Two Cooperating Manipulators with Fractional Controllers

This paper analyzes the dynamic performance of two cooperative robot manipulators. It is studied the implementation of fractional-order algorithms in the position/force control of two cooperating robotic manipulators holding an object. The simulations reveal that fractional algorithms lead to performances superior to classical integer-order controllers.


Introduction
Two robots carrying a common object are a logical alternative for the case in which a single robot is not able to handle the load.The choice of a robotic mechanism depends on the task or the type of work to be performed and, consequently, is determined by the position of the robots and by their dimensions and structure.In general, the selection is done through experience and intuition; nevertheless, it is important to measure the manipulation capability of the robotic system (Y.C. Tsai & A.H Soni., 1981) that can be useful in the robot operation.In this perspective it was proposed the concept of kinematic manipulability measure (T.Yoshikawa, 1985) and its generalization to dynamical manipulability (H.Asada, 1983) or, alternatively, the statistical evaluation of manipulation (J. A. Tenreiro. Machado & A. M. Galhano, 1997).Other related aspects such as the coordination of two robots handling objects, collision avoidance and free path planning have been also investigated (Y.Nakamura, K. Nagai, T. Yoshikawa, 1989) but they still require further study.With two cooperative robots the resulting interaction forces have to be accommodated and consequently, in addition to position feedback, force control is also required to accomplish adequate performances (T.J. Tarn, A. K. Bejczy, P. K., 1996) and (N.M. Fonseca Ferreira, J. A. Tenreiro Machado, 2000) and (A.K. Bejczy and T. Jonhg Tarn, 2000).There are two basic methods for force control, namely the hybrid position/force and the impedance schemes.The first method (M.H. Raibert and J. J. Craig, 1981) separates the task into two orthogonal sub-spaces corresponding to the force and the position controlled variables.Once established the subspace decomposition two independent controllers are designed.The second method (N.Hogan, 1985) requires the definition of the arm mechanical impedance.The impedance accommodates the interaction forces that can be controlled to obtain an adequate response.Others authors (Kumar, Manish;Garg, Devendra 2005, Ahin Yildirim, 2005, Jufeng Peng, Srinivas Akella, 2005) present advance methodologies to optimize the control of two cooperating robots using the neural network architecture and learning mechanism to train this architecture online.This paper analyzes the manipulation and the payload capability of two arm systems and we study the position/force control of two cooperative manipulators, using fractional-order (FO) algorithms (J. A. Tenreiro Machado, 1997) and (N.M. Fonseca Ferreira & J. A. Tenreiro Machado 2003, 2004and 2005).Bearing these facts in mind this article is organized as follows.Section two presents the controller architecture for the position/force control of two robotic arms.Based on these concepts, section three develops several simulations for the statistical analysis and the performance evaluation of FO and classical PID controllers, for robots having several types of dynamic phenomena at the joints.Finally, section four outlines the main conclusions.

Control of Two Arms
The dynamics of a robot with n links interacting with the environment is modelled as: where τ is the n × 1 vector of actuator torques, q is the n × 1 vector of joint coordinates, H(q) is the n × n inertia matrix, C(q,q) is the n × 1 vector of centrifugal/Coriolis terms and G(q) is the n × 1 vector of gravitational effects.
The n × m matrix J T (q) is the transpose of the Jacobian of the robot and F is the m × 1 vector of the force that the (mdimensional) environment exerts in the gripper.We consider two robots with identical dimensions (Fig. 1).The contact of the robot gripper with the load is modelled through a linear system with a mass M, a damping B and a stiffness K (Fig. 2).The numerical values adopted for the RR (where R denote rotational joints) robots and the object are Nm -1 .The controller architecture (Fig. 3), is inspired on the impedance and compliance schemes.Therefore, we establish a cascade of force and position algorithms as internal an external feedback loops, respectively, where xd and Fd are the payload desired position coordinates and contact forces.In the position and force control loops we consider FO controllers of the type C(s) = Kp + K α s α , −1 < α < 1, that are approximated by 4 th order discrete-time Pade expressions (ai, bi, ℜ, k = 4): ( ) We compare the response with the classical PD−PI algorithms therefore, in the position and force loops we consider, respectively.
( ) 1) ( ) Both algorithms were tuned by trial and error, having in mind getting a similar performance in the two cases (Tables 1 and 2).

Analysis of the system performance
In order to study the system dynamics we apply a small amplitude rectangular pulse δyd at the position reference and we analyze the system response.The simulations adopt a controller sampling frequency fc = 10 kHz, contact forces of the grippers {Fxj, Fyj} ≡ {0.5, 5} Nm, a operating point of the center of the object A ≡ {x, y} ≡ {0, 1} and a load orientation of α = 0º.
In a first phase we consider robots with ideal transmissions at the joints.Figure 4  (i = 1, 2), with gear clearance hi, the backlash reveals impact phenomena between the inertias, which obey the principle of conservation of momentum and the Newton law: ( ) ( ) where 0 ≤ ε ≤ 1 is a constant that defines the type of impact (ε = 0 inelastic impact, ε = 1 elastic impact) and i q and im q ( i q′ and im q′ ) are the velocities of the i th joint and motor before (after) the collision, respectively.The parameter Jii (Jim) stands for the link (motor) inertias of joint i.The numerical values adopted are hi = 1.8 10 − 4 rad and εi = 0.8 (i = 1, 2).
In a third phase (figure 6) we study the RR robot with compliant joints.For this case the dynamic model corresponds to model (1) augmented by the equations: K q q J q q q q G q C (8)  where Jm, Bm and Km the n × n diagonal matrices of the motor and transmission inertias, damping and stiffness, respectively.In the simulations we adopt Kmi = 2 10 6 Nm rad − 1 and Bmi = 10 4 Nms rad − 1 (i = 1, 2).The time response characteristics (Tables 3 and 4), namely the percent overshoot PO%, the steady-state error ess, the peak time Tp and the settling time Ts reveal that, if we consider similar performances for robots with ideal transmissions at the joints, the FO is superior to the PD−PI algorithms at the cases of robots with joint dynamic phenomena.I conclusion the FO have good dynamic response for position and force perturbations.

Conclusion
This paper studied the position/force control of two robots working in cooperation using fractional and integer order control algorithms.The system time response was analyzed for manipulators having several types of dynamical phenomena at the joints.The transient response of the system shows the superior performance of the FO controller.

Fig. 1 .
Fig. 1.Two RR robots working cooperation for the manipulation of an object with length l 0 and orientation α 0 .

Fig. 2 .
Fig. 2. The contact between the robot gripper and the object.

Fig. 4 .Fig. 5 .Fig. 6 .
Fig. 4. Time response for robots A and B with ideal joints under the action of the FO and the PD−PI algorithms for a pulse perturbation at the robot A position reference δyd = 10 − 3 m and a payload with M = 1 kg, Bi = 10 Nsm − 1 and Ki = 10 3 Nm − 1 .

Table 4 .
Time response characteristics for pulse δFd at the robot A force reference.