Elliot R. Johnson

Scalable Variational Integrators for Constrained Mechanical Systems in Generalized Coordinates

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.

Second-Order Switching Time Optimization for Nonlinear Time-Varying Dynamic Systems

This technical note gives a method for calculating the first and second derivatives of a cost function with respect to switching times for systems with piecewise second-differentiable dynamics. Differential equations governing the linear and bilinear operators required for calculating the derivatives are presented. Example optimizations of linear and nonlinear systems are presented as evidence for the value of second-order optimization methods. One example converges in 33 iterations using second-order methods whereas the first-order algorithm requires over 30 000 iterations.

Dynamic Modeling and Motion Planning for Marionettes: Rigid Bodies Articulated by Massless Strings

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

Structured Linearization of Discrete Mechanical Systems for Analysis and Optimal Control

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.

Discrete and continuous mechanics for tree representations of mechanical systems

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.

Constructing and Implementing Motion Programs for Robotic Marionettes

This technical note investigates how to produce control programs for complex systems in a systematic manner. In particular, we present an abstraction-based approach to the specification and optimization of motion programs for controlling robot marionettes. The resulting programs are based on the concatenation of motion primitives and are further improved upon using recent results in optimal switch-time control. Simulations as well as experimental results illustrate the operation of the proposed method.

A Variational Approach to Strand-Based Modeling of the Human Hand

This paper presents a numerical modeling technique for dynamically modeling a human hand. We use a strand-based method of modeling the muscles. Our technique represents a compromise between capturing the full dynamics of the tissue mechanics and the need for computationally efficient representations for control design and multiple simulations appropriate for statistical planning tools of the hand. We show how to derive a strand-based model in a variational integrator context. Variational integrators are particularly well-suited to resolving closed-kinematic chains, making them appropriate for hand modeling.We demonstrate the technique first with a detailed exposition of modeling an index finger, and then extend the model to a full hand with 19 rigid bodies and 23 muscle strands. We end with a discussion of future work, including the need for impact handling, surface friction representations, and system identification.

Second order switching time optimization for time-varying nonlinear systems

This paper presents second-order switching time optimization techniques for nonlinear time-varying systems that are piecewise second-differentiable. The techniques are based on direct analysis of flows rather than constrained optimization methods, making it possible to take higher order derivatives. The resulting algorithms also use the same number of integrations along the trajectory regardless of the number of switching times. Two examples are included-a linear example seen previously in the literature and a nonlinear example that emphasizes the fast convergence rates of second-order optimizations compared to first order techniques (e.g. convergence in 19 iterations compared to over 25,000).

Linearizations for mechanical systems in generalized coordinates

We describe an algorithm for calculating the linearization of the dynamics for arbitrary constrained mechanical systems in generalized coordinates without using symbolic equations. Linearizations of dynamics are useful tools for controllability and stability analysis and can be used to generate locally stabilizing controllers for linear and non-linear systems. However, the computational expense for finding linearizations of complex mechanical systems is often cited as a limiting factor that prevents their use. Recent work has introduced new methods of calculating the dynamics of arbitrary mechanical systems in generalized coordinates without deriving large, system-specific equations of motion. This paper extends that approach to calculate the linearizations of the dynamics without using the symbolic equations of motion. Using these ideas, it becomes practical to both simulate, analyze, and control more complex mechanical systems without sacrificing the benefits of generalized coordinates. Furthermore, this method addresses systems with closed kinematic chains, constraints, and external non-conservative forcing. The technique is applied to an example system with a closed kinematic chain and the resulting linearization agrees with results found by symbolically differentiating the full equations of motion.

Robotic Puppets and the Engineering of Autonomous Theater

This chapter outlines the design of software for embedded control of robotic marionettes using choreography. In traditional marionette puppetry, the puppets often possess dynamics that are quite different from the creatures they imitate. Puppeteers must therefore understand and leverage the inherent dynamics of the puppets to create believable and expressive characters. Because marionettes are actuated by strings, the mechanical description of the marionettes either creates a multiscale or degenerate system—making simulation of the constrained dynamics challenging. Moreover, marionettes have 40–50 degrees of freedom with closed kinematic chains. Generating puppet choreography that is mimetic (that is, recognizably human) results in a high-dimensional nonlinear optimal control problem that must be solved for each motion. In performance, these motion primitives must be combined in a way that preserves stability, resulting in an optimal timing control problem. Our software accounts for the efficient computation of the (1) discrete time dynamics that preserve the constraints and other integrals of motion, (2) nonlinear optimal control policies (including optimal control of LTV systems), and (3) optimal timing of choreography, all within a single framework. We discuss our current results and the potential application of our findings across disciplines, including the development of entertainment robots and autonomous theater.