Parametric uncertain identification of a robotic system

This paper presents the parametric uncertainties identification of a robotic system of one degree of freedom. A MSC-ADAMS / MATLAB co-simulation model was built to simulate the uncertainties that affect the robotic system. For a desired trajectory, a set of dynamic models of the system was identified in presence of variations in the mass, length and friction of the system employing least squares method. Using the input-output linearization technique a linearized model plant was defined. Finally, the maximum multiplicative uncertainty of the system was modelled giving the controller desired design conditions to achieve a robust stability and performance of the closed loop system.


Introduction
Usually, a parametric identification process is performed for a system under ideal conditions. That is, it does not take into account the presence of parametric uncertainness and external perturbations that alter the system dynamics. For a robotic system, parametric uncertain and external perturbations identification presents a challenge because these systems are non-linear, multivariable and exhibit a high degree of interaction between its links [2]- [4]. This paper proposes a parametric uncertainness identification for a robotic system of one degree of freedom (DOF). A MSC-ADAMS/MATLAB cosimulation model is built to simulate the robotic system dynamics. Then, the non-linear dynamic model parametric identification of the robot is performed using the recursive least square method [1]. After this, a linearized dynamic model of the robotic system is obtained applying the input-output linearization technique. This linear model belongs to the nominal plant and it represents a model of the system without considering external disturbances and parametric uncertainness. Then, the family of plants for the robotic system is calculated varying the mass, width and friction in the co-simulation model of the robotic system, applying the linearization technique for each condition. From the family of plants, the multiplicative uncertainty is computed for each member of the family and the maximum multiplicative uncertain profile of the system is determined. This maximum uncertain profile allows to find the desired design conditions for the controller which ensure the stability and robust performance of the robotic system [5]. The paper is structured as follows. First, a MSC-ADAMS/MATLAB co-simulation model is presented. Second, the parametric identification using recursive least square method is performed. Third, a linearized model of the robotic system is obtained by input-output linearization. Fourth, the family of plants is determined to obtain the maximum multiplicative uncertain profile and find the desired design controller specifications to reach a robust stability and performance.

ADAMS-MATLAB co-simulation dynamic model
In order to study the real-time dynamic behaviour of the robotic system is necessary to build a simulation model able to emulate all the dynamic actions applied over the robotic system during its operation, such as friction, Coriolis forces, gravity effect, and more. For this, MSC-ADAMS software was used. The robotic articulation of one DOF can be represented with a simple pendulum that receives an applied torque in order to generate the movement. The pendulum is placed in the same direction as the gravity, which must be compensated later for the development of the controller. The robotic system rotates around Z-axis, and the applied torque is perpendicular to the movement axis. Then, the MSC-ADAMS model is exported to MATLAB to modelling the parametric uncertain of the robotic system [6]. Figure  1a shows the MSC-ADAMS model of the robotic system.

Identification of the robotic-joint dynamic model
According to [1], the dynamic model for a robotic joint of 1DOF is given by (1) where, is the applied torque, 1 is the mass of the pendulum, is the gravity centre, g is the gravity force, ,̇ and ̈ are the position, speed and acceleration of the pendulum respectively, and 1 and 1 represent the viscous and Coulomb friction. (1) can be expressed in parametric form as is defined in (2).
The identification of the dynamic model constants 1 , 2 , 3 and 4 is performed using the linear regression method based on the recursive leastsquares (RLS) algorithm. Based on the co-simulation model and the applied torque defined in (3) the identification process is implemented using the experimental data from position, speed, acceleration and applied torque of the joint.

Robotic system linearization using the input-output technique
To obtain the nominal plant and the family of plants, is necessary a linear model of the system. Since the robot dynamic is non-linear system as is described in (2), it requires a linearization technique to obtain its linear model. In this case, the input-output linearization technique was used [7]. The representation of the system in state variables is shown in (4), with 1 = and 2 =̇.
The system (4) can be expressed as: The linearization control law is given by ( where is the new entry of the system, 0 and 1 are the constants of the desired polynomial that represents the open-loop behavior of the linearized system. ( ) , ( ) 2 and ( ) , are given by (7)-(10) using Lie's algebra.
The desired characteristic polynomial for the linearized system is described in (11): Parameters 1 , 0 , are computed so that the frequency response of linearized system ( ( )) be comparable to a second order linear model presented in (12).  Table 1 shows the original parameter of the robotic system and its percentage of variation due to parametric uncertainness described above. From each condition presented in Table 1, co-simulation model is modified to obtain the parameters 1 to 4 of the dynamic model (2) against each set of perturbations as show in Table 2. The + and -signs in Table 2 indicates the maximum and minimum values of the parameters mass (M), width (W) and joint friction (F). As shown in Table 2, the parameters of the dynamic model (2) varying according to the uncertain set presented. This indicates that the robotic system dynamics is different for each uncertain set applied. From the dynamic model parameters (2), the robotic system is linearized as a linear second order system descripted in (13) with , and calculated for each condition as shown in Table 3. Frequency response of the family of plants is presented in Figure 2.   As shown in Figure 2, the nominal plant represented by * is placed between the variation range presented by the uncertainness descripted in Table 1. Once the family of plants is defined, the uncertain of each member of the family of plants is calculated using the multiplicative uncertain model [8] showed in (14).  where ( ) correspond to the nominal plant descripted by (12) and ̃( ) is the ith member of the family of plants. Frequency response of the multiplicative uncertain profile of each member of the family of plants is shown in Figure 3a. To find the conditions for robust performance and stability is necessary to determinate the maximum multiplicative uncertain profile. For this, the maximum uncertain profile is estimated according to the frequency response of the multiplicative uncertain for the family of plants shown in Figure 3. The maximum multiplicative uncertain profile reached is given by (15) and its frequency response is shown in Figure 3b. Maximum multiplicative uncertain profile of the family of plants.
As observed in Figure 3b, the maximum profile of multiplicative uncertain represented by * covers all the multiplicative uncertainness descripted by the family of plants. From [8], the closed loop desired specifications are calculated to achieve a robust stability and performance according to (16).
where | ( )| is the magnitude of the complementary sensibility function and | ( )| correspond to the magnitude of maximum profile of multiplicative uncertain. Figure 4 shows the function |1/ max ( ) | which restrict the closed loop bandwidth of the system to 0.7 / . The bandwidth is related to the natural frequency by (17).
Assuming = 0.6 which is equivalent to an 10% overshoot, from (16) = 0.61 / , the desired closed loop function is (18). The frequency response of (18) is shown in Figure 5 which bandwidth is limited by the maximum inverse uncertain profile.  (20) Figure 4. Desired robust operation conditions for the robotic system

Conclusions
This paper presented a methodology for the parametric uncertainness identification of a robotic system. A MSC-ADAMS/MATLAB co-simulation model was built for the parametric identification of the dynamic model using least square algorithm. The dynamic model was linearized using input-output linearization, obtaining the nominal plant. After that, a set of parametric uncertainness is defined to obtain the family of plants for the robotic system. Then the multiplicative uncertain of the family of plants is determined to find the maximum uncertain profile, which one establishes the closed loop system specifications to reach a robust stability and performance. Results show that when a robotic system is affected by external disturbances and parametric uncertainness, it results in many dynamical behaviours of the system, which must be covered for a controller to get a robust performance of the system. Employing the proposed methodology the desired design controller specifications to achieve a robust stability and performance against the presence of external disturbances a parametric uncertainness can be found. As well, this methodology may be applied to any non-linear system in order to achieve a robust stability and performance.