We present CasADi, an open-source software framework for numerical optimization. CasADi is a general-purpose tool that can be used to model and solve optimization problems with a large degree of flexibility, larger than what is associated with popular algebraic modeling languages such as AMPL, GAMS, JuMP or Pyomo. Of special interest are problems constrained by differential equations, i.e. optimal control problems. CasADi is written in self-contained C++, but is most conveniently used via full-featured interfaces to Python, MATLAB or Octave. Since its inception in late 2009, it has been used successfully for academic teaching as well as in applications from multiple fields, including process control, robotics and aerospace. This article gives an up-to-date and accessible introduction to the CasADi framework, which has undergone numerous design improvements over the last 7 years.

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CasADi: a software framework for nonlinear optimization and optimal control

When I started out as a newly hatched PhD student, back in the day, one of the first articles I read and understood (or at least thought that I understood) was Ray Reiter’s classic article on default logic (Reiter, 1980).This was some years after the famous ‘non-monotonic logic’ issue of Artificial Intelligence in which that article appeared, but default logic was still one of the leading approaches, a tribute to the simplicity and power of the theory. As a result of reading the article, I became fascinated by both default logic and, more generally, non-monotonic logics. However, despite my fascination, these approaches never seemed terribly useful for the kinds of problem that I was supposed to be studying—problems like those in medical decision making—and so I eventually lost interest. In fact non-monotonic logics seemed to me, and to many people at the time I think, not to be terribly useful for anything. They were interesting, and clearly relevant to the long-term goals of Artificial Intelligence as a discipline, but not of any immediate practical importance. This verdict, delivered at the end of the 1980s, continued, I think, to be true for the next few years while researchers working in non-monotonic logics studied problems that to outsiders seemed to be ever more obscure. However, by the end of the 1990s, it was becoming clear, even to folk as short-sighted as I, that non-monotonic logics were getting to the point at which they could be used to solve practical problems. Knowledge in action shows quite how far these techniques have come. The reason that non-monotonic logics were invented was, of course, in order to use logic to reason about the world. Our knowledge of the world is typically incomplete, and so, in order to reason about it, one has to make assumptions about things one does not know. This, in turn, requires mechanisms for both making assumptions and then retracting them if and when they turn out not to be true. Non-monotonic logics are intended to handle this kind of assumption making and retracting, providing a mechanism that has the clean semantics of logic, but which has a non-monotonic set of conclusions. Much of the early work on non-monotonic logics was concerned with theoretical reasoning, that is reasoning about the beliefs of an agent—what the agent believes to be true. Theoretical reasoning is the domain of all those famous examples like ‘Typically birds fly. Tweety is a bird, so does Tweety fly?’, and the fact that so much of non-monotonic reasoning seemed to focus on theoretical reasoning was why I lost interest in it. I became much more concerned with practical reasoning—that is reasoning about what an agent should do—and non-monotonic reasoning seemed to me to have nothing interesting to say about practical reasoning. Of course I was wrong. When one tries to formulate any kind of description of the world as the basis for planning, one immediately runs into applications of non-monotonic logics, for example in keeping track of the state of a changing world. It is this use of non-monotonic logic that is at the heart of Knowledge in action. Building on the McCarthy’s situation calculus, Knowledge in action constructs a theory of action that encompasses a very large part of what an agent requires to reason about the world. As Reiter says in the final chapter,

Knowledge in Action: Logical Foundations for Specifying and Implementing Dynamical Systems

Continuum robotic manipulators articulate due to their inherent compliance. Tendon actuation leads to compression of the manipulator, extension of the actuators, and is limited by the practical constraint that tendons cannot support compression. In light of these observations, we present a new linear model for transforming desired beam configuration to tendon displacements and vice versa. We begin from first principles in solid mechanics by analyzing the effects of geometrically nonlinear tendon loads. These loads act both distally at the termination point and proximally along the conduit contact interface. The resulting model simplifies to a linear system including only the bending and axial modes of the manipulator as well as the actuator compliance. The model is then manipulated to form a concise mapping from beam configuration-space parameters to n redundant tendon displacements via the internal loads and strains experienced by the system. We demonstrate the utility of this model by implementing an optimal feasible controller. The controller regulates axial strain to a constant value while guaranteeing positive tendon forces and minimizing their magnitudes over a range of articulations. The mechanics-based model from this study provides insight as well as performance gains for this increasingly ubiquitous class of manipulators.

Mechanics Modeling of Tendon-Driven Continuum Manipulators

The mathematics of coordinated control of prosthetic arms and manipulators.

No wonder you activities are, reading will be always needed. It is not only to fulfil the duties that you need to finish in deadline time. Reading will encourage your mind and thoughts. Of course, reading will greatly develop your experiences about everything. Reading simulating hamiltonian dynamics is also a way as one of the collective books that gives many advantages. The advantages are not only for you, but for the other peoples with those meaningful benefits.

Simulating Hamiltonian dynamics.

Redundant actuation of parallel manipulators can lead to internal forces without generating end-effector forces (preload). Preload can be controlled in order to prevent backlash during the manipulator motion. Such control is based on the inverse dynamics. The general solution of the inverse dynamics of redundantly actuated parallel manipulators is given. For the special case of simple overactuation an explicit solution is derived in terms of a single preload parameter. With this formulation a computational efficient open-loop preload control is developed and applied to the elimination of backlash. Its simplicity makes it applicable in real-time applications. Results are given for a planar 4RRR manipulator and a spatial heptapod.

Internal Preload Control of Redundantly Actuated Parallel Manipulators&#8212;Its Application to Backlash Avoiding Control

Time-optimal motion control will only find industrial applications if the optimal motions can actually be performed by standard industrial robots. This is not ensured by any optimal motion planning scheme proposed up to now. The limiting aspect rendering all these schemes impractical is the insufficient continuity of the motion trajectories. In this paper, a time-optimal path following along a predefined end-effector path is addressed for kinematically redundant robots, where nonredundant robots are included as special cases. As prerequisite explicit expressions for the higher order inverse kinematics are presented. Kinematic redundancy is resolved and exploited within the trajectory planning using the joint space decomposition and a novel pseudoinverse-based solution of the higher order inverse kinematics. The approaches are demonstrated for two examples of kinematically redundant manipulators performing time-optimal motions along prescribed end-effector paths in compliance with technological constraints. The optimization results are experimentally validated.

On Higher Order Inverse Kinematics Methods in Time-Optimal Trajectory Planning for Kinematically Redundant Manipulators

A Lie-group formulation for the kinematics and dynamics ofholonomic constrained mechanical systems (CMS) is presented. The kinematics ofrigid multibody systems (MBS) is described in terms of the screw system of theMBS. Using Lie-algebraic properties of screw algebra, isomorphicto se(3), allows a purely algebraic derivation of the Lagrangian motion equations. As such the Lie-group SE(3) ⊗... ⊗ SE(3) (n copies) is theambient space of a MBS consisting of n rigid bodies. Any parameterizationof the ambient space corresponds to a chart on the MBS configuration space ℝn. The key to combine differential geometric and Lie-algebraic approaches is the existence of kinematic basic functions whichare push forward maps from the tangent bundle Tℝn to the Lie-algebra of the ambient space.

A Lie-Group Formulation of Kinematics and Dynamics of Constrained MBS and Its Application to Analytical Mechanics

Generic mobility of rigid body mechanisms

After three decades of computational multibody system (MBS) dynamics, current research is centered at the development of compact and user-friendly yet computationally efficient formulations for the analysis of complex MBS. The key to this is a holistic geometric approach to the kinematics modeling observing that the general motion of rigid bodies and the relative motion due to technical joints are screw motions. Moreover, screw theory provides the geometric setting and Lie group theory the analytic foundation for an intuitive and compact MBS modeling. The inherent frame invariance of this modeling approach gives rise to very efficient recursive $O ( n ) $
 algorithms, for which the so-called “spatial operator algebra” is one example, and allows for use of readily available geometric data. In this paper, three variants for describing the configuration of tree-topology MBS in terms of relative coordinates, that is, joint variables, are presented: the standard formulation using body-fixed joint frames, a formulation without joint frames, and a formulation without either joint or body-fixed reference frames. This allows for describing the MBS kinematics without introducing joint reference frames and therewith rendering the use of restrictive modeling convention, such as Denavit–Hartenberg parameters, redundant. Four different definitions of twists are recalled, and the corresponding recursive expressions are derived. The corresponding Jacobians and their factorization are derived. The aim of this paper is to motivate the use of Lie group modeling and to provide a review of different formulations for the kinematics of tree-topology MBS in terms of relative (joint) coordinates from the unifying perspective of screw and Lie group theory.

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Andreas Müller

Papers

Internal Preload Control of Redundantly Actuated Parallel Manipulators—Its Application to Backlash Avoiding Control

On Higher Order Inverse Kinematics Methods in Time-Optimal Trajectory Planning for Kinematically Redundant Manipulators

A Lie-Group Formulation of Kinematics and Dynamics of Constrained MBS and Its Application to Analytical Mechanics

Generic mobility of rigid body mechanisms

Screw and Lie group theory in multibody dynamics

Andreas Müller

Papers

Internal Preload Control of Redundantly Actuated Parallel Manipulators&#8212;Its Application to Backlash Avoiding Control

On Higher Order Inverse Kinematics Methods in Time-Optimal Trajectory Planning for Kinematically Redundant Manipulators

A Lie-Group Formulation of Kinematics and Dynamics of Constrained MBS and Its Application to Analytical Mechanics

Generic mobility of rigid body mechanisms

Screw and Lie group theory in multibody dynamics

Internal Preload Control of Redundantly Actuated Parallel Manipulators—Its Application to Backlash Avoiding Control