This paper develops a novel interval type-2 fuzzy Proportional-Derivative (PD) control scheme for electrically driven flexible-joint robots using the direct method of Lyapunov. The controller has a simple design in a decentralized structure. Compared to the previous controllers reported for the flexible-joint robots which use two control loops, it has a simpler structure using only one control loop. It guarantees stability and provides a good tracking performance. The controller considers the whole robotic system including the manipulator and motors by applying the voltage control strategy. Stability analysis is presented and the effectiveness of the proposed control approach is demonstrated by simulations using a three link flexible-joint robot driven by permanent magnet DC motors. Simulation results show that the interval type-2 fuzzy PD controller can handle external disturbance better than the type-1 fuzzy PD controller. In addition, it spends less control effort than the type-1 in order to deal with disturbance.
Fuzzy control as a model-free approach is simply designed to control complicated systems. In recent years, there has been an increasing attention to type 2 fuzzy logic system (FLS) in order to overcome the uncertainties. Type-1 FLS has difficulties in modeling and minimizing the effect of uncertainties [
Control of a flexible-joint robot as a complex system can highlight the capabilities of the T2FLS. In order to improve industrial productivity, it is required to reduce the weight of the arms and/or to increase their speed of operation. However, as a bad effect, the flexibility in joints and links may occur. On the other hand, compared to the conventional heavy and bulky robots, flexible manipulators have the potential advantage of lower cost, larger work volume, higher operational speed, greater payload-to-manipulator-weight ratio, smaller actuators, lower energy consumption, better maneuverability and better transportability due to reduced inertia [
The most important reason of joint flexibility is the essential use of power transmission systems which show the flexibility [
To solve these problems, voltage control strategy has been devoted to the electrically driven robot manipulators [
A model of robot may face uncertainties such as unmodelled dynamics, parametric uncertainty and external disturbances. In [
The contribution of this paper is to present an interval type-2 fuzzy PD (IT2PD) control approach for electrically driven flexible-joint robots. The proposed voltage control law has a simpler structure in the form of decentralized control yet more efficient than the torque control that is multivariable coupled control. As a result, the proposed control approach is free of many effects caused by manipulator dynamics. This is an important advantage of the proposed control approach over the torque based control approaches. Compared to the previous controllers reported in the literature for the flexible-joint robots which use two control loops, it has a simpler structure using only one control loop. This is the main novelty of this paper. Stability analysis is presented and the effectiveness of the proposed control approach is demonstrated by simulations.
The rest of the paper is organized as follows: Section 2 presents modeling of the flexible-joint robots. Section 3 introduces Interval type-2 fuzzy logic. Section 4 develops the proposed method. Section 5 presents the simulation results and finally, Section 6 concludes the paper.
In a simplified model of the flexible-joint robot [
D ( θ ) θ ¨ + C ( θ , θ ) θ ˙ + g ( θ ) = K ( r θ m − θ ) (1)
J θ ¨ m + B θ ˙ m + r K ( r θ m − θ ) = τ (2)
where θ ∈ R n is a vector of joint angles, θ m ∈ R n is a vector of rotor angles. Thus, this system possesses 2n coordinates as [ θ θ m ] . The matrix D ( θ ) is a n × n matrix of manipulator inertia, C ( θ , θ ) θ ˙ ∈ R n is the vector of centrifugal and Coriolis forces, g ( θ ) ∈ R n is a vector of gravitational forces and τ ∈ R n is a torque vector of motors. The diagonal matrices J , B and r represent coefficients of the motor inertia, motor damping and reduction gear, respectively. The diagonal matrix K represents the lumped flexibility provided by the joint and reduction gear. To simplify the model, both the joint stiffness and gear coefficients are assumed constant. The vector of gravitational forces g ( θ ) is assumed function of only the joint positions as used in the simplified model [
System (1)-(2) is highly nonlinear, extensively computational, heavily coupled and multi-input/multi-output system with the 2n coordinates. Complexity of the model has been a serious challenge in robot modeling and control in literature. It is expected to face a higher complexity if the proposed model includes the actuator dynamics. In order to obtain the motor voltages as inputs, consider electrical equation of the geared permanent magnet dc motors in the matrix form
u = R I a + L I ˙ a + K b θ ˙ m (3)
where u ∈ R n is a vector of motor voltages, I a ∈ R n is a vector of motor currents and θ ˙ m is a vector of rotor velocities. The diagonal matrices R , L and K b represent the coefficients of armature resistance, armature inductance and back-emf constant, respectively. The motor torques τ as input for dynamic Equation (2) is produced by the motor currents as
K m I a = τ (4)
where K m is a diagonal matrix of the torque constants. Equations (1)-(4) form the robotic system such that the voltage vector u is the input vector and the joint angle vector θ is the output vector.
The dynamics of the electrical robot (1)-(4) in the state space is formed as
x ˙ = f ( x ) + b u (5)
where
f ( x ) = [ x 2 D − 1 ( x 1 ) ( − g ( x 1 ) − K x 1 − C ( x 1 , x 2 ) x 2 + K r x 3 ) x 4 J − 1 ( r K x 1 − r 2 K x 3 − B x 4 + K m x 5 ) − L − 1 ( K b x 4 + R x 5 ) ] , b = [ 0 0 0 0 L − 1 ] , x = [ θ θ ˙ θ m θ ˙ m I a ]
A fuzzy logic system that uses at least one type-2 fuzzy set is called a type-2 fuzzy logic system. It is very useful in circumstances where determination of an exact membership grade for a fuzzy set is difficult [
An interval type-2 fuzzy set A ˜ in X is defined as [
A ˜ = ∫ x ∈ X [ ∫ u ∈ J x 1 u ] / x , J x ⊆ [ 0 , 1 ] (6)
where x is the primary variable with domain X; u is the secondary variable, which has domain J x ; J x is called the primary membership of x. Uncertainty about A ˜ is conveyed by the union of all of the primary memberships called the footprint of uncertainty (FOU) of A ˜ ; i.e.
FOU ( A ˜ ) = J x , x ∈ X (7)
The structure of a typical type-2 fuzzy logic system is shown in
In the following subsections the operations in an interval singleton type-2 FLS are described in details.
In practice the computations in an IT2FLS can be consisting of M rules assuming the following form:
R i : if x 1 is X ˜ 1 i and ⋯ x p is X ˜ p i , then y is Y i , i = 1 , 2 , ⋯ , M (8)
where x = ( x 1 , ⋯ , x p ) is the input vector, y are linguistic variables, X ˜ j i ( j = 1 , 2 , ⋯ , p ) is an interval type-2 fuzzy set and Y i = [ y l i , y r i ] , which can be understood as the simplest Takagi-Sugeno-Kang (TSK) model. The fuzzifier maps a crisp point x = ( x 1 , ⋯ , x p ) into a type-2 fuzzy set A ˜ [
The inference engine matches the fuzzy singletons with the fuzzy rules in the rule base. To compute unions and intersections of type-2 sets, compositions of type-2 relations are needed [
is to obtain the firing set ∏ j = 1 p μ X ˜ j i = F i ( x ) by performing the input and ante-
cedent operations. As only interval type-2 sets are used and the meet operation is implemented by the product t-norm, the firing set is the following type-1 interval set [
F i ( x ) = [ f _ i ( x ) , f ¯ i ( x ) ] = [ f _ i , f ¯ i ] (9)
where f _ i ( x ) = μ _ X ˜ 1 i ( x 1 ) × ⋯ × μ _ X ˜ p i ( x p ) and f ¯ i ( x ) = μ ¯ X ˜ 1 i ( x 1 ) × ⋯ × μ ¯ X ˜ p i ( x p ) the terms μ _ X ˜ j i and μ ¯ X ˜ j i are the lower and upper membership grades of μ X ˜ j i , respectively.
The type-2 fuzzy inference engine produces an aggregated output type-2 fuzzy set. The type reduction block operates on this set to generate a centroid type-1 fuzzy set known as the “type-reduced set” of the aggregate type-2 fuzzy set. Several type-reduction methods have been suggested in the literature, such as the center-of-sums, the height, the modified height and the center-of-sets, for example [
Y c o s ( x ) = [ y l , y r ] = ∪ f i ∈ F i ( x ) y i ∈ Y i ∑ i = 1 N f i y i ∑ i = 1 M f i (10)
where Y c o s is the interval set determined by two end points y l and y r , and firing strengths f i = [ f _ i , f ¯ i ] ∈ F i ( x ) . y l and y r can be expressed as [
y l = ∑ i = 1 L f ¯ i y l i + ∑ i = L + 1 N f _ i y l i ∑ i = 1 L f ¯ i + ∑ i = L + 1 N f _ i (11)
y r = ∑ i = 1 R f _ i y r i + ∑ i = R + 1 N f ¯ i y r i ∑ i = 1 R f _ i + ∑ r = R + 1 N f ¯ i (12)
Two end points y l and y r can be computed efficiently using the Karnik-Mendel (KM) algorithms [
1) Sort x _ i = ( i = 1 , ⋯ , N ) in increasing order
2) Initialize f i by setting f i = f _ i + f ¯ i 2 ( i = 1 , 2 , ⋯ , N ) and then compute y = ∑ i = 1 N x _ i f i ∑ i = 1 N f i
3) Find switch point k ( 1 ≤ k ≤ N − 1 ) such that x _ k ≤ y ≤ x ¯ k + 1
4) Set f i = f ¯ i ( i ≤ k ) and f i = f _ i ( i > k ) then compute y ′ = ∑ i = 1 N x _ i f i ∑ i = 1 N f i
5) Check if y ′ = y . If yes, stop, set y l = y , and call k, L. If no go to step 6
6) set y = y ′ and go to step 3
Since the type-reduced set is an interval type-1 set, the defuzzified output is [
y ( x ) = 0.5 ( y l + y r ) (13)
To control such a complicated system a novel simple controller is proposed using voltage control strategy. Electrical equation of a permanent magnet dc motor is written as
u = R I a + L I ˙ a + k b θ ˙ m + φ (14)
where R , L and k b denote the armature resistance, inductance, and back emf constant, respectively. u is the motor voltage, I a motor current, and θ m the rotor position. φ represents the external disturbance.
The motor angle θ m as an output can be controlled via the voltage u as an input. It is very interesting to note that (7) is a single-input/single-output (SISO) system while the robot manipulator is a multivariable multi-input system. The motor current I a contains effects of coupling between the motor and the manipulator.
From (2), we have
r θ m = K − 1 ( τ l ) + θ (15)
In addition, Equation (3) can be rewritten as
τ l = r − 1 ( τ + φ 1 ) (16)
where φ 1 include unmodeled dynamics. Substituting (16) into (15) and using (5) yields
r θ m = K − 1 ( r − 1 ( K m I a + φ 1 ) ) + θ (17)
Taking the time derivative of the above equation yields
θ ˙ m = K − 1 r − 2 K m I ˙ a + r − 1 θ ˙ + K − 1 r − 2 φ ˙ 1 (18)
Substituting (18) into (14) gives
u = L m I ˙ a + k b r − 1 θ ˙ + φ 2 (19)
where L m = L + k b K − 1 r − 2 K m and φ 2 = k b K − 1 r − 2 φ ˙ 1 + R I a + φ
The current of the motor can be directly controlled using a PI controller as follows:
I a = k p e + k i ∫ e d t (20)
where k d and k i are positive constant gains. e is tracking error expressed by e = θ d − θ . In the meantime, θ is the actual joint angle and θ d is the desired joint.
Substituting (20) into (19) yields
u = L m ( k p e ˙ + k i e ) + k b r − 1 ( θ ˙ ) + φ 2 (21)
Using (21) a control law is proposed as
u = L m ( k p e ˙ + k i e ) + k b r − 1 ( θ ˙ d + β ( θ d − θ ) ) + φ ^ 2 (22)
where β is a positive constant and φ ^ 2 is the estimation of φ 2 . After some manipulation, one can obtain
u = k b r − 1 θ ˙ d + ( k ′ p ) e + ( k ′ d ) e ˙ + φ ^ 2 (23)
where k ′ p = L m k i + k b r − 1 β and k ′ d = L m k p .
Equation (23) includes three terms. The first term is k b r − 1 θ ˙ d and the second term can be considered as a PD controller. The third term is the estimation of uncertainty. As a result, we can conclude from (23) that a flexible joint robot can be controlled directly using a simple PD controller plus uncertainty estimation with an extra term expressed by k b r − 1 θ ˙ d . It should be stated that the IT2FLC can be used instead of the PD control plus uncertainty estimation. This is why the IT2FLC can handle the uncertainty. Compared to the previous controllers reported for the flexible-joint robots which use two control loops, it has a simpler structure and more efficiency using only one control loop.
Suppose that y is the output of an IT2PD in the normalized form with the inputs of x 1 and x 2 . If three fuzzy sets are given to each fuzzy input, the whole control space will be covered by nine fuzzy rules. The linguistic fuzzy rules are proposed as
R i : if x 1 is X ˜ 1 i and x 2 is X ˜ 2 i then y i = a i 1 x 1 + a i 2 x 2 + a i 0 , i = 1 , 2 , ⋯ , 9 (24)
where R i denotes the ith fuzzy rule for i = 1 , ⋯ , 9 . In the ith rule, X ˜ 1 i and X ˜ 2 i are type-2 fuzzy membership functions belonging to the fuzzy variables x 1 and x 2 , respectively. a i 1 , a i 2 and a i 0 are the gain in consequent part and y i is the crisp output. The proposed interval type-2 fuzzy controller is for the case when antecedents are interval type-2 fuzzy sets (A2) and consequents are crisp numbers (C0). Three Gaussian membership functions with uncertain mean, μ X ˜ 1 i ( x 1 ) , named as Positive (P), Zero (Z), and Negative (N) are defined for input x 1 in the operating range of manipulator as shown in
In other words, y l in (11) can be rewritten as
y l = ∑ i = 1 L q ¯ l i ( a i 1 x 1 + a i 2 x 2 + a i 0 ) + ∑ i = L + 1 M q _ l i ( a i 1 x 1 + a i 2 x 2 + a i 0 ) (25)
where q ¯ l i = f ¯ i / D l and q _ l i = f _ i / D l . In the meantime, we have D l = ∑ i = 1 L f ¯ i + ∑ i = L + 1 M f _ i .
In the similar manner, y r in (12) can be rewritten as
y r = ∑ i = 1 R q _ r i ( a i 1 x 1 + a i 2 x 2 + a i 0 ) + ∑ i = R + 1 M q ¯ r i ( a i 1 x 1 + a i 2 x 2 + a i 0 ) (26)
where q ¯ r i = f ¯ i / D r and q _ r i = f _ i / D r . In the meantime, we have D r = ∑ i = 1 R f _ i + ∑ i = R + 1 M f ¯ i .
From (13) after some manipulation, one can obtain
y ( x ) = C 1 ( x ) x 1 + C 2 ( x ) x 2 + C 0 ( x ) (27)
where
C 1 ( x ) = 0.5 ( ∑ i = 1 L q ¯ l i a i 1 + ∑ i = L + 1 M q ¯ l i a i 1 ∑ i = 1 R q ¯ r i a i 1 + ∑ i = R + 1 M q ¯ r i a i 1 ) (28)
C 2 ( x ) = 0.5 ( ∑ i = 1 L q ¯ l i a i 2 + ∑ i = L + 1 M q ¯ l i a i 2 + ∑ i = 1 R q ¯ r i a i 2 + ∑ i = R + 1 M q ¯ r i a i 2 ) (29)
C 0 ( x ) = 0.5 ( ∑ i = 1 L q ¯ l i a i 0 + ∑ i = L + 1 M q ¯ l i a i 0 + ∑ i = 1 R q ¯ r i a i 0 + ∑ i = R + 1 M q ¯ r i a i 0 ) (30)
The obtained analytical structure of the fuzzy controller improves our study to develop the analysis and design. Using the scaling factors the input vector is formed as
x = [ k p i z 1 k d i z 2 ] T (31)
where for the ith joint z 1 and z 2 are defined as
x = [ k p i z 1 k d i z 2 ] T (31)
z 1 = θ d i − θ i (32)
z 2 = θ ˙ d i − θ ˙ i (33)
where θ d i and θ i are the desired and actual joint position, respectively. From (32) and (33) we have z ˙ 1 = z 2 .
Using x 1 = k p i z 1 and x 2 = k d i z 2 , one can obtain
x ˙ 1 = α x 2 (34)
where α = k p i / k d i > 0 .
Fuzzy controller by the use of scaling factors is formed as
u ( x ) = k 0 i ( C 1 ( x ) x 1 + C 2 ( x ) x 2 + C 0 ( x ) ) + k b r − 1 θ ˙ d (35)
This general structure shows a nonlinear variable gain controller that finds many applications in control. The nonlinear gain C i ( x ) covers the nonlinearity of controller by parameters in hand. The control purposes are simply described by linguistic rules in fuzzy controller transformed to a nonlinear function as stated by (35).
Substituting (35) into (3) forms the closed loop system as follows
( C 1 ( x ) x 1 + C 2 ( x ) x 2 + C 0 ( x ) ) k 0 i + k b r − 1 θ ˙ d = R I a + L I ˙ a + k b θ ˙ m (36)
Assume that the motor voltage u expressed by (3) is limited such that
| R I a + L I ˙ a + k b θ ˙ m | ≤ u max (37)
where u max > 0 is a maximum permitted voltage for the motor. This assumption is a technical regard to protect motor against over voltages. The complexity of design and analysis has been changed to simplicity for using the model of motor in place of model of manipulator. Here, we should know only the upper limits for the motor voltages as inputs of robotic system. Because electrical motors drive the electrical manipulator, the motor voltages are the system inputs. The desired trajectory should be planned with regarding the maximum permitted voltages for motors somehow each motor is so strong such that can track the desired trajectory under the permitted voltage. Moreover, the desired trajectory should be smooth such that its derivatives up to the required order are available and limited. To find a control law for the convergence of error, a positive definite function is proposed as
V = ∫ 0 x 1 C 2 ( x ) x 1 d x 1 (38)
where V is a positive definite function of x 1 if C 2 ( x ) is positive. To satisfy 0 ≤ C 2 ( x ) it is sufficient that 0 ≤ a i 2 .
Proof: Assume that 0 < C 2 < C 2 ( x ) where C 2 is a positive constant. Thus,
C 2 ∫ 0 x 1 x 1 d x 1 < ∫ 0 x 1 C 2 ( x ) x 1 d x 1 (39)
we have ∫ 0 x 1 C 2 x 1 d x 1 = 0.5 C 2 x 1 2 . Hence, 0.5 C 2 x 1 2 < ∫ 0 x 1 C 2 ( x ) x 1 d x 1 . Thus (39) implies that V > 0 for x 1 ≠ 0 . Since ∫ 0 0 C 2 ( x ) x 1 d x 1 = 0 and C 2 ( x ) x 1 is limited,
V = 0 if x 1 = 0 . Thus, V is a positive definite function of x 1 .
The time derivative of V is calculated as
V ˙ = C 2 ( x ) x 1 x ˙ 1 = α C 2 ( x ) x 1 x 2 (40)
From (36) we can write
C 2 ( x ) x 2 = − C 1 ( x ) x 1 − C 0 ( x ) + ( R I a + L I ˙ a + k b θ ˙ m ) / k 0 i − k b r − 1 θ ˙ d / k 0 i (41)
Substituting (41) into (40) yields
V ˙ = − α C 1 ( x ) x 1 2 − α C 0 ( x ) x 1 + α x 1 ( R I a + L I ˙ a + k b θ ˙ m ) / k 0 i − α x 1 k b r − 1 θ ˙ d / k 0 i (42)
Since − α C 1 ( x ) x 1 2 ≤ 0 for 0 ≤ C 1 ( x ) , to satisfy V ˙ ≤ 0 for stability, it is required that
x 1 ( R I a + L I ˙ a + k b θ ˙ m ) − x 1 k b r − 1 θ ˙ d ≤ k 0 i C 0 ( x ) x 1 (43)
Using the Cauchy-Schwartz inequality, one can obtain
x 1 ( R I a + L I ˙ a + k b θ ˙ m ) − x 1 k b r − 1 θ ˙ d ≤ | x 1 | | R I a + L I ˙ a + k b θ ˙ m | + k b r − 1 | x 1 | | θ ˙ d | ≤ | x 1 | u max (44)
Suppose that | θ ˙ d | ≤ γ where γ is a positive constant. To satisfy (44), it is sufficient that
| x 1 | ( u max + k b r − 1 γ ) ≤ k 0 i C 0 ( x ) x 1 (45)
Since k 0 > 0 , to guarantee stability 0 < x 1 C 0 ( x ) . This means that C 0 ( x ) must be designed with the same sign as x 1 . This condition is simply satisfied if a i 0 is selected with the same sign as x 1 .
From (45) and x 1 C 0 ( x ) > 0 , we obtain
( u max + k b r − 1 γ ) / | C 0 ( x ) | ≤ k 0 i (46)
From (30), one can obtain
c 0 , min ≤ C 0 ( x ) ≤ c 0 , max (47)
where c 0 , min and c 0 , max are to constant. To select a constant value, we should select a value for k 0 that satisfies (46) in all cases. The maximum permitted value for k 0 is then selected as
( u max + k b r − 1 γ ) / c 0 , max = k 0 i (48)
Therefore, stability is guaranteed under assumptions C 1 ( x ) > 0 , C 2 ( x ) > 0 , x 1 C 0 ( x ) > 0 and ( u max + k b r − 1 γ ) / c 0 , max = k 0 i .
In the rule, a i 0 is selected with the same sign as x 1 to satisfy x 1 C 0 ( x ) > 0 . We can select a i 1 ≥ 0 and a i 2 ≥ 0 in all subsections but ∃ i that a i 1 > 0 and a i 2 > 0 to satisfy C 2 ( x ) > 0 , x 1 C 0 ( x ) > 0 , respectively.
Fuzzy rules in the 9 subsections for i = 1 , ⋯ , 9 are designed where the following cases occur:
Case 1 Assume that x 1 x 2 < 0 resulting in asymptotic stability by V ˙ < 0 in (40). Thus, a small control effort is given to u .
Case 2 Assume that x 1 = 0 or x 2 = 0 resulting in Lyapunov stability by V ˙ = 0 in (40). Thus, a medium control effort is given to u .
Case 3 Assume that x 1 and x 2 both are positive or negative resulting in instability by V ˙ > 0 in (40). Thus, a very high effort is given to u .
The fuzzy rules for the first and second controllers are then given as:
Rule 1 : if x 1 is P and x 2 is P then y = 1
Rule 2 : if x 1 is P and x 2 is Z then y = 0.75
Rule 3 : if x 1 is P and x 2 is N then y = 0.25
Rule 4 : if x 1 is Z and x 2 is P then y = 0.5
Rule 5 : if x 1 is Z and x 2 is Z then y = 150 x 1 + 10 x 2
Rule 6 : if x 1 is Z and x 2 is N then y = − 0.5
Rule 7 : if x 1 is N and x 2 is P then y = − 0.25
Rule 8 : if x 1 is N and x 2 is Z then y = − 0.75
Rule 9 : if x 1 is N and x 2 is N then y = − 1
Therefore, using the above analysis the x 1 , x 2 are bounded. Then one can imply the boundedness of u because of boundedness x 1 and x 2 .
Proof: From (28), C 1 ( x ) , C 2 ( x ) and | C 0 ( x ) | are bounded as
| C 1 ( x ) | ≤ 0.5 α 1 (49)
| C 2 ( x ) | ≤ 0.5 α 2 (50)
| C 0 ( x ) | ≤ 0.5 α 0 (51)
where
| ∑ i = 1 L q ¯ l i a i 1 + ∑ i = L + 1 M q ¯ l i a i 1 + ∑ i = 1 R q ¯ r i a i 1 + ∑ i = R + 1 M q ¯ r i a i 1 | ≤ α 1 (52)
| ∑ i = 1 L q ¯ l i a i 2 + ∑ i = L + 1 M q ¯ l i a i 2 + ∑ i = 1 R q ¯ r i a i 2 + ∑ i = R + 1 M q ¯ r i a i 2 | ≤ α 2 (53)
| ∑ i = 1 L q ¯ l i a i 0 + ∑ i = L + 1 M q ¯ l i a i 0 + ∑ i = 1 R q ¯ r i a i 0 + ∑ i = R + 1 M q ¯ r i a i 0 | ≤ α 0 (54)
where α 1 , α 2 and α 0 are constant.
Considering (9) we have
f _ i ( x ) ≤ 1 (55)
f ¯ i ( x ) ≤ 1 (56)
Thus, one can imply that q ¯ l i , q _ l i and q ¯ r i , q _ r i are bounded. The coefficient a i 1 is a constant parameter. As a result, the inequality (52) is verified.
Similarly, the inequality (53) and (54) are proven. Therefore, u is bounded using (35) as
| u | ≤ k 0 i ( α 1 | x 1 | + α 2 | x 2 | + α 0 ) + k b r − 1 γ (57)
According to the proof given by [
Since the desired joint angle θ d and its time derivative θ ˙ d are bounded. The bound variables x 1 and x 2 imply that θ = θ d − x 1 and θ ˙ = θ ˙ d − x 2 are bounded.
Since I a is bounded, (4) implies that τ is bounded. From (2) we have
J θ ¨ m + B θ ˙ m + r 2 K θ m = τ + r K θ (58)
System (58) is a second order linear system with positive gains B , r 2 K , and a limited input τ + r K θ .This system is stable based on the Routh-Hurwitz criterion and implies that θ m , θ ˙ m and θ ¨ m are bounded.
Since all states associated with each joint i.e. θ , θ ˙ , θ m , θ ˙ m , and I a are bounded, vectors θ , θ ˙ , θ m , θ ˙ m and I a are bounded. As a conclusion, based on the stability analysis, all required signals are bounded.
The proposed type-2 PD fuzzy controller is simulated using an electrical flexible-joint articulated robot manipulator. The articulated manipulator is a serial link manipulator with three revolute joints as shown in
I i = [ I x x i I x y i I x z i I y x i I y y i I y z i I z x i I z y i I z z i ] (59)
The parameters of motors are given in
Simulation 1: In This simulation, the proposed IT2PD controller is simulated. The performance of control system is shown in
Simulation 2: in this case a comparison between type-1 PD (T1PD) controller and T2PD controller is presented. In order to consider the robustness evaluation of the controllers, external disturbances are added to the robot system. The disturbance is inserted to the input of each motor as a periodic pulse function with a period of 2 S, amplitude ∓ 4 V, time delay of 0.7 S, and pulse width 30% of period. This form of disturbance is an example of any form that can be applied but it includes jumps to cover the complex cases. To better assess the performance of both types of controllers in the face of external disturbance and unmodeled dynamics, the integral of squared errors (ISE) and the integral of absolute control input (ISU) are considered as a criterion
ISE = ∫ 0 10 ( e 1 2 + e 2 2 + e 3 2 ) d t (60)
ISU = ∫ 0 10 ( | u 1 | + | u 2 | + | u 3 | ) d t (61)
where e 1 , e 2 and e 3 are the tracking error of first, second and third link, respectively. In the meantime, u 1 , u 2 and u 3 are the voltage of first, second and third motor, respectively.
The responses of control system using T1PD controller and IT2PD controller in the presence of disturbance are shown in
A novel interval type-2 fuzzy PD control was developed for tracking control of a flexible-joint robot using the voltage control strategy. The proposed method is free from manipulator dynamics and very simple in the form of a decentralized control. In addition, there are no restrictions on the joint stiffness gains. The stability analysis has verified the control method and the simulation results have confirmed its effectiveness. Compared to the previous controllers reported for the flexible-joint robots which use two control loops, it has a simpler structure using only one control loop. A comparison between interval type-2 fuzzy PD and type-1 fuzzy PD controller has been done and simulation results confirmed that type-2 fuzzy PD controller can handle external disturbance better than the type-1 fuzzy PD controller. In addition, it spends less control effort than the type-1 in order to deal with disturbance. Note that in the present paper a novel control approach has been proposed whereas in [
The authors gratefully appreciate the support of the Behbahan Khatam Alanbia University of Technology.
Zirkohi, M.M. and Izadpanah, S. (2017) Interval Type-2 Fuzzy PD Tracking Control of Flexible-Joint Robots. Journal of Software Engineering and Applications, 10, 854-872. https://doi.org/10.4236/jsea.2017.1011048