This paper presents experimental results from the real-time parameter estimation of a system model and subsequent trajectory optimization for a dynamic task using the Baxter Research Robot from Rethink Robotics. An active estimator maximizing Fisher information is used in real time with a closed-loop, nonlinear control technique known as sequential action control. Baxter is tasked with estimating the length of a string connected to a load suspended from the gripper with a load cell providing the single source of feedback to the estimator. Following the active estimation, a trajectory is generated using the trep software package that controls Baxter to dynamically swing a suspended load into a box. Several trials are presented with varying initial estimates showing that the estimation is required to obtain adequate open-loop trajectories to complete the prescribed task. The result of one trial with and without the active estimation is also shown in the accompanying video.

This paper introduces and implements a method for real-time mode scheduling in linear time-varying switched systems subject to a quadratic cost functional. The execution time of switched system algorithms is often prohibitive for real-time applications and typically may only be reduced at the expense of approximation accuracy. We address this tradeoff by taking advantage of system linearity to formulate a projection-based approach, such that no simulation is required during open-loop optimization. A numerical example shows how the proposed open-loop algorithm outperforms the methods employing common numerical integration techniques. In addition, we follow a receding-horizon scheme to schedule the modes of a customized experimental setup in real time, using the robot operating system. In particular, we demonstrate-both in Monte Carlo simulation and in experiment-that optimal mode scheduling efficiently regulates a cart and suspended mass system and rejects disturbances online.

This paper presents the details and experimental results from an implementation of real-time trajectory generation and parameter estimation of a dynamic model using the Baxter Research Robot from Rethink Robotics. Trajectory generation is based on the maximization of Fisher information in real-time and closed-loop using a form of Sequential Action Control. On-line estimation is performed with a least-squares estimator employing a nonlinear state observer model computed with trep, a dynamics simulation package. Baxter is tasked with estimating the length of a string connected to a load suspended from the gripper with a load cell providing the single source of feedback to the estimator. Several trials are presented with varying initial estimates showing convergence to the actual length within a 6 second time-frame.

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.

When attempting to estimate parameters in a dynamical system, it is often beneficial to strategically design experimental trajectories that facilitate the estimation process. This paper presents an optimization algorithm which improves conditioning of estimation problems by modifying the experimental trajectory. An objective function which minimizes the condition number of the Hessian of the least-squares identification method is derived and a least-squares method is used to estimate parameters of the nonlinear system. A software-simulated example demonstrates that an arbitrarily designed trajectory can lead to an ill-conditioned least-squares estimation problem, which in turn leads to slower convergence to the best estimate and, in the presence of experimental uncertainties, may lead to no convergence at all. A physical experiment with a robot-controlled suspended mass also shows improved estimation results in practice in the presence of noise and uncertainty using the optimized trajectory.

Estimation of model parameters in a dynamic system can be significantly improved with the choice of experimental trajectory. For general nonlinear dynamic systems, finding globally "best" trajectories is typically not feasible; however, given an initial estimate of the model parameters and an initial trajectory, we present a continuous-time optimization method that produces a locally optimal trajectory for parameter estimation in the presence of measurement noise. The optimization algorithm is formulated to find system trajectories that improve a norm on the Fisher information matrix (FIM). A double-pendulum cart apparatus is used to numerically and experimentally validate this technique. In simulation, the optimized trajectory increases the minimum eigenvalue of the FIM by three orders of magnitude, compared with the initial trajectory. Experimental results show that this optimized trajectory translates to an order-of-magnitude improvement in the parameter estimate error in practice.

The underactuated system under consideration is a magnetically-suspended, differential drive robot utilizing a winch system to articulate a suspended mass. A dynamic model of the system is first constructed, and then a nonlinear, infinite-dimensional optimization algorithm is presented. The system model uses the principles of kinematic reduction to produce a mixed kinematic-dynamic model that isolates the modeling of the system actuators from the modeling of the rest of the system. In this framework, the inputs become generalized velocities instead of generalized forces facilitating real-world implementation with an embedded system. The optimization algorithm automatically deals with the complexities introduced by the nonlinear dynamics and underactuation to synthesize dynamically feasible system trajectories for a wide array of trajectory generation problems. Applying this algorithm to the mixed kinematic-dynamic model, several example problems are solved and the results are tested experimentally. The experimental results agree quite well with the theoretical showing promise in extending the capabilities of the system to utilize more advanced feedback techniques and to handle more complex, three-dimensional problems.