# Evaluation of Linearization Methods for Control of the Pendubot
## 1 Introduction  
Acrobot and Pendubot are well-known fundamental examples of under-actuated mechanical systems moving in the presence of gravity. Although their kinematic structures are relatively simple, still they can be considered as important benchmark systems which are of great importance for the development of nonlinear control strategies.  
Basically, a standard approach to stabilize both systems at an unstable equilibrium requires design of two controllers for the task of swing-up (when the robot hangs down or is far from equilibrium pose) and the balancing task around the equilibrium point, respectively [1]. In particular, the swing-up problem can be viewed as a challenging problem in non-linear control. Many solutions to it are based on partial feedback linearization, starting from fundamental works of [PERSON] and [PERSON] [2; 3; 4; 5]. Other approaches take advantage of an energy based control and the passivity properties of the system [1; 6], or a fuzzy logic control [7]. Among others a model orbit stabilization [8], or virtual holonomic constrains are proposed to swing up the Pendubot as well. Recently, an extended state observer based active disturbance rejection control scheme was applied to the Pendubot system for a trajectory tracking tasks in the presence of uncertain dynamics [9]. On the other hand, the stabilization task near the equilibrium pose are mainly supported by linear controllers including LQR-based feedbacks [5; 10; 11].  
The considered topics also cover essential aspects related to under-actuated robots, which are constantly being developed, bringing new values or detailing the problem of control of such systems [12; 13; 14].  
Recently, in [15] a new insight to control of under-actuated systems is proposed. The presented approach is based on finding the largest feedback linearizable subsystem for mechanical systems that are not fully feedback linearizable. Although in the mentioned paper the selection of a partially linearizing output that renders the zero dynamics asymptotically stable is thoroughly supported by mathematical formulas developed for double inverted pendulum with one actuator (Acrobot and Pendubot), no attempt is done to apply the discussed concept for a constructive design of a controller for these benchmark systems. Consequently, any simulation or experimental verification has not been reported yet. In this paper, however, authors attempt to fill this gap and extend preliminary resultsdiscussed in [16] for the Pendubot. Authors are also interested in a possible application of such an algorithm for the physically existing robot.  
An important issue of the linearization approach using state and feedback transformations discussed in [15] is the existence of singular points. Due to this obstruction, a feasible state space is constrained and a feedback cannot be designed in a global way. In the case of Pendubot, a severe linearization constraint comes from the requirement that the system should be in a moving phase. Thus, it is impossible to use this approach directly to stabilize the Pendubot at an equilibrium. In this paper, in order to overcome these limitations we propose a hybrid controller which is supported by a simple linear feedback. Then we consider the performance of closed-loop system based on numerical analysis. In particular, we search for admissible states of the Pendubot for which the hybrid algorithm ensures the convergence to the equilibrium point and makes this point asymptotically stable.  
The paper is organized as follows. Section 2 describes the mathematical model for the Pendubot system, together with constrains imposed on the model. In Section 3, the analyzed control algorithms along with the state transformation for the stabilization task is introduced. Section 4 describes the simulation results together with experimental validation of analysed algorithms, while the Section 5 gives final remarks.