Tracking fast inverted trajectories of the underactuated Acrobot

Abstract

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.

Nonlinear control of mechanical systems with one degree of underactuation

“…Remark 9: More generally, the underlying reason for the static feedback linearizability of the inertia wheel pendulum can be tied to be the following result, which applies to N DOF mechanisms (2). Consider again the one-formω = N −1 k=0 d 0,k (q 1 , · · · , q N −1 )dq k associated with the generalized conjugate momentum (9) and suppose thatω is closed.…”

“…Denote the Lagrangian by L = K − V and assume that the system is actuated according to (1). The model can then be written as in (2). Subsequent analysis and feedback design are more easily accomplished if the system is first transformed into the normal form [44,53] …”

“…Since the center of mass coordinates are unactuated, the control of the system (3) can be reduced to the control of a system having one degree of underactuation as in (2) by eliminating the trivial dynamics x c = 0,ÿ c = g 0 . In this sense, the two systems in Figures 1 and 2 are very similar: they give rise to control problems for systems with N ≥ 2 DOF, (N − 1) actuators, and the cyclic coordinate is unactuated.…”

“…Though not indicated in the figure, massless springs may be attached between links and between links and the inertial reference frame; prismatic joints between links are also allowed. This class of systems clearly includes the Acrobot [2,36,52], the brachiating robots of [15,34,35,46], the gymnast robots of [32,39,57] when pivoting on a high bar, and the stance phase models of Raibert's onelegged hopper [1,6,14,26,33,42] as well as RABBIT [7][8][9][10]41]. The control objectives will be to stabilize the system about an equilibrium point or to track a set of reference trajectories with internal stability.…”

Development of a real-time-position prediction algorithm for under-actuated robot manipulator by using of artificial neural network

“…In the case of under-actuated robots, this condition is not satisfied, which makes the behaviour of that class of robots very difficult to be predicted. Under-actuated robots can be a better design choice for robots in space and other industrial applications, and their advantages over fully actuated robots led to many studies to predict their behaviour [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. As a first advantage, a light-weight and low power consumption manipulator can be made.…”