## 3 Experimental Validation: FO-PI Design for a UR10 Robot  
Manipulator robots, Figure 2, represent one of the biggest class of robots which are used in the factory environment. The relationship between the actuation and the contact forces, which are acting on the robot, is given by the dynamic equation of motion. The kinematic model of a manipulated robot can be expressed as a series of rigid bodies connected by joints [20]. The dynamics of the robot can be defined using spatial vector notation or canonical equations (which are also used in this study). There are two canonical forms used to describe the equation of motion, the joint space formulation:  
\[Q=M(q)\ddot{q}+C(q,\dot{q})\dot{q}+G(q) \tag{9}\]  
and the operational space formulation:  
\[\mathrm{F}=\Lambda(x)\dot{\mathrm{v}}+\mathrm{Y}(\mathrm{x},\mathrm{v})+ \Phi(\mathrm{x}) \tag{10}\]  
where \(q,\dot{q},\ddot{q}\in\Re^{n}\) represent the joint variables (the generalized joint coordinates, the velocity and the acceleration), \(Mspace inertia matrix, \(Y\) is the vector of the velocity product and \(\Phi\) is the vector of gravity term. The two canonical forms are related and thus the following relation can be given:  
\[\left\{ \begin{array}{c}v=\dot{J}\dot{q}\\ Q=J^{T}F\\ Y=\Lambda(JM^{-1}C\dot{q}-\dot{J}\dot{q})\end{array} \right. \tag{11}\]  
where \(J\) is the Jacobian matrix of the manipulated robot. In most of the robotics applications, one needs to control the pose of the end-effector, which can be done through the direct kinematics formulation. The direct kinematics can be calculated via the Denavit-Hartenberg (D-H) formulation [21]. In this research study, a 6-degrees of freedom manipulator robot is considered, thus the direct kinematic equation is given as:  
\[{}^{0}T_{6}(q)={}^{0}A_{1}(q_{1}){}^{1}A_{2}(q_{2})\dots{}^{5}A_{6}(q_{6}) \tag{12}\]  
where \({}^{i-1}A_{i}(q_{i})\), \(i=1\), 6 represents the homogeneous transformation matrices defined as:  
\[{}^{i-1}A_{i}=Rot_{z,\theta_{i}}Trans_{z,d_{i}}Trans_{x,a_{i}}Rot_{x,\alpha_{i }}=\left[ \begin{array}{ccc}\cos\theta_{i}&-\sin\theta_{i}\cos\alpha_{i}&\sin \theta_{i}\sin\alpha_{i}\\ \sin\theta_{i}&\cos\theta_{i}\cos\alpha_{i}&-\cos\theta_{i}\sin\alpha_{i}\\ 0&\sin\alpha_{i}&\cos\alpha_{i}\\ 0&0&0\end{array} \right] \tag{13}\]  
where the parameters \(\theta_{i}\), \(a_{i}\), \(d_{i}\), \(\alpha_{i}\) represent the D-H parameters related to joint \(i\). These parameters associated to the UR10 robot considered in this study are given in Table 1.  
Figure 2: Schematic overview of the UR10 robot—CAD model. Nowadays, a large number of the industrial robots are driven by brushless servo motors. Several studies in the area of control of manipulator robots consider Cartesian coordinates as control variables. However, in order to obtain the desired Cartesian trajectory for the end-effector attached to the robot, each joint axis must follow a specific trajectory. The common approach to control the robot joints is based on the decentralized techniques, which consider an independent controller for each joint. In this study, the manipulated variable was considered as being the applied voltage \(V_{a}\), while the controlled variable is the position \(\theta(t)\). In the case of speed control, the controlled variable is the angular velocity \(\omega(t)\). The dynamics of a motor attached to a joint \(j\) can be defined as:  
\[J_{m}\dot{\omega}+B\omega+\tau_{c}(\omega)=K_{m}K_{a}V_{a} \tag{14}\]  
with \(\tau_{c}\) the [PERSON]'s law of friction, \(B\) the viscous friction, \(K_{m}\) is the motor torque, \(K_{a}\) is the transconductance of the amplifier and \(J_{m}\) the total inertia seen by the motor for joint \(j\). For simplification reasons, if the Coulomb friction from Equation (14) is ignored, the equivalent Laplace transform of (14) is given by:  
\[sJ_{m}\Omega(s)+B\Omega(s)=K_{m}K_{a}V_{a}(s) \tag{15}\]  
where \(\Omega(s)\) and \(V_{a}(s)\) are the Laplace transform of their corresponding signal from time domain \(\omega\) and \(V_{a}\).