Variational integrators are well-suited for simulation of mechanical systems because they preserve mechanical quantities about a system such as momentum, or its change if external forcing is involved, and holonomic constraints. While they are not energy-preserving they do exhibit long-time stable energy behavior. However, variational integrators often simulate mechanical system dynamics by solving an implicit difference equation at each time step, one that is moreover expressed purely in terms of configurations at different time steps. This paper formulates the first- and second-order linearizations of a variational integrator in a manner that is amenable to control analysis and synthesis, creating a bridge between existing analysis and optimal control tools for discrete dynamic systems and variational integrators for mechanical systems in generalized coordinates with forcing and holonomic constraints. The forced pendulum is used to illustrate the technique. A second example solves the discrete Linear Quadratic Regulator (LQR) problem to find a locally stabilizing controller for a 40 DOF system with six constraints.

We present a technique to implement scalable variational integrators for generic mechanical systems in generalized coordinates. Systems are represented by a tree-based structure that provides efficient means to algorithmically calculate values (position, velocities, and derivatives) needed for variational integration without the need to resort to explicit equations of motion. The variational integrator handles closed kinematic chains, holonomic constraints, dissipation, and external forcing without modification. To avoid the full equations of motion, this method uses recursive equations, and caches calculated values, to scale to large systems by the use of generalized coordinates. An example of a closed-kinematic-chain system is included along with a comparison with the open-dynamics engine (ODE) to illustrate the scalability and desirable energetic properties of the technique. A second example demonstrates an application to an actuated mechanical system.

We use a tree-based structure to represent mechanical systems comprising interconnected rigid bodies. Using this representation, we derive a simple algorithm to numerically calculate forward kinematic maps, body velocities, and their derivatives. The algorithm is computationally efficient and scales to large systems very well by using recursion to take advantage of the tree structure. Moreover, this method is less prone to modeling errors because each element of the graph is simple. The tree representation provides a natural framework to simulate mechanical dynamics with numeric computations rather than large symbolically-derived equations. In particular, the representation allows one to simulate systems in generalized coordinates using Lagrangian dynamics without symbolically finding the equations of motion. This method also applies to the relatively new variational integrators which numerically integrate dynamics in a way that preserve momentum and other symmetries. We show how to implement both integration schemes for an arbitrary system of interconnected rigid bodies in a computationally efficient way while avoiding symbolic equations of motion. We end with an example simulating a marionette; a mechanically complex, high degree-of-freedom system.

We consider the problem of modeling a robotic marionette. Marionettes are highly under-actuated systems that can only be controlled remotely by moving strings. We present a mixed dynamic-kinematic modeling technique that removes the controller dynamics from the marionette, resulting in a clean abstraction that represents the dynamics of the marionette in a natural way. As an example, a model is derived for a single arm moving in a plane. A model for a three-dimensional marionettes is also shown. Finally, an expansive-space tree (EST) motion planner is used to find a path from an input configuration to a goal for a puppet arm with seven degrees of freedom