# Influence of Joint Stiffness and Motion Time on the Trajectories of Underactuated Robots
## 1 Introduction  
### State of the Art  
Underactuated systems are mechanical systems in which the number of degrees of freedom is larger than the number of actuators [1]. Among the underactuated systems, in the last few decades, underactuated robots have gained popularity due to their ability to gain dexterity with low-cost and low-weight structures, which are of great importance in industrial applications. In fact, underactuated robots are manipulators with one or more nonactuated joints, and often, the actuators are substituted by torsional springs [2], even if interesting exceptions exist [3; 4]. Possible applications of underactuated robots include industrial robots, legged locomotion, and surgical robots [5; 6]. There are some studies on underactuated grasping tools [7; 8; 9]. Moreover, the interest in underactuated robots is increasing, since a fully actuated robot in case of actuator failure becomes an underactuated robot [1].  
The dynamics of a robot are typically nonlinear, and if the degrees of freedom (DOF) are more than the actuators, the system is usually hard to control due to the presence of nonholonomic second-order constraints [5]. Different approaches have been proposed by researchers to tackle this problem. Some researchers have proposed strategies to control nonlinear underactuated robots, such as:  
* Smooth second-order sliding mode control [1; 10];
* Dynamic feedback linearization [5; 11];
* Fuzzy control [1; 12; 13];
* Optimal control [1; 14; 15; 16].  
As a result, researchers have been able to control many underactuated systems [17], such as Planar RR robots [18], the Furuta pendulum [19], the flywheel inverted pendulum [20], the wheeled inverted Pendulum [21], and the inertia wheel inverted Pendulum [22].  
Other researchers have exploited the concept of differential flatness. Differentially flat systems have a set of outputs, so-called _flat outputs_, such that all states and inputs can be retrieved from flat outputs without integration [23]. The relationship connecting flat outputs with system states and inputs is referred to as a _diffeomorphism_. The number of flat outputs is equal to the number of inputs, i.e., the actuator torques. A differentially flat system gains controllability [24].  
A planar robot must satisfy some conditions to be differentially flat [25; 26; 27; 28]. Considering a \(n\)-DOF manipulator, the center of mass of the last link \(n\) has to be placed on joint axis \(n\), and the center of mass of links \(n\) and \(n-1\) has to be located on joint axis \(n-1\). These conditions sequentially repeat until the center of mass of the last \(j\) links (i.e., \(n,n-1,\ldots,n-j+1\)) is on the \(n-j+1\) joint axis, as described in [28]. The actuators of the system are placed on joints \(n-j+1\), whereas the \(j-1\) joints are equipped with torsional springs.  
The control of a differentially flat robot is simpler than the control of a generic underactuated robot. Moreover, if the robot moves in the horizontal plane, the dynamics become linear, with further simplifications.  
It is worth noting that differentially flat systems are not controllable along an entire trajectory. Only point-to-point motions are attainable, and the joint movement between the points is not a priori known, but can be calculated after the torques are retrieved from the flat outputs [29; 30]. Differentially flat systems are controlled by choosing appropriate flat output functions, based on the boundary conditions on the robot states; such functions are used to calculate the motor torques, which are the input of the dynamic system to calculate joint variables. In [5], a comparison between the approach based on differential flatness and the approach based on dynamic feedback linearization was made.