# Control of Flexible Robot by Harmonic Functions
## 1 Introduction  
This paper deals with the control of flexible structures as underactuated systems. The definition of underactuation should be based on the number of blocks in the Brunovsky canonical form [1]. The invariant control--its idea is presented in [2]--is used for flexible robot trajectory tracking with minimal residual oscillations. The desired trajectory is parametrized, exact input-output linearization is applied to the system coordinates (modal coordinates are excluded), and outputs are substituted by function of parametrization variables. This transformation assures that the robot endpoint follows the desired trajectory. Zero dynamics (system flexibility) oscillations are realized only in the perturbation of parametrization variables. The invariant control, based on the harmonic functions [3], is applied to the model of a flexible robotic arm, and its parameters are obtained using predictive optimization. Many authors have already investigated the control of underactuated mechanical systems. Neurodynamic programming, as a main part of the control in [4], is used for the control of the ball and beam system. In [5], a method of adding the actuators to the underactuated joints and running the swing-up motion with optimization minimizing the action of the added actuators is used. Sliding mode control extended with fuzzy logic is introduced in [6] to control the motion of overhead crane. An approach with optimal control is used in [7]. The adaptive approach used for tracking control is presented in [8]. Partly stable controllers derived using the dynamic model of the manipulator create basis for the optimal switching sequence control in [9]. An energy-based approach used in the design of stabilizing controllers is presented in [10]. Friction forces are used in the energy control strategy in [11]. Authors in [12] introduce an averaging theory for controlling the systems with and without drift terms. In [13], the trajectory tracking controller is used for following the upward trajectory created from the downward simulation (similar to the natural motion in [14] and the natural oscillations in [15] of a parallel robot) and stabilizing trajectory. An active disturbance rejection control is designed in [16] for a pendubot system. The walking-like trajectory tracking of an acrobot is continuously developed in [17]. A 3D bipedal gait walking robot is presented in [18], and the development of humanoid motion design is investigated in [19]. The robust sliding mode control is developed for stabilizing and trajectory tracking of a ballbot system in [20], and the wheel-legged biped robot motion is stabilized using active arm manipulator in [21]. The dynamic balancing of quadruped robot with optimal controller is presented in [22], and a different quadruped robot design is presented in [23]. Ref. [24] presents linear quadratic regulator applied to control vertical take-off and landing (VTOL) multi-rotor vehicle with passive rotor tilting mechanism. A control based on a function approximation technique using the variation of an approximated square system is presented in [25; 26]. The cyclic control using sinusoids for nonholonomic systems is presented in [27] (stabilization), and motion planning is presented in [28]. The control of a wheeled, inverted pendulum is transferred into dynamic trajectory planning with a boundary value problem in [29]. State feedback fuzzy system controller, based on \(H_{\infty}\), is investigated in [30]. The underactuated mechanisms in the gravity field can be controlled in their equilibrium positions (see [31]), but there are always singularity points between these equilibrium states. To move the mechanism through this region (see [32]) is a challenging task, and the work in [14] shows one possible way to overcome it. The investigation of system flexibility as a system underactuation is usually investigated with linearized stiffness; the authors of [33] deal with dynamic coupling of such underactuated manipulators.