Closure to “Discussion of ‘Geometric Algorithms for Robot Dynamics: A Tutorial Review'” (Park, F. C., Kim, B., Jang, C., and Hong, J., 2018, ASME Appl. Mech. Rev., 70(1), p. 010803)

摘要

We are grateful to Chirikjian for his in-depth analysis and insightful comments [1] on our tutorial review [2], which complement nicely our main discussion on how Lie group methods can be effectively used for robot dynamics. There is considerable machinery from the theory of Lie groups and differential geometry that impact robot dynamics, and more generally nonlinear mechanics, and Chirikjian's commentary offers a deeper but still very much readable discussion of Lie group essentials that our review paper did not cover. Chirikjian also provides important context to our review by further pointing out the past literature on robot dynamics that is not based on Lie group methods, e.g., recursive methods for inverse and forward dynamics based on classical Denavit–Hartenberg kinematic representations. Finally, the discussion and additional references pointed out by Chirikjian on Lie group methods for modeling constrained multibody systems, and connections with variational integrators and discrete Lagrangian mechanics, provide fitting closure to our review, by pointing the reader to the latest developments and trends in geometric methods for robot and multibody system dynamics. Hopefully having made the case that there are both important analytical and computational benefits to using Lie group methods for multibody dynamics, we would remark that there is still quite a ways to go in this program to “geometrize” classical mechanics, and plenty of new application domains and open problems.One such new domain is in fact alluded to in Chirikjian's discussion, and that is the use of Lie group methods for the analysis of biological and biomolecular systems, ranging from, e.g., proteins, polypeptide chains, and polymers, to the human body and other large-scale musculo-skeletal systems. Such systems are marked by a large number of degrees-of-freedom, where the effects of uncertainty and noise can be quite significant. Not surprisingly, stochastic methods have proven to be quite effective in addressing some of these inherent challenges. How to effectively combine the power of Lie group methods with stochastic models, and to apply these to real problems in science and engineering, raises important theoretical and implementational questions, e.g., how to model noise on the rotation group in a coordinate-invariant and physically meaningful way. Recent monographs such as Chirikjian's two-volume reference [3] describe not only the mathematical essentials for merging Lie group concepts with stochastic systems, but also essential algorithms and methods for forging these into a set of powerful computational tools.Mechanics lies at the heart of most systems that involve some form of interaction with the physical environment, and Lie groups—from the group of rigid body motions to volume-preserving and affine transformations, and many others—arise in numerous such contexts. More generally, solving practical problems invariably involves choosing coordinates and computing in those coordinates. Although it may sound paradoxical, the coordinate-invariant methods of Lie groups and differential geometry offer the best insight into the question of how to choose coordinates when necessary, e.g., choosing a coordinate frame to bring the stiffness matrix associated with a rigid body to normal form [4], or dimension reduction of the dynamics equations for systems with constraints and symmetries [5]. The pedagogical challenge is to bring Lie group concepts and methods into the mainstream of applied mechanics research and education, by minimizing abstruse jargon and notation, and embracing coordinates while emphasizing the geometric properties of whatever is being calculated. We hope our review takes a step toward this goal.