## 1 Introduction  
The concept of controllability denotes the ability to move a robot around in its entire configuration space using only certain admissible manipulations. The exact definition varies slightly within the framework or the kind of applied robots. Controllability and observability are dual aspects of the same problem. Many authors have proposed interesting controllers or observers such as [1, 2, 3, 4, 5, 6, 7, 8, 9]; however, in most of the studies, the robot models are assumed to be controllable or observable without any proof. A method to determine the controllability of a robot model is important because a robot model that is assured to be controllable can guarantee the existence of a controller to reach one of the objectives such as regulation, tracking, disturbance rejection, etc. A method to determine the observability of a robot model is important because a robot model that is assured to be observable can guarantee the existence of an observer to reach one of the objectives such as states estimation, disturbance estimation, etc. Hence, it would be interesting to study when the controllability and observability of robots is not assumed. There are some studies about controllability and observability. In [10, 11, 12, 13], the authors are focused on the local, linear, global, and exact controllability and observability. In [14, 15, 16], the authors discuss the controllability and observability of linear time-varyingsystems. In [17; 18; 19; 20], the authors use algorithms to determine controllability and observability. In [21; 22; 23], the authors use the geometric method to determine controllability and observability. In [24; 25; 26], the authors consider the fact that robot models are changed from being linearly uncontrollable to linearly controllable when the equilibrium point is moved from the origin to a different one. In [27; 28; 29], the authors propose feedback for the linearization of robot models to determine controllability. In [30; 31; 32], the authors present a controllability analysis proving that some robots are not controllable. In [33; 34; 35], the authors extract properties of the mechanical parameters to demonstrate that some robots are controllable and observable. In [36; 37; 38], the authors design elastic, flexible, or micro-robots, which, since their design, are controllable. In [39; 40], authors find a transformation to determine which states have the greatest contribution in controllability and observability. From the aforementioned studies, some authors use a linearization method [24; 25; 26; 27; 28; 29; 30; 31; 32], and the other use a structural properties method [33; 34; 35; 36; 37; 38; 39; 40] to determine the controllability and observability of robots. The linearization method is an approximation of the robot model around the origin, and the structural properties method is a different representation of the nonlinear model based on transformations. The linearization method has the problem that could produce erroneous results because of the approximation nature, and the structural properties method has the problem that the robot model requires to satisfy structural conditions. It would be interesting to propose a method to determine the controllability and observability of robots that evade the problems presented in the mentioned methods. In this study, we propose a transformed structural properties method to determine the controllability and observability of robots that is the combination of the linearization method [24; 25; 26; 27; 28; 29; 30; 31; 32] and the structural properties method [33; 34; 35; 36; 37; 38; 39; 40]. The proposed method uses a transformation in the robot model to obtain a linear robot model with the gravity terms and uses the linearization of the gravity terms to obtain a linear robot model; this linear robot model is used to determine controllability and observability.