Quotient method for controlling the acrobot

Abstract

Abstract-This paper describes a two-sweep control design method to stabilize the acrobot, an input-affine under-actuated system, at the upper equilibrium point. In the forward sweep, the system is successively reduced, one dimension at a time, until a two-dimensional system is obtained. At each step of the reduction process, a quotient is taken along one-dimensional integral manifolds of the input vector field. This decomposes the current manifold into classes of equivalence that constitute a quotient manifold of reduced dimension. The input to a given step becomes the representative of the previous-step equivalence class, and a new input vector field can be defined on the tangent of the quotient manifold. The representatives remain undefined throughout the forward sweep. During the backward sweep, the controller is designed recursively, starting with the twodimensional system. At each step of the recursion, a well-chosen representative of the equivalence class ahead of the current level of recursion is chosen, so as to guarantee stability of the current step. Therefore, this stabilizes the global system once the backward sweep is complete. Although stability can only be guaranteed locally around the upper equilibrium point, the domain of attraction can be enlarged to include the lowerequilibrium point, thereby allowing a swing-up implementation. As a result, the controller does not require switching, which is illustrated in simulation. The controller has four tuning parameters, which helps shape the closed-loop behavior.

A quotient method for designing nonlinear controllers

“…This is a consequence of Lemma 2 and Lemma 3. However, even if the distribution generated by the original system is not involutive, the approach can still be applied, as illustrated in [12] and [13].…”

“…This section presents an application of the algorithm to a FBL system. However, the algorithm is not restricted to FBL systems, and application of the quotient method to non-FBL system are illustrated in [12] and [13].…”

“…That is, non-FBL systems will necessarily have an iteration where a linear relationship cannot be established. Hence, in this case, the algorithm needs to be modified as illustrated in [12] and [13]. Remark 2: This algorithm can transform the system into a chain of integrators by choosing γ(x) = L f φ i such that L g L f φ i = 0 at every iteration.…”

“…Furthermore, with certain assumptions, the proposed approach can approximate non-FBL systems as FBL systems. For example, it has been possible to achieve both swingup and stabilization of the acrobot [12] and the inverted pendulum [13].…”

Numerical algorithm for feedback linearizable systems

“…Computing such an output requires a systematic procedure, such as the one proposed in [3,4], which proceeds by successively generating quotients that are desensitized with respect to the input vector field. This method has been extended to produce a control design technique applicable to FL input-affine single-input systems of the formFor non-FL systems, the method requires approximations [6,7]. However, for some non-FL systems, the system of equations turns out to be so complicated that, even after approximations, the method is not applicable due to the lack of closed-form solutions to some of the equations.…”