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# Accepted Manuscripts

BASIC VIEW  |  EXPANDED VIEW
Review Article
Appl. Mech. Rev   doi: 10.1115/1.4039078
We provide a tutorial and review of the state-of-the-art in robot dynamics algorithms that rely on methods from differential geometry, particularly the theory of Lie groups. After reviewing the underlying Lie group structure of the rigid body motions and the geometric formulation of the equations of motion for a single rigid body, we show how classical screw-theoretic concepts can be expressed in a reference frame-invariant way using Lie-theoretic concepts, and derive recursive algorithms for the forward and inverse dynamics and their differentiation. These algorithms are extended to robots subject to closed loop and other constraints, joints driven by variable stiffness actuators, and also to the modeling of contact between rigid bodies. We conclude with a demonstration of how the geometric formulations and algorithms can be effectively used for robot motion optimization.
TOPICS: Algorithms, Robot dynamics, Stiffness, Modeling, Optimization, Geometry, Dynamics (Mechanics), Robots, Screws, Equations of motion, Robot motion, Actuators
Discussion
Gregory S. Chirikjian
Appl. Mech. Rev   doi: 10.1115/1.4039080
Lie-theoretic methods provide an elegant way to formulate many problems in robotics, and the tutorial by Park et al is simultaneously a complete and concise introduction to these methods as they pertain to robot dynamics. The central reason why Lie groups are a natural mathematical tool for robotics is that rigid-body motions and pose changes can be described as Lie groups, and allow phenomena including robot kinematics and dynamics to be formulated in elegant notation without introducing superfluous coordinates. The emphasis of the tutorial by Park et al is robot dynamics from a Lie-theoretic point of view. Newton-Euler and Lagrangian formulation of robot dynamics algorithms with O(n) complexity were formulated more than 35 years ago using recurrence relations that use the serial structure of manipulator arms. This was done without using knowledge of Lie theory. But issues such as why the $\omega \times$ terms in rigid-body dynamics appear can be more easily understood in the context of this theory. The authors take great efforts to be understandable by nonexperts and present extensive references to the differential-geometric and Lie-group-centric formulations of manipulator dynamics. In the discussion presented here, connections are made to complementary methods that have been developed in other bodies of literature. This includes the multi-body dynamics, geometric mechanics, spacecraft dynamics, and polymer physics literature, as well as robotics works that present non-Lie-theoretic formulations in the context of highly parallelizable algorithms.
TOPICS: Algorithms, Robot dynamics, Dynamics (Mechanics), Robotics, Manipulator dynamics, Manipulators, Robot kinematics, Space vehicles, Physics, Polymers
Closure
Appl. Mech. Rev   doi: 10.1115/1.4039079
Review Article
Appl. Mech. Rev   doi: 10.1115/1.4038931
This paper presents a review of the literature related to the use of redundancy in parallel mechanisms. Two types of redundancies are considered, namely actuation redundancy and kinematic redundancy. The use of these concepts in the literature is highlighted. Each of the concepts is then formulated mathematically in order to clearly expose their characteristics and their properties. Two sub-classes of kinematically redundant parallel mechanisms are defined, namely those with serial redundant legs and those with parallel redundant legs. The force transmission in redundant parallel mechanisms is then discussed. Finally, a summary of the different approaches that can be used to implement redundancy in parallel mechanisms is given in order to identify the most promising synthesis avenues and to provide insight into their potential fields of application.
TOPICS: Redundancy (Engineering), Parallel mechanisms, Kinematics
Review Article
Appl. Mech. Rev   doi: 10.1115/1.4038699
The design and performance of liquid metal batteries, a new technology for grid-scale energy storage, depend on fluid mechanics because the battery electrodes and electrolytes are entirely liquid. Here we review prior and current research on the fluid mechanics of liquid metal batteries, pointing out opportunities for future studies. Because the technology in its present form is just a few years old, only a small number of publications have so far considered liquid metal batteries specifically. We hope to encourage collaboration and conversation by referencing as many of those publications as possible here. Much can also be learned by linking to extensive prior literature considering phenomena observed or expected in liquid metal batteries, including thermal convection, magnetoconvection, Marangoni flow, interface instabilities, the Tayler instability, and electro-vortex flow. We focus on phenomena, materials, length scales, and current densities relevant to the liquid metal battery designs currently being commercialized. We try to point out breakthroughs that could lead to design improvements or make new mechanisms important.
TOPICS: Fluid mechanics, Liquid metals, Flow (Dynamics), Design, Batteries, Electrodes, Energy storage, Vortices, Electrolytes, Collaboration, Convection