target and in whatever order deemed sufficient to defend the protected area. This must be performed in a real-time environment, where every second is crucial. Four different control methods in this thesis are developed. They are tested against each other in computer simulations to determine the survivability and thought process of the control algorithms. An auction based control algorithm finding targets of opportunity achieved the best results

Computational methods and software systems for dynamics and control of large space structures

Two key areas of crucial importance to the computer-based simulation of large space structures are discussed. The first area involves multibody dynamics (MBD) of flexible space structures, with applications directed to deployment, construction, and maneuvering. The second area deals with advanced software systems, with emphasis on parallel processing. The latest research thrust in the second area involves massively parallel computers

Proceedings of the Fifth NASA/NSF/DOD Workshop on Aerospace Computational Control

The Fifth Annual Workshop on Aerospace Computational Control was one in a series of workshops sponsored by NASA, NSF, and the DOD. The purpose of these workshops is to address computational issues in the analysis, design, and testing of flexible multibody control systems for aerospace applications. The intention in holding these workshops is to bring together users, researchers, and developers of computational tools in aerospace systems (spacecraft, space robotics, aerospace transportation vehicles, etc.) for the purpose of exchanging ideas on the state of the art in computational tools and techniques

A Weighted Least-Squares Method for Inverse Dynamic Analysis

Internal forces in the human body can be estimated from measured movements and external forces using inverse dynamic analysis. Here we present a general method of analysis which makes optimal use of all available data, and allows the use of inverse dynamic analysis in cases where external force data is incomplete. The method was evaluated for the analysis of running on a partially instrumented treadmill. It was found that results correlate well with those of a conventional analysis where all external forces are known