On nonlinear dynamics and control of a robotic arm with chaos

In this paper a robotic arm is modelled by a double pendulum excited in its base by a DC motor of limited power via crank mechanism and elastic connector. In the mathematical model, a chaotic motion was identified, for a wide range of parameters. Controlling of the chaotic behaviour of the system, were implemented using, two control techniques, the nonlinear saturation control (NSC) and the optimal linear feedback control (OLFC). The actuator and sensor of the device are allowed in the pivot and joints of the double pendulum. The nonlinear saturation control (NSC) is based in the order second differential equations and its action in the pivot/joint of the robotic arm is through of quadratic nonlinearities feedback signals. The optimal linear feedback control (OLFC) involves the application of two control signals, a nonlinear feedforward control to maintain the controlled system to a desired periodic orbit, and control a feedback control to bring the trajectory of the system to the desired orbit. Simulation results, including of uncertainties show the feasibility of the both methods, for chaos control of the considered system.


Introduction
Dynamically systems with pendulums have important applications.An auto parametric non-ideal system with pendulum was studied by [1], and an auto parametric system with two pendulums harmonically excited was studied by [2].The first detailed study on non-ideal vibrating systems was done by [3].After that publication, the problem of non-ideal vibrating systems has been investigated by a number of authors.A complete review of different theories on non-ideal vibrating systems is to be found in [4].Dynamic interactions between a parametric pendulum and an electro-dynamical shaker of limited power was investigated by [5].In That paper a mathematical model of the electromechanical shaker was described and its parameters were identified.The effectiveness of the nonlinear saturation control in vibration attenuation for a non-ideal portal frame was investigated by [6].The Optimal Linear Feedback Control was proposed by [7].In [7] the quadratic nonlinear Lyapunov function was proposed to solve the optimal nonlinear control design problem for a nonlinear system.[8] formulated the linear feedback control strategies for nonlinear systems, asymptotic stability of the closed-loop nonlinear system, guaranteeing both stability and optimality.
We organized this paper as follows.In section 2 we obtain the mathematical model and perform the analysis of the dynamic model considering: bifurcation diagrams, time histories, phase portraits, frequency spectrum, and 0-1 test for chaotic behaviour.In section 3, the nonlinear saturation control (NSC) and the optimal linear feedback control (OLFC) are implemented.In section 4, the efficiency and the robustness to parametric errors of each control technique are verified through computer simulations.Finally, some concluding remarks are given.

System description and governing equations
We consider a robotic arm modelled by a double pendulum excited in its base by a DC motor of limited power via a crank mechanism and a spring, displayed in  From the Lagrange's method, the equations of motion for the system are: to render the equations dimensionless, in state variables: sin cos cos sin sin The adopted state variables are: x θ = , 7 x ϕ = , 8 x ϕ = .For the numerical simulations, we used the following dimensionless parameters: . We used the following initial conditions: In Fig. 2, we can see that for some values of parameter 0 μ the system (9-11) has chaotic behaviour.To determine these values of parameter 0 μ we applied the 0- 1 test to verify chaotic behaviour of the system, as detailed in [9].The 0-1 test for chaos takes as input a time series of measurements and returns a single scalar value of either 0 for periodic attractors or 1 for chaotic attractors [9].According to [9] the value of c K is given by: where vectors X=[1,2,…,n max ], and is a fixed frequency arbitrarily chosen, and : ) Figures (4)(5)(6) show the chaotic behaviour of the system (5)-( 8).

PROPOSED CONTROL
With the objective of eliminate the chaotic behaviour of the system ( 9)-( 12), we considered the introduction of a control signal U to the system (9), as shown in Fig. 7Ç °°®

Formulation of nonlinear saturation control (NSC)
In this section, we implement the nonlinear saturation control: u is obtained from the following equation:

Control using optimal linear feedback control (OLFC)
Next, we present the optimal linear control strategy for nonlinear systems [10].It is important to observe that this approach is analytical, without dropping any nonlinear term [11].
The U vector control in equation ( 12) consists of two parts: , where ff u is the feed-forward control and fb u is the linear feedback control.We define the period orbit as a function is the solution of ( 12), without the control U , then 0 = fb u .In this way, the desired regime is obtained by the following equation: The feed-forward control ff u is given by: sin cos cos sin sin Substituting ( 16) into (12) and defining the deviation of the desired trajectory as: the system can be represented, in matrix form Control fb u can be found solving the following equation: P is a symmetric matrix symmetrically and may be found solving the Algebraic Riccati Equation: where: Q symmetrical, positive definite, and R positive definite, ensuring that the control ( 19) is optimal [12].We define the desired periodic orbits obtained with the nonlinear saturation control Fig. 8a through the use of Fourier series, calculated numerically as: and solving the Algebraic Riccati Equation (20), we obtain: Considering ( 25) and ( 16) we obtain U: In Fig. 9, one observes the controlled system (5)-( 8) in the orbit (21), with:  x As it can be seen, the proposed control (26), took the system to the desired orbit (21), with transient less than τ 2 .

Comparison between NSC control and OLFC Control
In Fig. 10 we can observe the behavior of the system ( 5)-( 8) using NSC control and OLFC control.As can be seen in Fig. 11, to eliminate the transient and maintain the system in a defined orbit, the OLFC control used a signal more intense than that used by NSC control.
We can also observe in Fig. 10 that even with 1 x being similar for both controls we do not get the same behavior for other states.

Control in the presence of parametric errors
To consider the effect of parameter uncertainties on the performance of the controller, the parameters used in the control will be considered having a random error of % 20 ± [13][14].A sensitivity analysis will be carried out considering the error: x is the state of the system with control without parametric error, and i x ~ for the control with parametric error.

Nonlinear saturation control (NSC) with parametric error
In Fig. 11 we can observe the periodic behavior for the system ( 5)-( 8) with the following control (  We can see from Fig. 11 that the NSC control is sensitive to parameter uncertainties taking the system to different periodic orbits from those obtained without parameter uncertainties.As it can be seen in Fig. 12, the OLFC control has proven to be robust to parameter uncertainties.

Conclusions
We consider a robotic arm modeled by a double pendulum excited in its base by a DC motor of limited power via a crank mechanism and a spring.An investigation of the nonlinear dynamics and chaos was carried out based on this model.The results obtained show chaotic behavior of the model and define the parameters for which chaos occurs.Two control strategies have shown to be effective in stabilizing the system in a periodic orbit.Associating the time delay control to get the desired orbit and the optimal control to maintain the desired orbit we got less transient time and more robustness.

Fig. 1 . 2 m 6
The supporting elastic substructure of the robotic arm consist of a mass, spring and damper ( m, k , c) whose motion is in the vertical direction.The length and mass of the two parts of the robotic arm are 1 , whose angular deflections are measured from the vertical line ( 1 θ , 2 θ ).The controlled torque of the unbalanced DC motor is considered as a linear function of its angular velocity, ( ) m m V C ϕ ϕ Γ = − .m V is set as our DOI: 10.1051/ C Owned by the authors, published by EDP Sciences, control parameter and it changes according to the applied voltage, m C is a constant for each model of motor considered.The coupling between the DC motor and robotic arm is acomplished by a crank mechanism of radius r and elastic connector of stiffness r k .

Fig. 7 .
Fig. 7. Control applied in pivot and joint of the robotic arm asymptotic growth rate

where cωγ and 2 γ 1 c
is the controller's natural frequency, 1 are positive constants.Internal resonance conditions are considered by letting 2

Fig. 11 .
Control signal U. (a) signal used in NSC control.(b) signal used in OLFC