On nonlinear dynamics and control of a robotic arm with chaos J. L. P. Felix1, E. L. Silva1, J. M. Balthazar2, A. M. Tusset3, A. M. Bueno4, R. M. L. R. F. Brasil2 1UNIPAMPA, Bagé, RS, Brazil 2UFABC, Santo Andre, SP, Brazil 3UTFPR, Ponta Grossa, PR, Brazil 4UNESP, Sorocaba, SP, Brazil Abstract. 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. 1 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. 2 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 Fig. 1. 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 1l , 2l , 1m , 2m , whose angular deflections are measured from the vertical line ( 1θ , 2θ ). The controlled torque of the unbalanced DC motor is considered as a linear function of its angular velocity, ( ) m mV Cϕ ϕΓ = −� � . mV is set as our DOI: 10.1051/ C© Owned by the authors, published by EDP Sciences, 2014 , / 050 02 (2014) 201conf Web of Conferences 4 050 02 1 1 MAT EC matec 6 6 This is an Open Access article distributed under the terms of the Creative Commons Attribution License 3.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Article available at http://www.matec-conferences.org or http://dx.doi.org/10.1051/matecconf/20141605002 http://www.matec-conferences.org http://dx.doi.org/10.1051/matecconf/20141605002 control parameter and it changes according to the applied voltage, mC 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 rk . Fig. 1. Robotic arm excited by a non-ideal motor via crank- spring mechanism. From the Lagrange’s method, the equations of motion for the system are: 1 2 1 1 2 1 1 2 2 2 2 2 1 2 1 1 1 2 2 2 2 2 ( ) ( ) sin sin ( ) cos cos sinr m m m y cy ky l m m l m m m l m l k r θ θ θ θ θ θ θ θ ϕ + + + + − + − − + − = ���� � �� � � (1) ( ) ( sin )cosrI k r y rϕ ϕ ϕ ϕ= Γ + −�� � (2) 2 1 2 1 1 2 1 2 2 1 2 2 2 1 2 2 2 1 1 2 1 1 1 1 2 2 1 1 2 1 1 ( ) cos( ) sin( ) ( ) sin ( ) ( ) sin m m l m l l m l l m m gl c c m m l y θ θ θ θ θ θ θ θ θ θ θ θ + + − − − + + + − − = + �� �� � � � � �� (3) 2 2 2 2 2 1 2 1 2 1 2 2 2 2 2 1 2 1 2 1 2 2 1 2 2 2 cos( ) sin sin( ) ( ) sin m l m l l m gl m l l c m l y θ θ θ θ θ θ θ θ θ θ θ + − + + − + − = �� �� � � � �� (4) Next, we use the following changes of variables: 0tτ ω= , 0 1 y y l = , 2 0 t k m ω = , 1 2tm m m m= + + , ˆ ( ) a bϕ ϕΓ = −� � , 2 1 1 g l ω = , 2 2 2 g l ω = , 1 2 1 t m m m α + = , 2 2 t m m α = , 2 1 l R l = , 1 1 rk r kl η = , 2 3 1 2 m m m α = + , 1 1 0 ω ω Ω = , 2 2 0 ω ω Ω = , 1 1 2 1 2 0 1( ) c m m l μ ω = + , 2 2 2 1 2 0 1( ) c m m l μ ω = + , 2 3 2 2 0 2 c m l μ ω = , 1 2 2 0 rk rl I η ω = , 2 3 2 0 rk r I η ω = . to render the equations dimensionless, in state variables: � � � � � +−−+ ++′=′ =′ 711205 2 62 3 2 41523412 21 sincos cossinsin xxxxxR xxxRxxx xx ημα ααα (5) � � � � � −+−−+ Ω−′+−′=′ =′ )()sin( sinsin)cos( 4624135 2 63 3 2 13235634 43 xxxxxxR xxxxxxRx xx μμα α (6) � � � � � � � � � −−−− Ω−′+−′−=′ =′ )()sin( 1 sinsin 1 )cos( 1 46335 2 4 5 2 2523546 65 xxxxx R xxx R xxx R x xx μ (7) � � � −+−=′ =′ 7731288 87 cos)sin( xxxbxax xx ηη (8) The adopted state variables are: 1 0x y= , 2 0x y= � , 3 1x θ= , 4 1x θ= � , 5 2x θ= , 6 2x θ= � , 7x ϕ= , 8x ϕ= � . For the numerical simulations, we used the following dimensionless parameters: 3.01 =α ; 17.02 =α ; 5.01 =α ; 1=R ; 01.01 =μ ; 01.02 =μ ; 01.03 =μ ; 22.1=a ; 2.1=b ; 9.01 =Ω ; 4.02 =Ω ; 05.01 =η ; 2.02 =η ; 3.02 =η . We used the following initial conditions: �� � � 000 180 5 0000 π . Figure 2 shows the bifurcation diagram for parameter 0μ . Fig. 2. Bifurcation diagram for 0μ . In Fig. 2, we can see that for some values of parameter 0μ the system (9-11) has chaotic behaviour. To determine these values of parameter 0μ 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]. Fig. 3. Asymptotic growth rate ( cK ) from 0-1 test as a function of parameter 0μ . 0.04 0.05 0.06 0.07 0.08 0.09 0.1 -1 0 1 2 3 x 1 μ0 0.04 0.06 0.08 0.1 0.986 0.988 0.99 0.992 0.994 X: 0.076 Y: 0.994 μ0 K c MATEC Web of Conferences 05002-p.2 According to [9] the value of cK is given by: ))(var()var( ))(,cov( cMX cMX Kc = (9) where vectors X=[1,2,…,nmax], and M(c)= [M(1,c),…, M(1, nmax)], ( )π,0∈c is a fixed frequency arbitrarily chosen, and : � = ∞→ � � � � � −++ +−+ = N j N jqnjq jpnjp N cnM 1 2 2 ))()(( ))()((1 lim),( (10) ( ) ( ) )sin()();cos()( 00 jc xx iqjc xx ip i j x j i j x j �� == − = − = σσ (11) where x and xσ are the mean value and square deviation of the examined ix series, and N is the length of the sampled points in the displacement time series. A value of 0≅cK indicates a non-chaotic data set while a value of 1≅cK indicates a chaotic data set. As we can see in Fig. 3, for 076.00 =μ system (5)-(8) has chaotic behaviour. In Figs. (4-6) one can observe the behaviour of the system (5)-(8) for 076.00 =μ . (a) (b) Fig. 4. Behaviour 1x . (a) Vertical movement. (b) Frequency Spectrum (a) (b) Fig. 5. Behaviour 3x . (a) Angular movement. (b) Frequency Spectrum (a) (b) Fig. 6. Behaviour 5x . (a) Angular movement. (b) Frequency Spectrum Figures (4-6) show the chaotic behaviour of the system (5)-(8). 3 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 � � � � � ++−−+ ++′=′ =′ UxxxxxR xxxRxxx xx 711205 2 62 3 2 41523412 21 sincos cossinsin ημα ααα (12) Fig. 7. Control applied in pivot and joint of the robotic arm asymptotic growth rate 3.1. Formulation of nonlinear saturation control (NSC) In this section, we implement the nonlinear saturation control: 2 1uU γ= (13) u is obtained from the following equation: uyuuu cc 02 2 γωμ =++ ��� (14) where cω is the controller’s natural frequency, 1γ and 2γ are positive constants. Internal resonance conditions are considered by letting 2 1cω ≈ and external resonance by 1ϕ ≈� . In Fig. 8 we can observe the behaviour of the system (5)- (8) with the proposed control (13) considering the following parameters: 01.01 =γ ; 07.02 =γ ; 01.0=cμ ; 01.0=cω and initial conditions: 1.0)0( =u and 0)0( =u� , excluding the transient behavior. 0 500 1000 1500 2000 -1 -0.5 0 0.5 1 τ x 1 0 5 10 15 20 0 0.05 0.1 Frequency (Hz) |x 1(f )| 0 500 1000 1500 2000 -2 -1 0 1 2 τ x 3 0 5 10 15 20 0 0.02 0.04 0.06 0.08 0.1 Frequency (Hz) |x 3(f )| 0 500 1000 1500 2000 -50 0 50 100 τ x 5 0 0.5 1 1.5 2 0 20 40 60 Frequency (Hz) |x 5(f )| CSNDD 2014 05002-p.3 (a) (b) (c) Fig. 8. (a) Vertical motion 1x . (b) Angular motion 3x . (c) Angular motion 5x 3.2. 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: fbff uuU += , where ffu is the feed-forward control and fbu is the linear feedback control. We define the period orbit as a function of [ ]Txx )()( * 2 * 1 ττ . If the function [ ]Txx )()( * 2 * 1 ττ is the solution of (12), without the control U , then 0=fbu . In this way, the desired regime is obtained by the following equation: � � � � � � � ++−−+ ++′= ′ = ′ ffuxxxxxR xxxRxxx xx 71 * 1 * 205 2 62 3 2 4152341 * 2 * 2 * 1 sincos cossinsin ημα ααα (15) The feed-forward control ffu is given by: 71 * 1 * 205 2 62 3 2 4152341 * 2 sincos cossinsin xxxxxR xxxRxxxuff ημα ααα −++− −−′− ′ = (16) Substituting (16) into (12) and defining the deviation of the desired trajectory as: � � � � − − = * 22 * 11 xx xx z (17) the system can be represented, in matrix form BuAzz +=′ : fbu z z z z � � � � +� � � � � � � � −− =� � � � ′ ′ 1 0 1 10 2 1 02 1 μ (18) Control fbu can be found solving the following equation: zPBRu T fb 1−−= (19) P is a symmetric matrix symmetrically and may be found solving the Algebraic Riccati Equation: 01 =+−+ − QPBPBRPAPA TT (20) 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: � � � �� � � = += ) 7 2 cos( 7 3498.0 ) 7 2 sin(1749.00393.0 * 2 * 1 τ ππ τ π x x (21) Matrices A and B are: � � � � −− = 076.01 10 A , � � � � = 1 0 B (22) Defining: � � � � = 100 0104 Q , ]10[=R (23) and solving the Algebraic Riccati Equation (20), we obtain: � � � � = 3467.14005.99 005.992713.1442 P (24) Substituting it into (20), we define the control: )(3467.14)(005.99 3467.14005.99 * 22 * 11 21 xxxx zzufb −−−− =−−= (25) Considering (25) and (16) we obtain U: 71 * 1 * 205 2 62 3 2 4152341 * 2 * 22 * 11 sincos cossinsin )(3467.14)(005.99 xxxxxR xxxRxxx xxxxU ημα ααα −++− −−′− ′ + −−−−= (26) In Fig. 9, one observes the controlled system (5)-(8) in the orbit (21), with: 6* 11 10<−xx , excluding the transient behavior. 2800 2850 2900 2950 3000 -0.2 -0.1 0 0.1 0.2 0.3 τ x 1 2800 2850 2900 2950 3000 -1 -0.5 0 0.5 1 x 10-3 τ x 3 2800 2850 2900 2950 3000 12.4 12.5 12.6 12.7 12.8 τ x 5 MATEC Web of Conferences 05002-p.4 (a) (b) (c) Fig. 9. (a) Vertical motion 1x . (b) Angular motion 3x . (c) Angular motion 5x As it can be seen, the proposed control (26), took the system to the desired orbit (21), with transient less than τ2 . 3.3. 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. (a) (b) (c) Fig. 10. Phase diagram. (a) Motion 1x . (b) Angular motion 3x . (c) Angular motion 5x In Fig. 11 we can see the variation of the control signal used to control NSC and OLFC control. (a) (b) Fig. 11. Control signal U. (a) signal used in NSC control. (b) signal used in OLFC 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 1x being similar for both controls we do not get the same behavior for other states. 4 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: iii xxe ~−= for 6:1=i . ix is the state of the system with control without parametric error, and ix~ for the control with parametric error. 4.1. 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 (13): )(004.0008.01 tr+=γ ; )(028.0056.02 tr+=γ ; )(004.0008.0 trc +=μ ; )(004.0008.0 trc +=ω , excluding the transient behavior. )(tr is a normally distributed random function. (a) (b) (c) Fig. 11. Sensitivity of NSC control to parametric errors In Fig. 12 we can observe the periodic behavior for the system (5)-(8) with the following control (26): )(12.024.01 tr+=α ; )(068.0136.02 tr+=α ; )(4.08.0 trR += ; )(0304.00608.00 tr+=μ ; )(02.004.01 tr+=η . (a) (b) 0 50 100 150 200 -0.2 -0.1 0 0.1 0.2 0.3 τ x 1 800 850 900 950 1000 -0.1 -0.05 0 0.05 0.1 τ x 3 800 850 900 950 1000 -1.5 -1 -0.5 0 0.5 1 1.5 τ x 5 -0.2 0 0.2 0.4 -0.2 -0.1 0 0.1 0.2 x 1 x 2 NSC OLFC -0.1 -0.05 0 0.05 0.1 -0.05 0 0.05 x 3 x 4 NSC OLFC -5 0 5 10 15 -0.5 0 0.5 x 5 x 6 NSC OLFC 0 1000 2000 3000 0 0.02 0.04 0.06 0.08 0.1 τ U N S C 0 200 400 600 800 1000 -100 -50 0 50 100 τ U O LF C 2800 2850 2900 2950 3000 -0.4 -0.2 0 0.2 0.4 τ e e 1 e 2 2800 2850 2900 2950 3000 -2 -1 0 1 2 x 10-3 τ e e 3 e 4 2800 2850 2900 2950 3000 -0.2 -0.1 0 0.1 0.2 τ e e 5 e 6 0 200 400 600 800 1000 -1 0 1 x 10-7 τ e 1 0 200 400 600 800 1000 -5 0 5 x 10-3 e 2 τ CSNDD 2014 05002-p.5 (c) (d) (e) (f) Fig. 12. Sensitivity of OLFC control to parametric errors 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. References 1. D. Sado and M. 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S., A hyperbolic tangent adaptive pid+lqr control applied to a step-down converter using poles placement design implemented in fpga, Mathematical problems in engineering (print), 2013 13. M. J. Shirazi, R. Vatankhah, M. Boroushaki, H. Salarieh and A. Alasty, Application of particle swarm optimization in chaos synchronization in noisy environment in presence of unknown parameter uncertainty. Communications in Nonlinear Science and Numerical Simulation, 17(2), 742-753 (2012). 14. A. M. Tusset, A. M. Bueno, C. B. Nascimento, M. S. Kaster, J. M. Balthazar, Nonlinear state estimation and control for chaos suppression in MEMS resonator. Shock and Vibration, 20, 749-761, (2013). 0 200 400 600 800 1000 -2 -1 0 1 2 x 10-6 τ e 3 0 200 400 600 800 1000 -2 -1 0 1 2 x 10-4 τ e 4 0 200 400 600 800 1000 -1 -0.5 0 0.5 1 x 10-5 τ e 5 0 200 400 600 800 1000 -4 -2 0 2 4 x 10-3 e 6 τ MATEC Web of Conferences 05002-p.6