The control of underactuated robot manipulators provides a significant challenge to the robotics engineer. The Acrobot is a simple underactuated system consisting of a double pendulum with an actuator at only the second joint. Previous work has shown how linearization methods can achieve the tracking of slow inverted trajectories. In this paper we derive a surprising set of exact trajectories of the nonlinear equations of motion, which involve inverted periodic motions. The trajectories can be made arbitrarily fast by appropriate choice of the Acrobot mass and length parameters. Next, we present a nonlinear control law (which has appeared elsewhere) and show how it can be applied to the Acrobot to track these trajectories. In simulations we compare tracking results for our controller and one based on pseudolinearization. The pseudolinearizing controller produces significant error for a 1 Hz trajectory, while ours produces none. Finally, we present experimental results which demonstrate that the assumptions of the theory were not overly restrictive. In particular, peak-to-peak oscillations of joint 2 as large as 85 were obtained, despite real-world effects, such as joint friction, inexact parameter values, and noisy and delayed joint velocity data.

In this paper we present a model for oscillating foil propulsion in which springs are used to transmit forces from the actuators to the foil. The expressions for hydrodynamic force and moment on the foil come from classical, linear, unsteady aerodynamics, and these are coupled to linearized rigid-body mechanics to obtain the complete model for swimming. The model is presented as a low-order set of ordinary dierential equations, which makes it suitable for the application of techniques from systems and control theory. The springs serve to reduce energy costs, and we derive explicit expressions for spring constants which are optimal in this sense. However, the use of springs can potentially lead to unstable dynamics. Therefore, we also derive a set of necessary and sucient conditions for stability. A detailed example is presented in which energy costs for one actuator are reduced by 33%.

In this paper, a 2-DOF model for quadrupedal running-in-place is presented, consisting of a rigid body on springy legs. "Energypumping " feedback is included to excite the natural dynamics of the system. The model exhibits at least two periodic solutions, which correspond to the bound and pronk gaits of four-legged animals. Approximate return maps are constructed around both trajectories, and these are used to derive explicit expressions for the amplitude and stability of the gaits. The pronk is shown to produce significantly more height than the bound for the same amount of effort. However, the bound has more desirable stability characteristics: simulations and analysis demonstrate that the bound is unstable if a dimensionless body inertia is greater than 1, which is an unlikely situation for either a robot or an animal. The pronk's stability, however, exhibits coupling between height and inertia: a given inertia is more likely to cause instability for large heights than for small heights. These results indicate that body inertia is a critical design parameter for running robots, and suggest passive mechanisms for balance and gait transition in legged robots and animals.

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