Figure 1 - uploaded by Tijana Ivancevic
Content may be subject to copyright.
Humanoid robot's configuration manifold M rob , modeled upon human skeleton. M rob is defined as a topological product of constrained rotational SO(3) groups, M rob = ∏ i SO(3) i .
Source publication
his paper reviews modern geometrical dynamics and control of humanoid robots.
This general Lagrangian and Hamiltonian formalism starts with a proper
definition of humanoid's configuration manifold, which is a set of all robot's
active joint angles. Based on the `covariant force law', the general humanoid's
dynamics and control are developed. Autono...
Contexts in source publication
Context 1
... of an ideal humanoid−robot motion (with human-like spine, see Figure 1) is rigorously defined in terms of rotational con- strained SO(3)−groups of motion [5−7,9] in all main robot joints. Therefore, the configuration manifold M rob for humanoid dynam- ics is defined as a topological product of all included SO(3) groups, M rob = ∏ i SO(3) i . ...
Context 2
... HBE is a sophisticated human neuro-musculo-skeletal dynamics simulator, The first version of the HBE simulator had the full human-like skeleton, driven by the generalized Hamiltonian dynamics (including muscular force-velocity and force-time curves) and two levels of reflex-like mo- tor control (simulated using the Lie derivative formalism) [1,2]. It had 135 purely rotational DOF, strictly following Figure 1. It was created for prediction and prevention of musculo-skeletal injuries occurring in the joints, mostly spinal (intervertebral). ...
Citations
... Practical: From the viewpoint of applications, the restriction of the recipe to Euclidean spaces dramatically restricts its scope, as many modern applications require P -preserving diffusions on manifolds. These include thermodynamic integration, free energy calculation and molecular simulations [106,109,111,153], spectral density estimation of partially observed models for Bayesian methods, directional statistics, and lattice QCD calculations on compact Lie groups [17,33,34,47,72,80,81,115,126,164], as well as applications that are built using Stein operators on manifolds, diffusions on spaces with symmetries used in robotics (such as coupled rigid body motion), or for learning to encode symmetries in neural nets [18,41,91,102,114]. ...
A complete recipe of measure-preserving diffusions in Euclidean space was recently derived unifying several MCMC algorithms into a single framework. In this paper, we develop a geometric theory that improves and generalises this construction to any manifold. We thereby demonstrate that the completeness result is a direct consequence of the topology of the underlying manifold and the geometry induced by the target measure $P$; there is no need to introduce other structures such as a Riemannian metric, local coordinates, or a reference measure. Instead, our framework relies on the intrinsic geometry of $P$ and in particular its canonical derivative, the deRham rotationnel, which allows us to parametrise the Fokker--Planck currents of measure-preserving diffusions using potentials. The geometric formalism can easily incorporate constraints and symmetries, and deliver new important insights, for example, a new complete recipe of Langevin-like diffusions that are suited to the construction of samplers. We also analyse the reversibility and dissipative properties of the diffusions, the associated deterministic flow on the space of measures, and the geometry of Langevin processes. Our article connects ideas from various literature and frames the theory of measure-preserving diffusions in its appropriate mathematical context.
... In doing so we seek to give an intuitive appreciation of the terminology and structure of Riemannian geometry rather than a mathematically rigorous one. Detailed mathematical descriptions can be found in texts such as Abraham and Marsden (1978), Arnold (1989), Bullo and Lewis (2005), Darling (1994), Isidori (1995, Ivancevic and Ivancevic (2007 Ivancevic ( , 2010 Ivancevic ( , 2011), Jurdjevic (1997), Lang (1999), Lee (1997, 2011), Marsden and Ratiu, (1999), Ortega and Ratiu (2004), and Szekeres (2004) . In a nutshell we can say that the Riemannian framework consists of multiple interconnected spaces or 'manifolds' and the relationships between them. ...
