Advanced Strategies For Robot Manipulators Part 6 docx

Sliding Mode Control of Robot Manipulators via Intelligent Approaches 141 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 output variable K Degree of membership SM B Fig. 2. The membership functions of the output f uzz K PB PS Z NS NB s s  B S M B B N B M S M B Z B B M S B P Table 1. The fuzzy rule base for tuning f uzz K Simulation example 2.1. In order to show the effectiveness of the proposed control law, it is applied to a two-link robot with the following parameters: 22 2 2cos cos () cos qq Mq q αβ γ βγ βγ β ++ + ⎡ ⎤ = ⎢ ⎥ + ⎣ ⎦ )22( 22 12 2 12 sin ( )sin (,) sin 0 qq qq q Cqq qq γγ γ −−+ ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦    )23( 111 12) 112 cos cos( () cos( ) qqq Gq qq αδ γδ γδ ++ ⎡ ⎤ = ⎢ ⎥ + ⎣ ⎦ )24( where 2 121 ()mma α =+ , 2 22 ma β = , 212 maa γ = , 1 ga δ = , and 1 m , 2 m , 1 .7a = , 2 .5a = are the masses and lengths of the first and second links, respectively. The masses are assumed to be in the end of the arms and the gravity acceleration is considered as 9.8g = . Moreover, the masses are considered with 10% uncertainty as follow: 0 0 11 1 1 22 2 2 , .4 , .2 mm m m mm m m = +Δ Δ ≤ = +Δ Δ ≤ (25) where 0 1 4m = and 0 2 2m = , and ˆ M , ˆ C , and ˆ G are estimated. The desired state trajectory is: 1cos 2cos d t q t π π − ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ (26) Advanced Strategies for Robot Manipulators 142 and the disturbance torque is considered as: 0.5sin 2 0.5sin 2 d t t π τ π ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ (27) which leads to 0.5 0.5 D T ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ . The design parameters are determined as follow: 1 15 0 015 λ ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ , 2 40 0 040 λ ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ (28) Values of ϕ and η are selected as 0.167 ϕ = and [] 0.1 0.1 T η = . Moreover, the factors N and v N are selected as: 50 0 05 N ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ , 50 010 v N ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ (29) In order to show the improvement due to the proposed method, the simulation results of applying this method are compared with the related results of the conventional SMC. The tracking error and control law in the case of conventional SMC have been shown in Fig. 3 and Fig. 4, respectively. The corresponding graphs for the case of applying fuzzy SMC-PID are also provided in Fig. 5 and 6. 0 2 4 6 8 10 -0.05 0 0.05 0.1 0.15 time(sec) Error1(rad) 0 2 4 6 8 10 -0.5 0 0.5 1 1.5 2 time(sec) Error2(rad) Fig. 3. The tracking errors in the case of using conventional SMC As it can be seen from these figures, the proposed fuzzy SMC-PID has faster response and less tracking error in comparison with conventional SMC. In order to show more clearly the difference between the tracking errors in two cases, the enlarged graphs have been provided in Fig. 7 and 8. Sliding Mode Control of Robot Manipulators via Intelligent Approaches 143 0 2 4 6 8 10 -50 0 50 100 150 time(sec) input1(N.m) 0 2 4 6 8 10 -50 0 50 100 time(sec) input2(N.m) Fig. 4. The control inputs in the case of using conventional SMC 0 2 4 6 8 10 -0.05 0 0.05 0.1 0.15 time(sec) Error1(rad) 0 2 4 6 8 10 -0.5 0 0.5 1 1.5 2 time(sec) Error2 (rad) Fig. 5. The tracking errors in the case of using Fuzzy SMC-PID Advanced Strategies for Robot Manipulators 144 0 2 4 6 8 10 -100 0 100 200 time(sec) input1 (N.m) 0 2 4 6 8 10 -100 -50 0 50 100 time(sec) input2 (N.m) Fig. 6. The control inputs in the case of using Fuzzy SMC-PID 0 2 4 6 8 10 -0.01 -0.005 0 0.005 0.01 time(sec) Error1(rad) 0 2 4 6 8 10 -5 0 5 x 10 -3 time(sec) Error2(rad) Fig. 7. The enlargement of the tracking errors in the case of using conventional SMC 0 2 4 6 8 10 -5 0 5 x 10 - 4 time(sec) Error1 (rad) 0 2 4 6 8 10 -1 -0.5 0 0.5 1 x 10 -3 time(sec) Error2 (rad) Fig. 8. The enlargement of the tracking errors in the case of using Fuzzy SMC-PID Sliding Mode Control of Robot Manipulators via Intelligent Approaches 145 2.2 Incorporating sliding mode and fuzzy control In this section, a combined controller includes SMC term and fuzzy term is proposed for set- point tracking of robot manipulators. Some practical issues, such as existence of joint frictions, restriction on input torque magnitude due to saturation of actuators, and modeling uncertainties have been considered here. Design procedure contains two steps. First, SMC design is accomplished and system stability in this case is provided by Lyapunov direct method. When the tracking error would be less than predefined value then a sectorial fuzzy controller (SFC), (Calcev, 1998), is responsible for control action. Designing of this kind of fuzzy controller is exactly the same as in which has performed in (Santibanez et al., 2005). This proposed controller has following advantages. 1) There are less tracking errors versus traditional SMC in condition that the control input is limited, 2) the chattering is avoided, 3) convergence of tracking error is more rapid than fuzzy controller designed in (Santibanez et al., 2005) and modeling uncertainty is considered here (Shafiei & Sepasi, 2010). 2.2.1 Mathematical model and problem formulation This time the friction of joint is considered and is added to dynamical equation (1) as: () (,) () (,)Mqq Cqqq Gq Fq τ τ + ++ =     (30) where ( , ) n F q R τ ∈  stands for the friction vector which is as follows (Cai & Song, 1994): (, ) s g n( ) 1 s g n( ) ( ; ) ii iici i i isi fq bq f q q sat f ττ ⎡⎤ =+ +− ⎣⎦    (31) where (, ) ii fq τ  , 1,2, ,in =  , denotes the i-th element of (,)Fq τ  vector. i b , ci f and si f are the viscous, Coulomb and static friction, respectively. The sat(·; ·) indicates saturation function with following equation. (;) rif xr sat x r x if r x r rif x r > ⎧ ⎪ = −≤ ≤ ⎨ ⎪ − <− ⎩ In the following, () M q , (,)Cqq  and ()Gq might be shown by M , C , and G , respectively in where it would be requisite. Now, the boundedness properties are defined as below: { } sup ( ) , 1, , n ii qR gq g i n ∈ ≤= (32) where i g stands for the i-th element of ()Gq and i g is finite nonnegative constant. Assume that the maximum torque that joint actuator can supply is max τ . Therefore: max ,1,, ii in ττ ≤= (33) and each actuator satisfies the following condition: max iisi gf τ >+ (34) Advanced Strategies for Robot Manipulators 146 In robot modeling, one can well determine the terms () M q and ()Gq but it is difficult in most cases obtaining the parameters of (,)Cqq  and (,)Fq τ  exactly. So, in present section, the matrix C is considered as follows: ˆ CC C = +Δ (35) where ˆ C denotes estimation of C , and C Δ is bounded estimation error which has the following relation: ,, 0.1 i j i j CCΔ≤ (36) where ,i j C stands for elements of the matrix C . Also the vector F is supposed as an external disturbance with the following unknown upper bound: u p FF≤ (37) where the operator ⋅ denotes Euclidean norm. If one considers the desired point which joint position must be held on it as d q , then the position error could be defined as: d qq q = −  (38) Here, the set-point tracking problem refers to define the control law such that error e would be driven toward the inside of an arbitrary small region around zero with maintaining the torques within the constraints (33). In succeeding subsections, this aim will be attained. 2.2.2 Sliding mode controller design The following sliding surface is considered for designing SMC controller. se e λ = +  (39) where d eqqq=− = −  is error vector and λ is supposed symmetric positive definite matrix such that s=0 would become a stable surface. The reference velocity vector " r q  " is defined as in (Slotin & Li, 1991): rd qq e λ = −  (40) Thus, one can interpret sliding surface as: r sqq = −  (41) Here, the SMC controller design is expressed by lemma 2.2. Lemma 2.2. Consider the system with dynamic equation (30) and sliding surface and reference velocity defined by (39) and (40), respectively. If one chooses the control law below, ˆ s g n( )Ks ττ =− (42) Sliding Mode Control of Robot Manipulators via Intelligent Approaches 147 such that ˆ ˆ rr Mq C q G τ = ++   (43) and iri KCq≥Δ +Γ  (44) then the sliding condition (10) is satisfied. In the last inequality, K i denotes the element of sliding gain vector K and Γ is design parameter vector which must be selected such that iu p i F η Γ≥ + . Proof: Consider the following Lyapunov function candidate: 1 2 T VsMs= (45) Since M is positive definite, for 0s ≠ we have 0V > and by taking time derivative of the relation (45) and regarding the symmetric property of M, it can be written: 1 2 TT VsMs sMs=+   (46) from (40), gives: 1 () 2 TT r V s Mq Mq s Ms=−+    (47) By substituting (30) in (47) and considering asymmetry property (2)0 T sM Cs − =  , we have: () T rr Vs C q GFM q τ =−−−−   (48) Now, applying (42) and (43) yields: 1 () n T rii i Vs C q FKs = =Δ +− ∑   (49) Finally, from relation (44) it can be concluded that: 1 n ii i Vs η = ≤− ∑  (50) This indicates that V is a Lyapunov function and the sliding condition (10) has been satisfied. Note that, in general, the sign function is replaced by saturation function as ( ) sat /s ϕ , where ϕ denotes boundary layer thickness. 2.2.3 Fuzzy controller design In this section, the SFC class of fuzzy controller studied in (Santibanez et al., 2005) is considered which has two-input one-output rules used in the formulation of the knowledge base. These IF-THEN rules have following form: Advanced Strategies for Robot Manipulators 148 12 12 11 22 IF is and is THEN is ll ll xA xA y B (51) where [] 2 12 1 2 T xxx UUU=∈=×⊂ℜ and yV∈⊂ℜ. For each input fuzzy set j l j A in jj xU⊂ and output fuzzy set 12 ll B in y V⊂ , exist an input membership function ( ) l j j j A x μ and output membership function 12 () ll B y μ shown in Fig. 10 and Fig. 11, respectively. Fig. 9. Input membership functions Fig. 10. Output membership functions The fuzzy system considered here has following specifications: Singleton fuzzifier, triangular membership functions for each inputs, singleton membership functions for the output, rule base defined by (51), (see Table. 2), product inference and center average defuzzifier. PB PS ZE NS NB 1 x 2 x ZE ZE NS NB NB NB ZE ZE NS NB NB NS PS PS ZE NS NS ZE PB PB PS ZE ZE PS PB PB PS ZE ZE PB Table 2. The fuzzy rule base for obtaining output y Thus, one can compute the output y in terms of inputs as follows (Wang, 1997): 12 12 12 2 1 12 2 1 () () ( , ) () l j j l j j ll j A j ll j A j ll y x yx x x x μ ϕ μ = = ⎛⎞ ⎜⎟ ⎝⎠ == ⎛⎞ ⎜⎟ ⎝⎠ ∑∑ ∑∑ ∩ ∩ (52) Sliding Mode Control of Robot Manipulators via Intelligent Approaches 149 Special properties of this input-output mapping () y x for x 1 , x 2 are given in (Santibanez et al., 2005). Lemma 2.3. For the system with dynamical equation (30), if one chooses the following control law, (,) () qq G q τϕ =+   (53) where q  is defined as (38) and d qq q = −    is velocity error vector, then the closed-loop system shown in Fig. 11 becomes stable. Proof: the stability analysis is based on the study performed in (Calcev 1998) and is fully discussed in (Santibanez et al., 2005), so it is omitted here. Note that for constant set-point we have 0 d q =  , hence qq = −   . Fig. 11. Closed-loop system in the case of fuzzy controller (Santibanez et al., 2005) 2.2.4 Incorporating SMC and SFC Each of the two controllers explained in last two subsections drives the robot joint angles to desired set-point in finite time and according to the Lemma 2.2 and 2.3 the closed-loop system is stable in both cases. In this section, for utilizing advantages of both sliding mode control and sectorial fuzzy control, and also minimizing the drawbacks of both of them, the following control law is proposed: e e ˆ sgn( ) when q (,) ()whenq ee Ks yq q Gq τ α τ α ⎧ − ≥ ⎪ = ⎨ + < ⎪ ⎩  (54) where α is strictly positive small parameter which can be determined adaptively or set to a constant value. So, while the magnitude of error is greater than or equal to α , SMC drives the system states, errors in our case, toward sliding surface and as soon as the magnitude of error becomes less than α , then the SFC which is designed independent of initial conditions, controls the system. Since the SMC shows faster transient response, the response of the system controlled by (54) is faster than the case of SFC. Additionally, in spite of the torque boundedness, since the SFC controls the system in the steady state, the proposed controller (54) has less set-point tracking error. Also, since near the sliding surface the proposed controller switch from SMC to SFC, therefore, the chattering is avoided here. Advanced Strategies for Robot Manipulators 150 Simulation example 2.2. In order to show the effectiveness of the proposed control law, it is applied to a two-link direct drive robot arm with the following parameters (Santibanez et al., 2005): 22 2 2.351 0.168cos( ) 0.102 0.084cos( ) () 0.102 0.084cos( ) 0.102 qq Mq q ++ ⎡ ⎤ = ⎢ ⎥ + ⎣ ⎦ 22 2 1 2 21 0.084sin( ) 0.084sin( )( ) ˆ (,) 0.084sin( ) 0 qq q q q Cqq qq −−+ ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦    112 12 3.921sin( ) 0.186sin( ) ( ) 9.81 0.186sin( ) qqq Gq qq ++ ⎡ ⎤ = ⎢ ⎥ + ⎣ ⎦ 1111 2222 2.288 8.049s g n( ) 1 s g n( ) sat( ;9.7) () 0.186 1.734s g n( ) 1 s g n( ) sat( ;1.87) qqq Fq qqq τ τ ⎡ ⎤ ⎡⎤ ++− ⎣⎦ ⎢ ⎥ = ⎢ ⎥ ⎡⎤ ++− ⎣⎦ ⎣ ⎦    ˆ CC C = +Δ (55) According to the actuators manufacturer, the direct drive motors are able to supply torques within the following bounds: max 11 max 22 150[Nm] 15[Nm] ττ ττ ≤= ≤= (56) The desired set-point is, [] T d q π π =− (57) which is applied as a step function at time zero. The SMC design parameters are as below: 10 0 010 λ ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ , 140 8 ⎡⎤ Γ= ⎢⎥ ⎣⎦ and 5 φ = (58) For SFC case, according to Fig. 9 and Fig. 11, 21012 {,,,,} j x jjjjj p p pppp = −− is fuzzy partition of the input universe of discourse and 21012 {,,,,} y p y yyyy = −− is for output universe of discourse. Now, SFC design parameters are given by following equations (Santibanez et al., 2005): 1 2 { 180, 4,0,4,180} { 180, 2,0,2,180} q q p p = −− =− −   1 2 { 360, 270,0,270,360} { 360, 270,0,270,360} q q p p =− − =− −     1 2 { 109, 90,0,90,109} { 13, 9,0,9,13} y y p p = −− =− − (59) For our proposed controller (54), the constant 0.3 α = is supposed. Additionally, to show the improvement achieved from applying the proposed method of this section (incorporating [...]... Robot Manipulators via Intelligent Approaches (a) (b) Fig 23 (sim1) Tracking error of joints, (a) FSM_PID (b) NNSM_PID 165 166 Advanced Strategies for Robot Manipulators (a) (b) Fig 24 (sim1) Control commands (a) FSM_PID (b) NNSM_PID Sliding Mode Control of Robot Manipulators via Intelligent Approaches Fig 25 (sim1) NN control effort 167 168 Advanced Strategies for Robot Manipulators (a) (b) Fig 26. .. gears as follows: q = grθ (65 ) 1 56 Advanced Strategies for Robot Manipulators and τ m = grτ (66 ) where g r is the diagonal matrix of reduction ratio In the following a practical constraint is considered Constraint 3.1 The maximum voltage that joint actuator can supply is V max So, we have: Vi ≤ Vimax , i = 1, ,n It should be noted that, the applicable control input for driving robot arm is the armature... + tr W T W + tr V TV + K T K α β γ 2 (95) 160 Advanced Strategies for Robot Manipulators By substituting (90) in to the first part of (95) and by using (87) one can obtain ˆ ˆ ST Ds = sT [ −Vms + f − U ] = sT [ −Vms + f − K v s − f − K sgn(s )] ˆ ˆ ˆ ˆ ˆ ˆ = sT [ −V s − K s + W Tσ − W Tσ ′V T x + W Tσ ′V T x + ε − K sgn( s)] m v ( 96) N Some useful relations for manipulating last tow equations are provided... Manipulators (a) (b) Fig 26 (sim2) Tracking error of joints (a) FSM_PID (b) NNSM_PID Sliding Mode Control of Robot Manipulators via Intelligent Approaches (a) (b) Fig 27 (sim2) Control commands (a) FSM_PID (b) NNSM_PID 169 170 Advanced Strategies for Robot Manipulators Fig 28 (sim2) NN control effort Fig 29 (sim2) Matrix norm of adaptive weights W and V ... value of K increases to force them back to sliding manifold, and when the states are close to the sliding manifold, ˆ the absolute value of K decreases accordingly This feature beside the replacing saturation function, act as what is heuristically designed by fuzzy system in (Ataei & Shafiei, 2008) Furthermore, the system stability is addressed here  162 Advanced Strategies for Robot Manipulators Simulation... considered with 10% uncertainty The design parameters of the FSM_PID controller are 164 Advanced Strategies for Robot Manipulators ⎡ 3.2 0 ⎤ ⎡0.8 0 ⎤ , Nf = ⎢ N vf = ⎢ ⎥ 0 3.5 ⎥ ⎣ ⎦ ⎣ 0 0.7 ⎦ (113) Simulation 1─ In this case, the friction term is neglected, mass variation occurs at 3 sec and external disturbance is injected at 6 sec The desired trajectory is depicted in Fig 22 The vectors of tracking errors... equations (61 )- (66 ) and neglecting the inductance L , because of its tiny amount, the following equation is achieved − V = RK m1 {[ J m gr−1 + gr M ]q + ( Bm gr−1 + grC + K m R −1K b gr−1 )q + gr G + gr F(q ) + grτ d } (67 ) The previous equation can be expressed in a compact form as: U = Dq + H + d (68 ) with U = V is the control command and the other parameters are − D = RK m1 ( J m gr−1 + gr M ) (69 ) −... chattering will be occurred The corresponding graphs for the case of applying SFC are also provided in Fig 14, and Fig 15 In the case of control law proposed in the present section, Fig 16 and Fig 17 illustrate the error vector and control law, respectively The tracking error is about 0.002 in this state of affairs 152 Advanced Strategies for Robot Manipulators As it can be seen from these results,... approach 4 3 Error (rad) 2 1 0 -1 -2 -3 -4 0 0.5 1 1.5 2 2.5 Time(sec) 3 3.5 4 Fig 16 Error vector in the case of incorporating SMC and SFC 150 Input torques (Nm) 100 50 0 -50 -100 0 0.5 1 1.5 2 2.5 Time(sec) 3 3.5 Fig 17 The control torques in the case of incorporating SMC and SFC 4 154 Advanced Strategies for Robot Manipulators 4 3 2 Error (rad) 1 0 -1 -2 -3 -4 0 0.5 1 1.5 2 2.5 Time (sec) 3 3.5 4... −4 s/rad.V −4 Table 3 Parameters of two-link elbow robot and actuators The external disturbances can be considered as external forces injected into the robotic system, and are supposed to have following expression τ d = [sin 4t sin 4t ]T (1 06) Also, the friction term is considered here as (Wai & Chen, 20 06) : F (q ) = [20q1 + 0.8 sgn(q1 ) 4q2 + 0. 16 sgn(q2 )] T (107) In order to show the effectiveness . speed of the manipulators, motors are equipped with the high reduction gears as follows: r qg θ = (65 ) Advanced Strategies for Robot Manipulators 1 56 and mr g τ τ = (66 ) where r g. has two-input one-output rules used in the formulation of the knowledge base. These IF-THEN rules have following form: Advanced Strategies for Robot Manipulators 148 12 12 11 22 IF is. actuator satisfies the following condition: max iisi gf τ >+ (34) Advanced Strategies for Robot Manipulators 1 46 In robot modeling, one can well determine the terms () M q and ()Gq but