PID position domain control for contour tracking
International Journal of Systems Science, 2013 http://dx.doi.org/10.1080/00207721.2013.775385
PID position domain control for contour tracking P.R. Ouyang∗ , V. Pano and T. Dam Department of Aerospace Engineering, Ryerson University, Toronto, Canada
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(Received 6 January 2012; final version received 2 February 2013) Contour error reduction for modern machining processes is an important concern in multi-axis contour tracking applications in order to ensure the quality of final products. Many control methods were developed in time domain to deal with contour tracking problems, and a proportional–derivative (PD) position domain control (PDC) was also proposed by the authors. It is well known that proportional–integral–differential (PID) control is the most popular control in applications of control theory. In this paper, a PID PDC is proposed for reducing contour tracking errors and improving contour tracking performances. To determine proper control gains, system stability analysis is conducted for the proposed PDC. Several experiments are conducted to evaluate the performance of the developed approach and are compared with the PID time domain control (TDC) and the cross-coupled control. Different control gains are used in the simulations to explore the robustness of PID PDC. Comparison results demonstrate the effectiveness and good contour performances of PID PDC for contour tracking applications. Keywords: position domain; PID control; contour tracking; contour error; stability
1. Introduction In manufacturing processes, one of the most important issues is the reduction of machining errors to ensure the quality of final products. To achieve this goal, a good control system is required in a machining system. Despite the development of various advanced control algorithms in industry and academia over past a few decades, it is well known that proportional–integral–differential (PID) control (Radke and Isermann 1987; Mills and Goldenbery 1988; Qu and Dorsey 1991; Rocco 1996; Visioli and Legnani 2002; Astrom and Hagglund 2006; Jin, Yang, and Chang 2013) is the most popular one in applications of control theory. Due to its simple structure, easy implementation and robust operation, PID control is widely used for the following applications: process control, robot manipulations, motor drives, automotive control, flight control, instrumentation operations, etc. Most industrial robot manipulators are equipped with PID controllers that do not require information about the robot dynamics in their control laws. A linear and decoupled PID control with appropriate control gains can achieve the desired position without steady-state error. That is the main reason for the wide applications of PID control. It is thought that PID control is the best solution from an industrial point of view (Khalil 2002). Many studies have been conducted for stability analysis of PID control using different approaches (Qu and Dorsey 1991; Mills and Goldenbery 1988; Rocco 1996).
∗
Corresponding author. Email: [email protected]
C 2013 Taylor & Francis
There are two different control tasks in applications of robot manipulators: one is set-point (position) control that does not specify the path of the end-effector, and the other is trajectory tracking control that requires a robot manipulator following a specifically defined path. According to the controlled target, robot control can be classified into two different categories: one is in the joint level, and the other is in the end-effect or task space level. For the purpose of control, it should be noted that the task space control should be mapped into the joint level control of the actuators through inverse kinematic analysis. A contour error is defined as the error between the desired contour and the real contour in an orthogonal direction. The contour error is an important index to measure the quality of a machined product or the contour tracking performance. In a conventional approach, for example, the decoupled PID control, the contour error is controlled and improved by a tracking controller for each individual axis. A common method to improve the contour accuracy is to achieve high tracking accuracy of each individual axis. Many efforts have been made to improve the tracking performance through developing many advanced control systems, such as adaptive control (Radke and Isermann 1987), robust control (Jin et al. 2013), sliding mode control (Zhao and Zou 2012), fuzzy logic control (Tzafestas and Papanikolopoulos 1990), neural network control (Miller, Hewes, Glanz, and Kraft 1990), iterative learning control (Bouakrif, Boukhetala, and Boudjea 2013), etc.
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P.R. Ouyang et al.
It should be noted that a good tracking performance for each individual axis does not guarantee the reduction of contour errors for a multi-axis motion system, as poor synchronisations of relevant motion axes result in diminished contour accuracy of the contour tracking (Koren 1980; Fang and Chen 2002; Barton and Alleyne 2008; Hu, Yao, and Wang 2009). To solve this problem, a so-called crosscoupled control (CCC) was proposed by Koren (1980) and extensively discussed in Fang and Chen (2002), Barton and Alleyne (2008) and Hu et al. 2009) for further improvements. Actually, the majority of CCC is a combination of proportional–derivative (PD)/PID control for individual axis and a coupled error feedback control for multiple axes. In our previous work (Ouyang, Dam, Huang, and Zhang 2012), a PD-type contour tracking control law established in position domain was proposed, focusing on contour tracking performance improvement. In order to further improve the contour accuracy, a PID contour tracking control in position domain is developed in this paper. The stability analysis is conducted using the Lyapunov method, and the principle for selecting control gains in position domain is provided based on the stability analysis. After that, some complex contour tracking problems are examined and compared with the traditional PID control and CCC in time domain. Simulation results demonstrate the effectiveness and high contour performances of the proposed PID position domain control (PDC).
Figure 1.
A 2-DOF robot.
is demonstrated that a global stability of trajectory tracking in each axis can be achieved (Mills and Goldenbery 1988; Qu and Dorsey 1991; Rocco 1996) under PID feedback control, as shown in Equation (2):
2. PID PDC and contour error 2.1. System description and dynamic model in time domain and position domain In this paper, we use a 2-DOF (degree of freedom) Cartesian robot (P´erez, Reinoso, Garc´ıa, Sabater, and Gracia 2006; Spong, Hutchinson, and Vidyasagar 2006), shown in Figure 1, as an example of a multi-axis motion system to discuss the contour tracking control problem in position domain. The dynamic model including the mechanical system and the actuator for each axis can be expressed as a general second-order differential equation (P´erez et al., 2006; Spong et al. 2006) as follows:
mx x¨ + cx x˙ + kx x + fx = Fx , my y¨ + cy y˙ + ky y + fy = Fy
(1)
˙ x¨ and y¨ are position, velocity, where x and y, x˙ and y, and acceleration of the X-axis and the Y-axis, respectively. mi , ci , and ki are the mass, damping, and stiffness of the ith axis motion, fx and fy are the disturbances and uncertainties of the system model, and Fx and Fy are the control input forces, respectively. From Equation (1), one can see that the dynamic model of this Cartesian robot for these two axes is decoupled and PID control can be applied for the control of each axis. It
Fx Fy
=
Kpx 0 0 Kpy
+ +
Kix 0 0 Kiy
Kdx 0 0 Kdy
ex ey ⎛ t ⎜ 0 ⎜ ⎝ t
⎞ ex (t)dt
ey (t)dt e˙x , e˙y
⎟ ⎟ ⎠
0
(2)
where ex , ey and e˙x , e˙y are the tracking errors (the desired value – the real value) and the derivative of tracking errors, respectively. Kpj , Kij , and Kdj are the proportional gain, integral gain, and derivative gain, respectively. To develop PID PDC, we assume that the X-axis is a reference (master) axis and the Y-axis is the serve (slave) axis. The Y-axis dynamic model can be rewritten related to the X-axis position. In this paper, for simplicity, we assume that the X-axis motion direction is unchanged for a controlled contour segment. Without loss of generality, we assume that the X-axis motion is in the positive direction with a monotonically increasing function of time that means x˙ ≥ 0. Also, the X-axis position is sampled equidistantly by a sensor such as an encoder. It should be noted that the developed new control law also can be applied to x˙ ≤ 0, as shown in the following simulations.
International Journal of Systems Science If we define a space velocity (the first derivative) and a space acceleration (the second derivative) of y with respect to variable x (the master motion) as: dy , dx d 2y dy y = = , dx dx 2
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y =
(3) (4)
where y is called the relative position domain velocity of the Y-axis with respect to the X-axis, and y is the relative position domain acceleration, as both are defined in position domain with respect to the X-axis. It is clear that y is the tangent of a tracking contour at a point of the reference x or the contour motion direction. To develop the dynamic model in position domain, a relationship between the absolute velocity and the relative velocity for the Y-axis motion can be expressed as: y˙ =
dy dx dy ˙ = = y x, dt dx dt
(5)
where x˙ > 0. Similarly, a relationship between the absolute acceleration and the relative acceleration can be obtained: dy d x˙ d y˙ ¨ . = x˙ + y = x˙ 2 y + xy y¨ = dt dt dt
(6)
Applying Equations (5) and (6) to the original dynamic model, Equation (1), the dynamic model for the Y-axis motion in Equation (1) can be rewritten in position domain as (Ouyang et al. 2012):
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But there are some significant differences. First, these two control laws are defined in two different domains. Second, the derivative gains have different meanings because of the differences between ey (x)and e˙ (t). Remark 2: The tracking error, Equation (9), forms the contour error in position domain where the X-axis is the reference axis with zero bias. That is, ec (x) = Cy ey (x) for PDC. According to the definition of y (x) in Equation (3), there is no need to get the velocity information for the X-axis. Only the position information of the X-axis is needed in order to define the contour tracking error. As the contour tracking error and the contour tracking error derivative in Equation (9) are defined in position domain; therefore, we call the new developed PID control as a PID PDC. Applying Equations (8) and (9) to Equation (7), the dynamic model based on PID PDC can be expressed as ˙ (x) + ky y (x) + fy (x) my x˙ 2 y (x) + (my x¨ + cy x)y x (yd (s) − y (s))ds = Kpy (yd (x) − y (x)) + Kiy + Kdy (yd (x) − y (x)).
0
(10)
Examine the control law for the Y-axis motion in Equation (8). Substituting Equation (5) in Equation (8), we have the following equation:
x
ey (x)ds + Kdy ey (x) x (yd (s) − y (s))ds = Kpy (yd (x) − y (x)) + Kiy
Fy (x) = Kpy ey (x) + Kiy
0
0
¨ (x) + cy xy ˙ (x) + ky y (x) my x˙ 2 y (x) + my xy + fy = Fy (x) , (7) where Fy (x) is the control input described in position domain. 2.2. PID PDC For the position domain dynamic model expressed in Equation (7), we propose a new PID PDC law as follows: Fy (x) = Kpy ey (x) + Kiy 0
x
ey (s) ds + Kdy ey (x). (8)
The contour tracking error and its relative derivative in position domain are defined as:
ey (x) = yd (x) − y (x) . ey (x) = yd (x) − y (x)
Kdy e˙y (x) + x˙
(11)
Remark 3: It is clearly shown in Equation (11) that a position domain linear PID control is equivalent to a nonlinear PID control in time domain (Ouyang, Zhang, and Wu 2002) when the speed of motion in the X-axis is not constant. It should be noted that the transformation of dynamics from time domain to position domain is a one-to-one nonlinear mapping. The dynamic model in Equation (7) described in position domain is equivalent to the dynamic model, Equation (1), in time domain. Therefore, the proposed PID control in position domain has the same stability property as the PID control developed in time domain. It means that the developed PID PDC is stable for contour control of the robotic system. A detailed discussion about the stability analysis will be presented in the next section.
(9)
Remark 1: The PID PDC law in Equation (8) is similar in formula to the traditional PID control law in Equation (2).
2.3. CCC and contour error In this paper, to demonstrate the effectiveness and success of PID PDC as compared with time domain control (TDC)
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P.R. Ouyang et al. 3. Stability analysis 3.1. Preparation and Lemma Consider a dynamic system described in position domain by y (x) = f (x, y (x)),
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Figure 2.
where x ∈ R is the ‘position’ of the master motion or the independent variable, and y (x) ∈ R n is the state.
Tracking and contour errors.
methods, CCC applied in time domain is also used and its control laws are selected (Koren 1980; Fang and Chen 2002; Barton and Alleyne 2008; Hu et al. 2009) as: t Fcx = Kpx ex +Kix ex ds + Kdx e˙x −Cx (Kpc ec + Kdc e˙c ) 0 , t Fcy = Kpy ey +Kiy ey ds + Kdy e˙y + Cy (Kpc ec + Kdc e˙c ) 0
ec = −Cx ex + Cy ey , e˙c = −Cx e˙x + Cy e˙y
(13)
where Cx = sin θ, Cy = cos θ . θ is the angle between the X-axis and the desired line measured in a counter-clockwise direction. For a circular motion, the contour errors are defined as (Koren 1980; Fang and Chen 2002; Barton and Alleyne 2008; Hu et al. 2009):
ec = −Cx ex + Cy ey , e˙c = −Cx e˙x + Cy e˙y − C˙ x ex + C˙ y ey
Lemma 1: Let D ⊂ R n be a domain that contains the origin and V: [0, ∞) × D → R be a continuously differentiable function such that γ1 ( y ) ≤ V (y) ≤ γ2 ( y ), V (y) ≤ −W (y),
(16)
∀ y ≥ μ > 0, ∀ x ≥ 0, ∀ y ∈ D, (17)
(12)
where ec and e˙c are the contour error and its derivative, respectively. Contour error is defined as the shortest distance from the actual position to the desired contour. The difference between the tracking error and contour error can be identified through Figure 2. From this figure, one can see that the contour error is less than the tracking error. For a linear motion, the contour errors are defined as (Koren 1980; Fang and Chen 2002; Barton and Alleyne 2008; Hu et al. 2009):
(15)
(14)
where Cx = sin θ − ex /2R, Cy = cos θ + ey /2R, C˙ x = θ˙ cos θ − e˙x /2R, C˙ y = −θ˙ sin θ + e˙y /2R, and R is the radius of the tracked circular motion. It should be mentioned that the angle θ is not constant in a circular contour motion. In summary, we developed a PID PDC for contour tracking of a Cartesian robot in this section. In the following sections, we will fulfil the stability analysis and conduct some simulation tests to demonstrate the effectiveness of the proposed PID PDC based on the comparison with PID TDC and CCC.
where γ1 and γ2 are class K functions and W (y) is a continuous positive definite function. Take r > 0 such that Br ⊂ D, and suppose that μ < γ2−1 (γ1 (r)) .
(18)
Then, there exists a class KL function φ, and for every initial state y(x0 ) satisfying y(x0 ) ≤ γ2−1 (γ1 (r)), there is X ≥ 0 such that the solution of dynamic equation satisfies y(x) ≤ φ( y(x0 ) , x − x0 ), y(x) ≤ γ1−1 (γ2 (μ)),
∀ x0 ≤ x ≤ x0 + X, (19) ∀x ≥ x0 + X.
(20)
Moreover, if D = R n and γ1 belongs to class K∞ , then Equations (19) and (20) hold for any initial statey (x0 ), with no restriction on how large μ is. Proof 1: See reference Khalil (2002).
This lemma proves that the dynamic system is globally uniformly exponentially convergent to a closed ball for any initial conditions if one can find a positive definite function V(y) so that Equations (16) and (17) hold. 3.2. Theorem To facilitate the discussion, we have the following assumptions: (A1). The desired contour of yd (x)is of the third-order continuity for x ∈ [xini , xf in ]. (A2). The real tracking path in the X-axis x (t) is of the third-order continuity for the contour. (A3). The disturbance fy (x)is bounded in the full contour tracking process.
International Journal of Systems Science For the briefness of the stability analysis, some notations are introduced and used in the following sections: z = max |zi |, 1≤i≤n
ρ=
my x˙ 2 yd (x)
˙ d (x) + ky yd (x) + fy , + (my x¨ + cy x)y
|ρ|max
= my x˙ 2 yd (x)+ my x¨ +cy x˙ yd (x)+ky yd (x) + fy max ¨ + cy x˙ 2 yd (x) , ≤ my x˙ 2 yd (x) + my x + ky yd (x) + fy
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my x¨ + cy x˙ Kiy α ρe = Kpy + k y + − − , and 2 β 2β
β α, my x¨ + cy x˙ − βmy x˙ 2 − ρe = Kdy + 1 + 2 2 where α and β are positive constants. According to the assumptions (A1∼A3), one can prove that the parameter |ρ|max is bounded. Parameter ρe is related to Kdy ,while ρe is related to all other two control gains. For the developed position domain PID controller, Equation (8), we have the following theorem: Theorem 1: A robot system represented in position domain by Equation (7), where the desired contour shape satisfies assumptions A1 and A2, is controlled by the proposed PID PDC law in Equation (8). The contour tracking error and its relative derivative are bounded, and the boundednesses are given by ⎧ ⎪ ⎪ ⎪ e ≤ 2 1 + 1 |ρ|max , ⎪ ⎪ ⎨ y ρe2 βρe ρe . ⎪ ⎪ β 1 ⎪ ⎪ ⎪ ⎩ ey ≤ 2 ρ ρ + ρ 2 |ρ|max e e e
for ρe and ρe , and the better the contour tracking performance, which will be demonstrated through the following simulation study. Specifically, the derivative gain Kdy determines the parameter ρe that is dominantly controlling the boundary value of the relative derivative error ey , and the proportional–integral (PI) gains determine the parameter ρe that is dominantly controlling the boundary value of the tracking error ey . Basically, to determine the control gains, first we choose these two constants α and β, then determine the maximum value of control gain Kiy , followed by the selection of Kpy and Kdy . As there is no restriction for the selection of α and β; therefore, the selection of three PID control gains is very easy and simple.
3.3. Stability proof A stability analysis for the proposed PID PDC is conducted based on the Lyapunov function method. First, we define ⎧ ⎨
σy (x) =
x
ey (s)ds . ⎩ σ (x) = e 0 (x) y
(23)
y
Using Equation (9), the dynamic model with position domain PID controller in Equation (10) can be re-described in an error function format as follows:
my x˙ 2 ey (x) + my x¨ + cy x˙ + Kdy ey (x) + Kiy σy (x)
+ Kpy + ky ey (x) = ρ. (24) For the dynamic system described in position domain, we define the following two functions:
(21)
Provided that the control gains and the positive constant parameters are selected properly such that ⎧ α>β>0 ⎪ ⎪ ⎪ ⎪ ¨ + cy x ) ˙ ≥0 = α − β(my x ⎪ ⎪ ⎪ ⎪ ⎨ ˙ 2 Kpy > α + + βmy x , 2 ⎪ ⎪ K < αβ iy ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ K > + m x ˙ + βmy x ˙ 2 dy y ¨ + cy x 2
5
(22)
it is noted that Equation (22) provides some guidelines about the choices of control gains for the developed PID PDC. In order to reduce the tracking errors, we need to increase the values of two parameters: ρe and ρe . These two parameters are related to the PID control gains. Generally speaking, the larger the control gains, the larger the values
⎧ 1
Kpy + ky βmy x˙ 2 ⎪ ⎪ ey e V1 ey (x) , ey (x) = ⎪ ⎪ βmy x˙ 2 my x˙ 2 2⎛ ⎞y ⎪ ⎪ ⎪ ⎪ ey ⎪ ⎪ ⎪ ⎨ × ⎝ e ⎠ y
⎪ ⎪ ⎪ 1 ey α Kiy ⎪ ⎪ V2 (ey (x), σy (x)) = ( ey σy ) ⎪ ⎪ K βK σ 2 iy iy y ⎪ ⎪ ⎪ β ⎪ 2 ⎩ + Kdy ey 2
.
(25) Based on Equation (25), we define the Lyapunov function as V (ey (x), ey (x), σy (x)) = V1 (ey (x) , ey (x)) + V2 (ey (x) , σy (x)).
(26)
If the control gains are properly chosen according to Equation (22), the following inequality holds: ˙ 2 > β 2 my x˙ 2 − ky . Kpy > α + βmy x
(27)
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P.R. Ouyang et al.
According to Equation (27), we can prove that V1 ey (x) , ey (x) is a positive definite function. Also, from Equation (22), we have Kiy < αβ, and it is easy to prove that V2 ey (x) , σy (x) is also a positive definite function according to Sylvester’s criterion. Therefore, the Lyapunov function V ey (x) , ey (x) , σy (x) in Equation (26) is a positive definite function. It means that V ≥ 0. It is easy to demonstrate that
According to Equation (22), we have α > ˙ ≥ β(my x¨ + cy x). ˙ ¨ + cy x ) ˙ ≥ β( my x¨ + cy x ) β(my x From Equation (28), we can prove that
⎧ 1 ⎪ ⎨ ab ≥ − (a 2 + b2 ) 2 . 1 ⎪ ⎩ ab ≤ (a 2 + b2 ) 2
Applying Equation (32) to Equation (31), we have (28)
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Applying Equation (28) to Equation (25) and (26), the following inequalities can be obtained: 1 1 (Kpy + ky − βmy x˙ 2 )ey2 + (1 − β)my x˙ 2 ey2 2 2 + V2 (ey (x), σy (x)). ≤ V (ey (x), ey (x), σy (x)) 1 1 ≤ (Kpy + ky + βmy x˙ 2 )ey2 + (1 + β)my x˙ 2 ey2 2 2 + V2 (ey (x), σy (x)) (29)
˙ y e y (α − β(my x¨ + cy x))e 1 2 2 ˙ ≤ (α − β(my x¨ + cy x))(e y + ey ). 2
(32)
β α my x¨ + cy x˙ − βmy x˙ 2 − V ≤ − Kdy + 1 + 2 2 ¨ ˙ x + c x K m α y y iy − − × ey2 − β Kpy + k y + 2 β 2β
2 × ey + ey + βey ρ. (33) From Equation (22), if we choose Kpy > α + ¨ and Kiy < αβ, one can prove that βmy x
2β
+
Kiy Kiy my x¨ + cy x˙ α − − >α− 2 β 2β β ¨ + βmy x ¨ ≥ ky + ky + +βmy x ¨ > 0. (34) +βmy x
ρe = Kpy + k y +
From Equation (29), we can see that the defined Lyapunov function satisfies Equation (16). In PDC, the reference position x of the X-axis motion is an independent variable that has a similar meaning of t in time domain. ey and ey are functions of the independent variable x. Therefore, the derivative of the Lyapunov function V is related to variable x in this stability analysis. Rewritting Equation (24), we have
Similarly, from Equation (22), if we choose Kdy > 2 + ¨ + cy x ˙ + βmy x ˙ 2 , then we have my x β α my x¨ + cy x˙ − βmy x˙ 2 − > 0. ρe = Kdy + 1 + 2 2 (35)
my x˙ 2 ey (x) = ρ − (my x¨ + cy x˙ + Kdy )ey (x) − Kiy σy (x)
Applying Equations (34) and (35) to Equation (33), we obtain
− (Kpy + ky )ey (x).
(30)
Differentiating Equation (26) with respect to the variable x and using Equation (30), we have Kpy + ky βmy x˙ 2 ey V = ey βmy x˙ 2 my x˙ 2
α Kiy ey + βKdy ey ey . + ey ey Kiy βKiy σy
ey
ey
= (Kpy + ky )ey ey + (ey + βey )my x˙ 2 ey + βmy x˙ 2 ey2 + (βKdy + α)ey ey + Kiy ey σy + βKiy ey σy + Kiy ey2 = (Kpy + ky + α)ey ey + βmy x˙ 2 ey2 + (ey + βey )Kiy σy
+ Kiy ey2 + (ey + βey ) ρ − (my x¨ + cy x˙ + Kdy )ey −(Kpy + ky )ey − Kiy σy
= − my x¨ + cy x˙ + Kdy − βmy x˙ 2 ey2
− β Kpy + ky − Kiy ey2
+ α − β my x¨ + cy x˙ ey ey + ey + βey ρ (31)
V ≤ − βρe ey2 − ρe ey2 + βey + ey ρ ≤ −βρe ey2
+ βey |ρ|max − ρe ey2 + ey |ρ|max .
(36)
Applying another inequality: az − bz2 ≤ we have ⎧ ⎪ ⎪ ⎨ β |ρ|
a2 1 − bz2 b 4
for
a>0
and
b > 0, (37)
βρe 2 β |ρ|2max e + 4 y ρe . ρe 2 |ρ|2max ⎪ ⎪ 2 ⎩ |ρ|max ey − ρe ey ≤ − ey + 4 ρe 2 max ey − βρe ey ≤ −
(38)
Applying Equation (38) to Equation (36), we get V ≤ −
βρe 2 ρe 2 e − e + 4 y 4 y
β 1 + ρe ρe
|ρ|2max .
(39)
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International Journal of Systems Science
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Figure 3. Contour tracking results for a zigzag motion. (a) Tracking errors for the zigzag motion. (b) Desired and actual motions with 10 × amplified errors.
Therefore, According to Lemma 1, we can demonstrate that both the contour tracking error and the derivative of the contour tracking error are bounded as follows: ⎧ ⎪ 1 1 ⎪ ⎪ ⎪ |ρ|max e ≤ 2 + ⎪ ⎨ y ρe2 βρe ρe . ⎪ ⎪ β 1 ⎪ ⎪ e ≤ 2 + 2 |ρ|max ⎪ ⎩ y ρe ρe ρe
motion contours. It is assumed that the 2-DOF robotic system has the following parameters: mx = 10 kg, cx = 10 N · s/m, kx = 20 N/m my = 5 kg, cy = 10 N · s/m, ky = 20 N/m.
(40)
According to Equation (40), one can see that the contour error and its derivative are bounded. From Equation (40), it is also shown that the maximum errors can be reduced to very small values by increasing control gains Kpy and Kiy (related to ρe ), and Kdy (related to ρe ). Therefore, the tracking errors will be reduced by choosing high PID control gains according to Equation (40).
4. Simulation tests In this section, two different types of contour tracking problems are simulated: linear motion contours and nonlinear
4.1. Linear motion contour tracking In the first two simulation examples, linear motion contours with positive and negative speed of the master motions are considered to verify the effectiveness of the proposed PID PDC for contour tracking.
4.1.1. Zigzag motion contour tracking In this simulation, a zigzag linear motion contour is tracked for 8 seconds, where x ∈ [0, 4] and y ∈ [0, 2]. To define the trajectories in two axes controlled in time domain, we assume that each linear motion segment for the zigzag shape is defined by a fifth-order polynomial (Spong et al. 2006) in 2 seconds. In PDC, we assume that the X-axis is controlled by a traditional PID control, and we measured its
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displacement by a sensor, and used it as a reference for the Y-axis PDC. According to the trajectory planning method, we have x ˙ < 1 m/s, and x ¨ < 1 m/s2 . If we chooseα = 1000 and β = 10, according to Equation (22), the control gains for the proposed PID PDC can be estimated as follows:
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⎧ ⎨ Kpy > 1975 Kiy < 10, 000 . ⎩ Kdy > 490 Of course, as there is no restriction for the selection of parameters α and β, the control gains can be easily determined based on Equation (22) and selected to some other values. For each segment, as the motion range in the Y-axis is double that in the X-axis, we set control gains for the X-axis to be half of the control gains used for the Y-axis. Therefore, the following control gains are chosen for time domain and position domain controllers for all simulation examples: Kpx = 2000, Kpy = 4000, Kpc = 2000,
Kdx = 1200, Kdy = 2400, Kdc = 1200
Kix = 2000 Kiy = 4000 .
For the zigzag motion, the motion in the X-axis is in one direction, with positive speeds from one segment to another segment. Figure 3 shows the contour tracking performances for PID TDC, CCC, and the proposed PID PDC, respectively. On the tracking error side, it is clearly shown that TDC in the Y-axis and PDC obtained good tracking results, whereas CCC obtained acceptable result. The tracking performance for TDC in the X-axis is the worst. From the amplified contour tracking performances shown in Figure 3(b), it can be seen that PDC obtained the best contour performance and demonstrated the effectiveness of the proposed PDC. On the contour tracking error side, Figure 4 shows the contour errors for three different control methods. It shows the significant reductions in the contour errors based
Table 1.
Figure 4.
Comparison of contour errors for zigzag motion.
on PDC as compared with TDC and CCC. Comparing Figure 4 with Figure 3, one can see that the contour errors for these three control methods are less than the maximum tracking errors. Such results are expected, as the contour error is defined as the shortest distance between the actual position and the desired contour. Another observation from Figure 4 is that the contour errors controlled by TDC and CCC are discontinuous at the connecting points of the segments. The actual positions for both these controllers are behind the desired positions shown in Figure 3. The actual positions are above the desired contours for the odd segments that produce negative contour errors, and the actual positions are below the desired contours for the even segments that obtain positive contour errors. Therefore, there is a sign change at the connecting points that makes the contour errors discontinuous. For the contour tracking based on PDC, all the actual positions are below the desired contour, so there is no sign change for the contour error that makes the contour errors smooth.
Zigzag contour performance comparison under different control gains. TDC (×10−3 )
Factor_p = 0.5 Factor_p = 1.0 Factor_p = 1.5 Factor_i = 0.5 Factor_i = 1.0 Factor_i = 1.5 Factor_d = 0.5 Factor_d = 1.0 Factor_d = 1.5
CCC (×10−3 )
PDC (×10−3 )
Max.
Min.
Max.
Min.
Max.
Min.
17.46 12.60 10.66 18.65 12.60 9.65 13.00 12.60 11.84
–8.96 –8.97 –7.55 –13.14 –8.97 –6.02 –9.09 –8.97 –8.64
12.79 10.15 8.69 14.44 10.15 8.00 10.87 10.15 9.80
–6.17 –5.71 –5.04 –7.55 –5.71 –5.18 –7.08 –5.71 –4.92
3.53 2.57 1.94 2.82 2.57 2.35 2.98 2.57 2.19
–2.30 –0.94 –0.57 –0.13 –0.94 –1.48 –1.77 –0.94 –0.28
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tracking errors, Figure 6(b) shows the actual contours based on these three control methods, while Figure 6(c) shows the contour errors. It can be seen from Figure 6(a) that the tracking errors of the Y-axis for all three controllers are almost at the same level. But the tracking errors in the Xaxis for TDC are little higher than those for CCC. TDC is the worst control in terms of tracking errors and contour errors; CCC and PDC have the same level of tracking errors, but have significant differences in contour errors, as shown in Figure 6(c). It is shown in Figure 6(c) that PDC ensures the best contour tracking performance. One can see that the maximum contour error for PDC is about one-third of that for TDC and half of the maximum contour error for CCC. It also demonstrates that CCC has some advantages compared with TDC for linear motion where the angle for calculating the contour errors in each segment is constant. Figure 5. Comparison of contour errors between PID and PD control for zigzag motion.
To examine the effects of control gains on contour tracking performances, different factors on PID control gains selected before are used in the simulations, and the maximum and minimum contour errors are obtained and listed in Table 1. From this table, one can see that the larger the control gains, the lesser the contour tracking errors. Also, it is shown that the proportional and derivative gains have large contributions to the final contour tracking performances for all three control methods, especially the proportional gain. Overall, PDC has the lowest contour tracking errors. To demonstrate the effectiveness of the proposed PID PDC, a comparison study is also conducted with PD PDC. Figure 5 presents the contour errors controlled by PD and PID PDCs. It shows the better contour performance controlled by PID PDC. From this figure, one can see that the contour errors are distributed on the positive side controlled by PD PDC, while the contour errors are distributed on both sides controlled by PID PDC. Such a result is contributed by the effect of integral control.
4.1.2. Diamond contour tracking A desired diamond contour is required to track for x ∈ [0, 2] and y ∈ [−2, 2]. For TDC and CCC, the time duration for tracking the diamond contour in both axes is assumed to be 8 seconds: 4 seconds for the positive motions of the X-axis and 4 seconds for the negative motion. This experiment demonstrates the effectiveness of PDC for both positive and negative speed cases in the X-axis. In this simulation study, the control gains for three different control algorithms are chosen the same as those in zigzag motion. Figure 6 shows the tracking performances under three different control methods where Figure 6(a) depicts the
4.2. Nonlinear motion contour tracking In the previous section, simulation results show that PDC is better than the controllers based on time domain for linear motions. In this part, we use some nonlinear motion contour tracking examples to verify the effectiveness of PDC. 4.2.1. Circular motion contour tracking A full circular contour with radius of 0.5 m is simulated using three different control methods. For TDC and CCC, the time durations for path tracking are assumed to be 10 seconds. The control gains for both axes are selected the same as the control gains for the Y-axis used in the linear motion contour tracking examples. It should be noted that for the full circular contour tracking problem, the speed of the X-axis is positive for the first half-circle and negative for the second half-circle. Figure 7 shows the tracking errors and contour tracking results using three different control methods. From this figure, specifically for the amplified actual contour shown in Figure 7(b), one can see that PDC obtained the best contour tracking result, CCC obtained a good contour tracking result, and TDC got the worst tracking result. Such a conclusion is the same as for the linear motion contour tracking. For the circular contour tracking problem, to examine the robustness of the proposed PDC, two speed cases are simulated. The low-speed case is the case where the time duration is 10 seconds, while the high-speed case is when the time duration is reduced by half to 5 seconds for both the X-axis and the Y-axis. The same control gains are used for both cases in the three control methods. The simulation result of contour tracking errors is shown in Figure 8. From Figure 8, one can see that PDC is ensured the smallest contour tracking errors for both low-speed and high-speed motions. Table 2 shows the maximum and minimum contour tracking errors under different gain factor conditions. It still
P.R. Ouyang et al.
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Figure 6. Contour tracking control results for a diamond contour. (a) Tracking errors under different control laws. (b) Actual contour tracking results for a diamond shape. (c) Contour errors based on different control laws.
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International Journal of Systems Science
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Figure 7. Contour tracking control results for a circular contour. (a) Tracking errors for a circular contour based on three control methods. (b) Actual contour tracking results for a circular contour (10×).
demonstrates that the larger the control gains, the smaller the contour errors for all three control methods, and the best tracking performance is obtained by PDC.
Figure 8.
In Table 2, the maximum contour errors controlled by PD-based controllers are also included when we set Factor_i = 0 for all three controllers. Comparing PID PDC
Contour tracking errors for a full circular motion. (a) Low-speed case. (b) High-speed case.
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Figure 9. Contour tracking results for an ellipse motion. (a) Tracking errors for an ellipse contour. (b). Contour tracking results with amplified tracking errors (10×). (c) Contour tracking errors for an ellipse motion contour.
with PD PDC, one can see that the contour performance controlled by PID PDC is better than that controlled by PD PDC. Also, it shows the improvement in contour tracking with the increase in integral gain in PID PDC. 4.2.2. Ellipse contour tracking Finally, an ellipse motion contour tracking example is simulated using the same control gains listed before and the
results are shown in Figure 9, where the tracking errors are shown in Figure 9(a), the amplified contour errors are presented in Figure 9(b), and the contour tracking errors are shown in Figure 9(c) for the three different control methods. From Figure 9(a), it can be clearly seen the best tracking performance is by PDC. From Figure 9(b), it can be clearly seen that the best contour tracking performance is by PDC and the worst contour tracking performance is by TDC. From Figure 9(c), one can see that the biggest contour
International Journal of Systems Science Table 2.
Circular contour performance comparison under different control gains. TDC (×10−4 )
Factor_p = 0.5 Factor_p = 1.0 Factor_p = 1.5 Factor_i = 0 Factor_i = 0.5 Factor_i = 1.0 Factor_i = 1.5 Factor_d = 0.5 Factor_d = 1.0 Factor_d = 1.5
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CCC (×10−4 )
PDC (×10−4 )
Max.
Min.
Max.
Min.
Max.
Min.
23.74 14.23 10.21 20.28 16.71 14.23 12.66 15.24 14.23 13.37
–57.38 –24.64 –15.25 –23.93 –24.57 –24.64 –22.77 –26.41 –24.64 –21.52
14.92 10.19 7.30 14.61 11.92 10.19 9.66 10.52 10.19 9.16
–32.77 –15.12 –10.35 –18.54 –19.08 –15.12 –10.98 –14.12 –15.12 –15.55
7.95 9.11 8.33 8.49 8.89 9.11 9.14 14.79 9.11 5.30
–14.99 –11.68 –9.52 –12.13 –11.90 –11.68 –11.54 –15.28 –11.68 –9.32
there is no tracking error for the master motion. Only the slave motion tracking errors will affect the final contour tracking performance. The guideline for determining control gains is set up through the stability analysis. It is demonstrated that PID PDC is stable and can obtain very good contour tracking performance. The new developed PD PDC is applied to a robotic system for improving the contour tracking performance. Linear and nonlinear motion contours are used to verify the effectiveness of the new controller through comparison with PID TDC and CCC. Also, it is demonstrated that PID PDC is better than its PD counterpart. More advanced control in position domain for multiple DOF system control would be a future research direction.
Figure 10. Comparison of contour errors between PID and PD control for an ellipse motion.
error controlled by PDC is about half of that by CCC and one-third of that by TDC. Figure 10 shows the comparison results of contour tracking errors controlled by PD and PID in position domain. Once again, it demonstrates that PID PDC is better than PD PDC in terms of reduction in contour errors.
5. Conclusions Many machining applications involve the generation of complex profiles and need to synchronise the motions of different axes in order to obtain good contour tracking performance. In this paper, a PID PDC is proposed for contour tracking control as an alternative to traditional PID TDC. The proposed PDC adopts the master–slave synchronised control concept. In the developed PDC, the master motion is sampled equidistantly and used as a reference; therefore,
Acknowledgement This research is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) through a Discovery Grant.
Notes on contributors Puren Ouyang is an assistant professor in the Department of Aerospace Engineering at Ryerson University, Toronto, Canada. He has been working in the areas of robotics and control, mechatronics, and precision manipulator and devices for more than 20 years. Dr Ouyang has also specialised in robotic systems with compliant mechanisms. He has developed a novel position domain control method that can significantly improve the contour tracking performance as compared with time domain control methods. This research was recognised in an IEEE conference in 2011 through the best paper award. In the robotics and control area, he has developed a series of robust and adaptive online learning control methods. Dr Ouyang has published more than 40 refereed journal papers and 30 conference articles.
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P.R. Ouyang et al. Vangjel Pano received his Bachelor’s degree in Aerospace Engineering from Ryerson University in Canada in 2011. Currently, he is studying as an MASc student in the Aerospace Engineering Department of Ryerson University. His research interests include robotics and advanced control.
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Truong Dam graduated from Ryerson University in Toronto, Canada. He received his BEng and MSc for aerospace engineering in 2010 and 2012, respectively. He is currently employed in the aerospace sector.
References Astrom, K.J., and Hagglund, T. (2006), Advanced PID Control, Research Triangle Park, NC: Instrumentation, Systems, and Automation Society (ISA). Barton, K.L., and Alleyne, A.G. (2008), A ‘Cross-Coupled Iterative Learning Control Design for Precision Motion Control’, IEEE on Control Systems Technology, 16(6), 1218– 1230. Bouakrif, F., Boukhetala, D., and Boudjea, F. (2013), ‘Velocity Observed-Based Iterative Learning Control for Robot Manipulators’, International Journal of Systems Science, 44(2), 214–222. Fang, R.W., and Chen, J.S. (2002), ‘Cross-Coupling Control for a Direct-Drive Robot’, JSMA, Series C, 45(3), 749–757. Hu, C., Yao, B., and Wang, Q. (2009), ‘Coordinated Contouring Controller Design for an Industrial Biaxial Linear Motor Driven Gantry’, in IEEE/ASME International Conference on Advanced Intelligent Mechatronics, pp. 1810– 1815. Jin, X.Z., Yang, G.H., and Chang, X.H. (2013), ‘Robust H∞ and Adaptive Tracking Control Against Actuator Faults With a Linearised Aircraft Application’, International Journal of Systems Science, 44(1), 151–165.
Khalil, H.K. (2002), Nonlinear Systems (3rd Version), Upper Saddle River, NJ: Prentice Hall. Koren, Y. (1980), ‘Cross-Coupled Biaxial Computer Control for Manufacturing Systems’, ASME Journal of Dynamic Systems, Measurement, and Control, 102(4), 265–272. Miller, W.T., Hewes, R.P., Glanz, F.H., and Kraft, L.G. (1990), ‘Real-Time Dynamic Control of an Industrial Manipulator Using a Neural-Network-Based Learning Controller’, IEEE Transactions on Robotics and Automation, 6(1), 1–9. Mills, J.K., and Goldenbery, A.A. (1988), ‘Global Connective Stability of a Class of Robot Manipulators’, Automatica, 24(6), 835–839. Ouyang, P.R., Dam, T., Huang, J., and Zhang, W.J. (2012), ‘Contour Tracking Control in Position Domain’, Mechatronics, 22(7), 934–944. Ouyang, P.R., Zhang, W.J., and Wu, F.X. (2002), ‘Nonlinear PD Control for Trajectory Tracking With Consideration of the Design for Control Methodology’, Proceedings of 2002 IEEE ICRA, 4126–4131. P´erez, C., Reinoso, O., Garc´ıa, N., Sabater, J.M., and Gracia, L. (2006), ‘Modelling of a Direct Drive Industrial Robot’, World Academy of Science, Engineering and Technology, 24, 109– 114. Qu, Z., and Dorsey, J. (1991), ‘Robust PID Control of Robots’, International Journal of Robotic and Automation, 6(4), 228– 235. Radke, F., and Isermann, R. (1987), ‘A Parameter-Adaptive PIDController With Stepwise Parameter Optimization’, Automatica, 23(4), 449–457. Rocco, P. (1996), ‘Stability of PID Control for Industrial Robot Arms’, IEEE Transactions On Robotics and Automation, 12(4), 606–614. Spong, M.W., Hutchinson, S., and Vidyasagar, M. (2006), Robot Modeling and Control, Hoboken, NJ: John Wiley. Tzafestas, S., and Papanikolopoulos, N.P. (1990), ‘Incremental Fuzzy Expert PID Control’, IEEE Transactions on Industrial Electronics, 37(5), 365–371. Visioli, A., and Legnani, G. (2002), ‘On the Trajectory Tracking Control of Industrial SCARA Robot Manipulators’, IEEE Transactions Industrial Electronics, 49(1), 224–232. Zhao, D.Y., and Zou, T. (2012), ‘A Finite-Time Approach to Formation Control of Multiple Mobile Robots With Terminal Sliding Mode’, International Journal of Systems Science, 43(11), 1998–2014.
PID position domain control for contour tracking P.R. Ouyang∗ , V. Pano and T. Dam Department of Aerospace Engineering, Ryerson University, Toronto, Canada
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(Received 6 January 2012; final version received 2 February 2013) Contour error reduction for modern machining processes is an important concern in multi-axis contour tracking applications in order to ensure the quality of final products. Many control methods were developed in time domain to deal with contour tracking problems, and a proportional–derivative (PD) position domain control (PDC) was also proposed by the authors. It is well known that proportional–integral–differential (PID) control is the most popular control in applications of control theory. In this paper, a PID PDC is proposed for reducing contour tracking errors and improving contour tracking performances. To determine proper control gains, system stability analysis is conducted for the proposed PDC. Several experiments are conducted to evaluate the performance of the developed approach and are compared with the PID time domain control (TDC) and the cross-coupled control. Different control gains are used in the simulations to explore the robustness of PID PDC. Comparison results demonstrate the effectiveness and good contour performances of PID PDC for contour tracking applications. Keywords: position domain; PID control; contour tracking; contour error; stability
1. Introduction In manufacturing processes, one of the most important issues is the reduction of machining errors to ensure the quality of final products. To achieve this goal, a good control system is required in a machining system. Despite the development of various advanced control algorithms in industry and academia over past a few decades, it is well known that proportional–integral–differential (PID) control (Radke and Isermann 1987; Mills and Goldenbery 1988; Qu and Dorsey 1991; Rocco 1996; Visioli and Legnani 2002; Astrom and Hagglund 2006; Jin, Yang, and Chang 2013) is the most popular one in applications of control theory. Due to its simple structure, easy implementation and robust operation, PID control is widely used for the following applications: process control, robot manipulations, motor drives, automotive control, flight control, instrumentation operations, etc. Most industrial robot manipulators are equipped with PID controllers that do not require information about the robot dynamics in their control laws. A linear and decoupled PID control with appropriate control gains can achieve the desired position without steady-state error. That is the main reason for the wide applications of PID control. It is thought that PID control is the best solution from an industrial point of view (Khalil 2002). Many studies have been conducted for stability analysis of PID control using different approaches (Qu and Dorsey 1991; Mills and Goldenbery 1988; Rocco 1996).
∗
Corresponding author. Email: [email protected]
C 2013 Taylor & Francis
There are two different control tasks in applications of robot manipulators: one is set-point (position) control that does not specify the path of the end-effector, and the other is trajectory tracking control that requires a robot manipulator following a specifically defined path. According to the controlled target, robot control can be classified into two different categories: one is in the joint level, and the other is in the end-effect or task space level. For the purpose of control, it should be noted that the task space control should be mapped into the joint level control of the actuators through inverse kinematic analysis. A contour error is defined as the error between the desired contour and the real contour in an orthogonal direction. The contour error is an important index to measure the quality of a machined product or the contour tracking performance. In a conventional approach, for example, the decoupled PID control, the contour error is controlled and improved by a tracking controller for each individual axis. A common method to improve the contour accuracy is to achieve high tracking accuracy of each individual axis. Many efforts have been made to improve the tracking performance through developing many advanced control systems, such as adaptive control (Radke and Isermann 1987), robust control (Jin et al. 2013), sliding mode control (Zhao and Zou 2012), fuzzy logic control (Tzafestas and Papanikolopoulos 1990), neural network control (Miller, Hewes, Glanz, and Kraft 1990), iterative learning control (Bouakrif, Boukhetala, and Boudjea 2013), etc.
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It should be noted that a good tracking performance for each individual axis does not guarantee the reduction of contour errors for a multi-axis motion system, as poor synchronisations of relevant motion axes result in diminished contour accuracy of the contour tracking (Koren 1980; Fang and Chen 2002; Barton and Alleyne 2008; Hu, Yao, and Wang 2009). To solve this problem, a so-called crosscoupled control (CCC) was proposed by Koren (1980) and extensively discussed in Fang and Chen (2002), Barton and Alleyne (2008) and Hu et al. 2009) for further improvements. Actually, the majority of CCC is a combination of proportional–derivative (PD)/PID control for individual axis and a coupled error feedback control for multiple axes. In our previous work (Ouyang, Dam, Huang, and Zhang 2012), a PD-type contour tracking control law established in position domain was proposed, focusing on contour tracking performance improvement. In order to further improve the contour accuracy, a PID contour tracking control in position domain is developed in this paper. The stability analysis is conducted using the Lyapunov method, and the principle for selecting control gains in position domain is provided based on the stability analysis. After that, some complex contour tracking problems are examined and compared with the traditional PID control and CCC in time domain. Simulation results demonstrate the effectiveness and high contour performances of the proposed PID position domain control (PDC).
Figure 1.
A 2-DOF robot.
is demonstrated that a global stability of trajectory tracking in each axis can be achieved (Mills and Goldenbery 1988; Qu and Dorsey 1991; Rocco 1996) under PID feedback control, as shown in Equation (2):
2. PID PDC and contour error 2.1. System description and dynamic model in time domain and position domain In this paper, we use a 2-DOF (degree of freedom) Cartesian robot (P´erez, Reinoso, Garc´ıa, Sabater, and Gracia 2006; Spong, Hutchinson, and Vidyasagar 2006), shown in Figure 1, as an example of a multi-axis motion system to discuss the contour tracking control problem in position domain. The dynamic model including the mechanical system and the actuator for each axis can be expressed as a general second-order differential equation (P´erez et al., 2006; Spong et al. 2006) as follows:
mx x¨ + cx x˙ + kx x + fx = Fx , my y¨ + cy y˙ + ky y + fy = Fy
(1)
˙ x¨ and y¨ are position, velocity, where x and y, x˙ and y, and acceleration of the X-axis and the Y-axis, respectively. mi , ci , and ki are the mass, damping, and stiffness of the ith axis motion, fx and fy are the disturbances and uncertainties of the system model, and Fx and Fy are the control input forces, respectively. From Equation (1), one can see that the dynamic model of this Cartesian robot for these two axes is decoupled and PID control can be applied for the control of each axis. It
Fx Fy
=
Kpx 0 0 Kpy
+ +
Kix 0 0 Kiy
Kdx 0 0 Kdy
ex ey ⎛ t ⎜ 0 ⎜ ⎝ t
⎞ ex (t)dt
ey (t)dt e˙x , e˙y
⎟ ⎟ ⎠
0
(2)
where ex , ey and e˙x , e˙y are the tracking errors (the desired value – the real value) and the derivative of tracking errors, respectively. Kpj , Kij , and Kdj are the proportional gain, integral gain, and derivative gain, respectively. To develop PID PDC, we assume that the X-axis is a reference (master) axis and the Y-axis is the serve (slave) axis. The Y-axis dynamic model can be rewritten related to the X-axis position. In this paper, for simplicity, we assume that the X-axis motion direction is unchanged for a controlled contour segment. Without loss of generality, we assume that the X-axis motion is in the positive direction with a monotonically increasing function of time that means x˙ ≥ 0. Also, the X-axis position is sampled equidistantly by a sensor such as an encoder. It should be noted that the developed new control law also can be applied to x˙ ≤ 0, as shown in the following simulations.
International Journal of Systems Science If we define a space velocity (the first derivative) and a space acceleration (the second derivative) of y with respect to variable x (the master motion) as: dy , dx d 2y dy y = = , dx dx 2
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y =
(3) (4)
where y is called the relative position domain velocity of the Y-axis with respect to the X-axis, and y is the relative position domain acceleration, as both are defined in position domain with respect to the X-axis. It is clear that y is the tangent of a tracking contour at a point of the reference x or the contour motion direction. To develop the dynamic model in position domain, a relationship between the absolute velocity and the relative velocity for the Y-axis motion can be expressed as: y˙ =
dy dx dy ˙ = = y x, dt dx dt
(5)
where x˙ > 0. Similarly, a relationship between the absolute acceleration and the relative acceleration can be obtained: dy d x˙ d y˙ ¨ . = x˙ + y = x˙ 2 y + xy y¨ = dt dt dt
(6)
Applying Equations (5) and (6) to the original dynamic model, Equation (1), the dynamic model for the Y-axis motion in Equation (1) can be rewritten in position domain as (Ouyang et al. 2012):
3
But there are some significant differences. First, these two control laws are defined in two different domains. Second, the derivative gains have different meanings because of the differences between ey (x)and e˙ (t). Remark 2: The tracking error, Equation (9), forms the contour error in position domain where the X-axis is the reference axis with zero bias. That is, ec (x) = Cy ey (x) for PDC. According to the definition of y (x) in Equation (3), there is no need to get the velocity information for the X-axis. Only the position information of the X-axis is needed in order to define the contour tracking error. As the contour tracking error and the contour tracking error derivative in Equation (9) are defined in position domain; therefore, we call the new developed PID control as a PID PDC. Applying Equations (8) and (9) to Equation (7), the dynamic model based on PID PDC can be expressed as ˙ (x) + ky y (x) + fy (x) my x˙ 2 y (x) + (my x¨ + cy x)y x (yd (s) − y (s))ds = Kpy (yd (x) − y (x)) + Kiy + Kdy (yd (x) − y (x)).
0
(10)
Examine the control law for the Y-axis motion in Equation (8). Substituting Equation (5) in Equation (8), we have the following equation:
x
ey (x)ds + Kdy ey (x) x (yd (s) − y (s))ds = Kpy (yd (x) − y (x)) + Kiy
Fy (x) = Kpy ey (x) + Kiy
0
0
¨ (x) + cy xy ˙ (x) + ky y (x) my x˙ 2 y (x) + my xy + fy = Fy (x) , (7) where Fy (x) is the control input described in position domain. 2.2. PID PDC For the position domain dynamic model expressed in Equation (7), we propose a new PID PDC law as follows: Fy (x) = Kpy ey (x) + Kiy 0
x
ey (s) ds + Kdy ey (x). (8)
The contour tracking error and its relative derivative in position domain are defined as:
ey (x) = yd (x) − y (x) . ey (x) = yd (x) − y (x)
Kdy e˙y (x) + x˙
(11)
Remark 3: It is clearly shown in Equation (11) that a position domain linear PID control is equivalent to a nonlinear PID control in time domain (Ouyang, Zhang, and Wu 2002) when the speed of motion in the X-axis is not constant. It should be noted that the transformation of dynamics from time domain to position domain is a one-to-one nonlinear mapping. The dynamic model in Equation (7) described in position domain is equivalent to the dynamic model, Equation (1), in time domain. Therefore, the proposed PID control in position domain has the same stability property as the PID control developed in time domain. It means that the developed PID PDC is stable for contour control of the robotic system. A detailed discussion about the stability analysis will be presented in the next section.
(9)
Remark 1: The PID PDC law in Equation (8) is similar in formula to the traditional PID control law in Equation (2).
2.3. CCC and contour error In this paper, to demonstrate the effectiveness and success of PID PDC as compared with time domain control (TDC)
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P.R. Ouyang et al. 3. Stability analysis 3.1. Preparation and Lemma Consider a dynamic system described in position domain by y (x) = f (x, y (x)),
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Figure 2.
where x ∈ R is the ‘position’ of the master motion or the independent variable, and y (x) ∈ R n is the state.
Tracking and contour errors.
methods, CCC applied in time domain is also used and its control laws are selected (Koren 1980; Fang and Chen 2002; Barton and Alleyne 2008; Hu et al. 2009) as: t Fcx = Kpx ex +Kix ex ds + Kdx e˙x −Cx (Kpc ec + Kdc e˙c ) 0 , t Fcy = Kpy ey +Kiy ey ds + Kdy e˙y + Cy (Kpc ec + Kdc e˙c ) 0
ec = −Cx ex + Cy ey , e˙c = −Cx e˙x + Cy e˙y
(13)
where Cx = sin θ, Cy = cos θ . θ is the angle between the X-axis and the desired line measured in a counter-clockwise direction. For a circular motion, the contour errors are defined as (Koren 1980; Fang and Chen 2002; Barton and Alleyne 2008; Hu et al. 2009):
ec = −Cx ex + Cy ey , e˙c = −Cx e˙x + Cy e˙y − C˙ x ex + C˙ y ey
Lemma 1: Let D ⊂ R n be a domain that contains the origin and V: [0, ∞) × D → R be a continuously differentiable function such that γ1 ( y ) ≤ V (y) ≤ γ2 ( y ), V (y) ≤ −W (y),
(16)
∀ y ≥ μ > 0, ∀ x ≥ 0, ∀ y ∈ D, (17)
(12)
where ec and e˙c are the contour error and its derivative, respectively. Contour error is defined as the shortest distance from the actual position to the desired contour. The difference between the tracking error and contour error can be identified through Figure 2. From this figure, one can see that the contour error is less than the tracking error. For a linear motion, the contour errors are defined as (Koren 1980; Fang and Chen 2002; Barton and Alleyne 2008; Hu et al. 2009):
(15)
(14)
where Cx = sin θ − ex /2R, Cy = cos θ + ey /2R, C˙ x = θ˙ cos θ − e˙x /2R, C˙ y = −θ˙ sin θ + e˙y /2R, and R is the radius of the tracked circular motion. It should be mentioned that the angle θ is not constant in a circular contour motion. In summary, we developed a PID PDC for contour tracking of a Cartesian robot in this section. In the following sections, we will fulfil the stability analysis and conduct some simulation tests to demonstrate the effectiveness of the proposed PID PDC based on the comparison with PID TDC and CCC.
where γ1 and γ2 are class K functions and W (y) is a continuous positive definite function. Take r > 0 such that Br ⊂ D, and suppose that μ < γ2−1 (γ1 (r)) .
(18)
Then, there exists a class KL function φ, and for every initial state y(x0 ) satisfying y(x0 ) ≤ γ2−1 (γ1 (r)), there is X ≥ 0 such that the solution of dynamic equation satisfies y(x) ≤ φ( y(x0 ) , x − x0 ), y(x) ≤ γ1−1 (γ2 (μ)),
∀ x0 ≤ x ≤ x0 + X, (19) ∀x ≥ x0 + X.
(20)
Moreover, if D = R n and γ1 belongs to class K∞ , then Equations (19) and (20) hold for any initial statey (x0 ), with no restriction on how large μ is. Proof 1: See reference Khalil (2002).
This lemma proves that the dynamic system is globally uniformly exponentially convergent to a closed ball for any initial conditions if one can find a positive definite function V(y) so that Equations (16) and (17) hold. 3.2. Theorem To facilitate the discussion, we have the following assumptions: (A1). The desired contour of yd (x)is of the third-order continuity for x ∈ [xini , xf in ]. (A2). The real tracking path in the X-axis x (t) is of the third-order continuity for the contour. (A3). The disturbance fy (x)is bounded in the full contour tracking process.
International Journal of Systems Science For the briefness of the stability analysis, some notations are introduced and used in the following sections: z = max |zi |, 1≤i≤n
ρ=
my x˙ 2 yd (x)
˙ d (x) + ky yd (x) + fy , + (my x¨ + cy x)y
|ρ|max
= my x˙ 2 yd (x)+ my x¨ +cy x˙ yd (x)+ky yd (x) + fy max ¨ + cy x˙ 2 yd (x) , ≤ my x˙ 2 yd (x) + my x + ky yd (x) + fy
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my x¨ + cy x˙ Kiy α ρe = Kpy + k y + − − , and 2 β 2β
β α, my x¨ + cy x˙ − βmy x˙ 2 − ρe = Kdy + 1 + 2 2 where α and β are positive constants. According to the assumptions (A1∼A3), one can prove that the parameter |ρ|max is bounded. Parameter ρe is related to Kdy ,while ρe is related to all other two control gains. For the developed position domain PID controller, Equation (8), we have the following theorem: Theorem 1: A robot system represented in position domain by Equation (7), where the desired contour shape satisfies assumptions A1 and A2, is controlled by the proposed PID PDC law in Equation (8). The contour tracking error and its relative derivative are bounded, and the boundednesses are given by ⎧ ⎪ ⎪ ⎪ e ≤ 2 1 + 1 |ρ|max , ⎪ ⎪ ⎨ y ρe2 βρe ρe . ⎪ ⎪ β 1 ⎪ ⎪ ⎪ ⎩ ey ≤ 2 ρ ρ + ρ 2 |ρ|max e e e
for ρe and ρe , and the better the contour tracking performance, which will be demonstrated through the following simulation study. Specifically, the derivative gain Kdy determines the parameter ρe that is dominantly controlling the boundary value of the relative derivative error ey , and the proportional–integral (PI) gains determine the parameter ρe that is dominantly controlling the boundary value of the tracking error ey . Basically, to determine the control gains, first we choose these two constants α and β, then determine the maximum value of control gain Kiy , followed by the selection of Kpy and Kdy . As there is no restriction for the selection of α and β; therefore, the selection of three PID control gains is very easy and simple.
3.3. Stability proof A stability analysis for the proposed PID PDC is conducted based on the Lyapunov function method. First, we define ⎧ ⎨
σy (x) =
x
ey (s)ds . ⎩ σ (x) = e 0 (x) y
(23)
y
Using Equation (9), the dynamic model with position domain PID controller in Equation (10) can be re-described in an error function format as follows:
my x˙ 2 ey (x) + my x¨ + cy x˙ + Kdy ey (x) + Kiy σy (x)
+ Kpy + ky ey (x) = ρ. (24) For the dynamic system described in position domain, we define the following two functions:
(21)
Provided that the control gains and the positive constant parameters are selected properly such that ⎧ α>β>0 ⎪ ⎪ ⎪ ⎪ ¨ + cy x ) ˙ ≥0 = α − β(my x ⎪ ⎪ ⎪ ⎪ ⎨ ˙ 2 Kpy > α + + βmy x , 2 ⎪ ⎪ K < αβ iy ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ K > + m x ˙ + βmy x ˙ 2 dy y ¨ + cy x 2
5
(22)
it is noted that Equation (22) provides some guidelines about the choices of control gains for the developed PID PDC. In order to reduce the tracking errors, we need to increase the values of two parameters: ρe and ρe . These two parameters are related to the PID control gains. Generally speaking, the larger the control gains, the larger the values
⎧ 1
Kpy + ky βmy x˙ 2 ⎪ ⎪ ey e V1 ey (x) , ey (x) = ⎪ ⎪ βmy x˙ 2 my x˙ 2 2⎛ ⎞y ⎪ ⎪ ⎪ ⎪ ey ⎪ ⎪ ⎪ ⎨ × ⎝ e ⎠ y
⎪ ⎪ ⎪ 1 ey α Kiy ⎪ ⎪ V2 (ey (x), σy (x)) = ( ey σy ) ⎪ ⎪ K βK σ 2 iy iy y ⎪ ⎪ ⎪ β ⎪ 2 ⎩ + Kdy ey 2
.
(25) Based on Equation (25), we define the Lyapunov function as V (ey (x), ey (x), σy (x)) = V1 (ey (x) , ey (x)) + V2 (ey (x) , σy (x)).
(26)
If the control gains are properly chosen according to Equation (22), the following inequality holds: ˙ 2 > β 2 my x˙ 2 − ky . Kpy > α + βmy x
(27)
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According to Equation (27), we can prove that V1 ey (x) , ey (x) is a positive definite function. Also, from Equation (22), we have Kiy < αβ, and it is easy to prove that V2 ey (x) , σy (x) is also a positive definite function according to Sylvester’s criterion. Therefore, the Lyapunov function V ey (x) , ey (x) , σy (x) in Equation (26) is a positive definite function. It means that V ≥ 0. It is easy to demonstrate that
According to Equation (22), we have α > ˙ ≥ β(my x¨ + cy x). ˙ ¨ + cy x ) ˙ ≥ β( my x¨ + cy x ) β(my x From Equation (28), we can prove that
⎧ 1 ⎪ ⎨ ab ≥ − (a 2 + b2 ) 2 . 1 ⎪ ⎩ ab ≤ (a 2 + b2 ) 2
Applying Equation (32) to Equation (31), we have (28)
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Applying Equation (28) to Equation (25) and (26), the following inequalities can be obtained: 1 1 (Kpy + ky − βmy x˙ 2 )ey2 + (1 − β)my x˙ 2 ey2 2 2 + V2 (ey (x), σy (x)). ≤ V (ey (x), ey (x), σy (x)) 1 1 ≤ (Kpy + ky + βmy x˙ 2 )ey2 + (1 + β)my x˙ 2 ey2 2 2 + V2 (ey (x), σy (x)) (29)
˙ y e y (α − β(my x¨ + cy x))e 1 2 2 ˙ ≤ (α − β(my x¨ + cy x))(e y + ey ). 2
(32)
β α my x¨ + cy x˙ − βmy x˙ 2 − V ≤ − Kdy + 1 + 2 2 ¨ ˙ x + c x K m α y y iy − − × ey2 − β Kpy + k y + 2 β 2β
2 × ey + ey + βey ρ. (33) From Equation (22), if we choose Kpy > α + ¨ and Kiy < αβ, one can prove that βmy x
2β
+
Kiy Kiy my x¨ + cy x˙ α − − >α− 2 β 2β β ¨ + βmy x ¨ ≥ ky + ky + +βmy x ¨ > 0. (34) +βmy x
ρe = Kpy + k y +
From Equation (29), we can see that the defined Lyapunov function satisfies Equation (16). In PDC, the reference position x of the X-axis motion is an independent variable that has a similar meaning of t in time domain. ey and ey are functions of the independent variable x. Therefore, the derivative of the Lyapunov function V is related to variable x in this stability analysis. Rewritting Equation (24), we have
Similarly, from Equation (22), if we choose Kdy > 2 + ¨ + cy x ˙ + βmy x ˙ 2 , then we have my x β α my x¨ + cy x˙ − βmy x˙ 2 − > 0. ρe = Kdy + 1 + 2 2 (35)
my x˙ 2 ey (x) = ρ − (my x¨ + cy x˙ + Kdy )ey (x) − Kiy σy (x)
Applying Equations (34) and (35) to Equation (33), we obtain
− (Kpy + ky )ey (x).
(30)
Differentiating Equation (26) with respect to the variable x and using Equation (30), we have Kpy + ky βmy x˙ 2 ey V = ey βmy x˙ 2 my x˙ 2
α Kiy ey + βKdy ey ey . + ey ey Kiy βKiy σy
ey
ey
= (Kpy + ky )ey ey + (ey + βey )my x˙ 2 ey + βmy x˙ 2 ey2 + (βKdy + α)ey ey + Kiy ey σy + βKiy ey σy + Kiy ey2 = (Kpy + ky + α)ey ey + βmy x˙ 2 ey2 + (ey + βey )Kiy σy
+ Kiy ey2 + (ey + βey ) ρ − (my x¨ + cy x˙ + Kdy )ey −(Kpy + ky )ey − Kiy σy
= − my x¨ + cy x˙ + Kdy − βmy x˙ 2 ey2
− β Kpy + ky − Kiy ey2
+ α − β my x¨ + cy x˙ ey ey + ey + βey ρ (31)
V ≤ − βρe ey2 − ρe ey2 + βey + ey ρ ≤ −βρe ey2
+ βey |ρ|max − ρe ey2 + ey |ρ|max .
(36)
Applying another inequality: az − bz2 ≤ we have ⎧ ⎪ ⎪ ⎨ β |ρ|
a2 1 − bz2 b 4
for
a>0
and
b > 0, (37)
βρe 2 β |ρ|2max e + 4 y ρe . ρe 2 |ρ|2max ⎪ ⎪ 2 ⎩ |ρ|max ey − ρe ey ≤ − ey + 4 ρe 2 max ey − βρe ey ≤ −
(38)
Applying Equation (38) to Equation (36), we get V ≤ −
βρe 2 ρe 2 e − e + 4 y 4 y
β 1 + ρe ρe
|ρ|2max .
(39)
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Figure 3. Contour tracking results for a zigzag motion. (a) Tracking errors for the zigzag motion. (b) Desired and actual motions with 10 × amplified errors.
Therefore, According to Lemma 1, we can demonstrate that both the contour tracking error and the derivative of the contour tracking error are bounded as follows: ⎧ ⎪ 1 1 ⎪ ⎪ ⎪ |ρ|max e ≤ 2 + ⎪ ⎨ y ρe2 βρe ρe . ⎪ ⎪ β 1 ⎪ ⎪ e ≤ 2 + 2 |ρ|max ⎪ ⎩ y ρe ρe ρe
motion contours. It is assumed that the 2-DOF robotic system has the following parameters: mx = 10 kg, cx = 10 N · s/m, kx = 20 N/m my = 5 kg, cy = 10 N · s/m, ky = 20 N/m.
(40)
According to Equation (40), one can see that the contour error and its derivative are bounded. From Equation (40), it is also shown that the maximum errors can be reduced to very small values by increasing control gains Kpy and Kiy (related to ρe ), and Kdy (related to ρe ). Therefore, the tracking errors will be reduced by choosing high PID control gains according to Equation (40).
4. Simulation tests In this section, two different types of contour tracking problems are simulated: linear motion contours and nonlinear
4.1. Linear motion contour tracking In the first two simulation examples, linear motion contours with positive and negative speed of the master motions are considered to verify the effectiveness of the proposed PID PDC for contour tracking.
4.1.1. Zigzag motion contour tracking In this simulation, a zigzag linear motion contour is tracked for 8 seconds, where x ∈ [0, 4] and y ∈ [0, 2]. To define the trajectories in two axes controlled in time domain, we assume that each linear motion segment for the zigzag shape is defined by a fifth-order polynomial (Spong et al. 2006) in 2 seconds. In PDC, we assume that the X-axis is controlled by a traditional PID control, and we measured its
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displacement by a sensor, and used it as a reference for the Y-axis PDC. According to the trajectory planning method, we have x ˙ < 1 m/s, and x ¨ < 1 m/s2 . If we chooseα = 1000 and β = 10, according to Equation (22), the control gains for the proposed PID PDC can be estimated as follows:
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⎧ ⎨ Kpy > 1975 Kiy < 10, 000 . ⎩ Kdy > 490 Of course, as there is no restriction for the selection of parameters α and β, the control gains can be easily determined based on Equation (22) and selected to some other values. For each segment, as the motion range in the Y-axis is double that in the X-axis, we set control gains for the X-axis to be half of the control gains used for the Y-axis. Therefore, the following control gains are chosen for time domain and position domain controllers for all simulation examples: Kpx = 2000, Kpy = 4000, Kpc = 2000,
Kdx = 1200, Kdy = 2400, Kdc = 1200
Kix = 2000 Kiy = 4000 .
For the zigzag motion, the motion in the X-axis is in one direction, with positive speeds from one segment to another segment. Figure 3 shows the contour tracking performances for PID TDC, CCC, and the proposed PID PDC, respectively. On the tracking error side, it is clearly shown that TDC in the Y-axis and PDC obtained good tracking results, whereas CCC obtained acceptable result. The tracking performance for TDC in the X-axis is the worst. From the amplified contour tracking performances shown in Figure 3(b), it can be seen that PDC obtained the best contour performance and demonstrated the effectiveness of the proposed PDC. On the contour tracking error side, Figure 4 shows the contour errors for three different control methods. It shows the significant reductions in the contour errors based
Table 1.
Figure 4.
Comparison of contour errors for zigzag motion.
on PDC as compared with TDC and CCC. Comparing Figure 4 with Figure 3, one can see that the contour errors for these three control methods are less than the maximum tracking errors. Such results are expected, as the contour error is defined as the shortest distance between the actual position and the desired contour. Another observation from Figure 4 is that the contour errors controlled by TDC and CCC are discontinuous at the connecting points of the segments. The actual positions for both these controllers are behind the desired positions shown in Figure 3. The actual positions are above the desired contours for the odd segments that produce negative contour errors, and the actual positions are below the desired contours for the even segments that obtain positive contour errors. Therefore, there is a sign change at the connecting points that makes the contour errors discontinuous. For the contour tracking based on PDC, all the actual positions are below the desired contour, so there is no sign change for the contour error that makes the contour errors smooth.
Zigzag contour performance comparison under different control gains. TDC (×10−3 )
Factor_p = 0.5 Factor_p = 1.0 Factor_p = 1.5 Factor_i = 0.5 Factor_i = 1.0 Factor_i = 1.5 Factor_d = 0.5 Factor_d = 1.0 Factor_d = 1.5
CCC (×10−3 )
PDC (×10−3 )
Max.
Min.
Max.
Min.
Max.
Min.
17.46 12.60 10.66 18.65 12.60 9.65 13.00 12.60 11.84
–8.96 –8.97 –7.55 –13.14 –8.97 –6.02 –9.09 –8.97 –8.64
12.79 10.15 8.69 14.44 10.15 8.00 10.87 10.15 9.80
–6.17 –5.71 –5.04 –7.55 –5.71 –5.18 –7.08 –5.71 –4.92
3.53 2.57 1.94 2.82 2.57 2.35 2.98 2.57 2.19
–2.30 –0.94 –0.57 –0.13 –0.94 –1.48 –1.77 –0.94 –0.28
International Journal of Systems Science
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tracking errors, Figure 6(b) shows the actual contours based on these three control methods, while Figure 6(c) shows the contour errors. It can be seen from Figure 6(a) that the tracking errors of the Y-axis for all three controllers are almost at the same level. But the tracking errors in the Xaxis for TDC are little higher than those for CCC. TDC is the worst control in terms of tracking errors and contour errors; CCC and PDC have the same level of tracking errors, but have significant differences in contour errors, as shown in Figure 6(c). It is shown in Figure 6(c) that PDC ensures the best contour tracking performance. One can see that the maximum contour error for PDC is about one-third of that for TDC and half of the maximum contour error for CCC. It also demonstrates that CCC has some advantages compared with TDC for linear motion where the angle for calculating the contour errors in each segment is constant. Figure 5. Comparison of contour errors between PID and PD control for zigzag motion.
To examine the effects of control gains on contour tracking performances, different factors on PID control gains selected before are used in the simulations, and the maximum and minimum contour errors are obtained and listed in Table 1. From this table, one can see that the larger the control gains, the lesser the contour tracking errors. Also, it is shown that the proportional and derivative gains have large contributions to the final contour tracking performances for all three control methods, especially the proportional gain. Overall, PDC has the lowest contour tracking errors. To demonstrate the effectiveness of the proposed PID PDC, a comparison study is also conducted with PD PDC. Figure 5 presents the contour errors controlled by PD and PID PDCs. It shows the better contour performance controlled by PID PDC. From this figure, one can see that the contour errors are distributed on the positive side controlled by PD PDC, while the contour errors are distributed on both sides controlled by PID PDC. Such a result is contributed by the effect of integral control.
4.1.2. Diamond contour tracking A desired diamond contour is required to track for x ∈ [0, 2] and y ∈ [−2, 2]. For TDC and CCC, the time duration for tracking the diamond contour in both axes is assumed to be 8 seconds: 4 seconds for the positive motions of the X-axis and 4 seconds for the negative motion. This experiment demonstrates the effectiveness of PDC for both positive and negative speed cases in the X-axis. In this simulation study, the control gains for three different control algorithms are chosen the same as those in zigzag motion. Figure 6 shows the tracking performances under three different control methods where Figure 6(a) depicts the
4.2. Nonlinear motion contour tracking In the previous section, simulation results show that PDC is better than the controllers based on time domain for linear motions. In this part, we use some nonlinear motion contour tracking examples to verify the effectiveness of PDC. 4.2.1. Circular motion contour tracking A full circular contour with radius of 0.5 m is simulated using three different control methods. For TDC and CCC, the time durations for path tracking are assumed to be 10 seconds. The control gains for both axes are selected the same as the control gains for the Y-axis used in the linear motion contour tracking examples. It should be noted that for the full circular contour tracking problem, the speed of the X-axis is positive for the first half-circle and negative for the second half-circle. Figure 7 shows the tracking errors and contour tracking results using three different control methods. From this figure, specifically for the amplified actual contour shown in Figure 7(b), one can see that PDC obtained the best contour tracking result, CCC obtained a good contour tracking result, and TDC got the worst tracking result. Such a conclusion is the same as for the linear motion contour tracking. For the circular contour tracking problem, to examine the robustness of the proposed PDC, two speed cases are simulated. The low-speed case is the case where the time duration is 10 seconds, while the high-speed case is when the time duration is reduced by half to 5 seconds for both the X-axis and the Y-axis. The same control gains are used for both cases in the three control methods. The simulation result of contour tracking errors is shown in Figure 8. From Figure 8, one can see that PDC is ensured the smallest contour tracking errors for both low-speed and high-speed motions. Table 2 shows the maximum and minimum contour tracking errors under different gain factor conditions. It still
P.R. Ouyang et al.
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Figure 6. Contour tracking control results for a diamond contour. (a) Tracking errors under different control laws. (b) Actual contour tracking results for a diamond shape. (c) Contour errors based on different control laws.
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Figure 7. Contour tracking control results for a circular contour. (a) Tracking errors for a circular contour based on three control methods. (b) Actual contour tracking results for a circular contour (10×).
demonstrates that the larger the control gains, the smaller the contour errors for all three control methods, and the best tracking performance is obtained by PDC.
Figure 8.
In Table 2, the maximum contour errors controlled by PD-based controllers are also included when we set Factor_i = 0 for all three controllers. Comparing PID PDC
Contour tracking errors for a full circular motion. (a) Low-speed case. (b) High-speed case.
P.R. Ouyang et al.
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Figure 9. Contour tracking results for an ellipse motion. (a) Tracking errors for an ellipse contour. (b). Contour tracking results with amplified tracking errors (10×). (c) Contour tracking errors for an ellipse motion contour.
with PD PDC, one can see that the contour performance controlled by PID PDC is better than that controlled by PD PDC. Also, it shows the improvement in contour tracking with the increase in integral gain in PID PDC. 4.2.2. Ellipse contour tracking Finally, an ellipse motion contour tracking example is simulated using the same control gains listed before and the
results are shown in Figure 9, where the tracking errors are shown in Figure 9(a), the amplified contour errors are presented in Figure 9(b), and the contour tracking errors are shown in Figure 9(c) for the three different control methods. From Figure 9(a), it can be clearly seen the best tracking performance is by PDC. From Figure 9(b), it can be clearly seen that the best contour tracking performance is by PDC and the worst contour tracking performance is by TDC. From Figure 9(c), one can see that the biggest contour
International Journal of Systems Science Table 2.
Circular contour performance comparison under different control gains. TDC (×10−4 )
Factor_p = 0.5 Factor_p = 1.0 Factor_p = 1.5 Factor_i = 0 Factor_i = 0.5 Factor_i = 1.0 Factor_i = 1.5 Factor_d = 0.5 Factor_d = 1.0 Factor_d = 1.5
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CCC (×10−4 )
PDC (×10−4 )
Max.
Min.
Max.
Min.
Max.
Min.
23.74 14.23 10.21 20.28 16.71 14.23 12.66 15.24 14.23 13.37
–57.38 –24.64 –15.25 –23.93 –24.57 –24.64 –22.77 –26.41 –24.64 –21.52
14.92 10.19 7.30 14.61 11.92 10.19 9.66 10.52 10.19 9.16
–32.77 –15.12 –10.35 –18.54 –19.08 –15.12 –10.98 –14.12 –15.12 –15.55
7.95 9.11 8.33 8.49 8.89 9.11 9.14 14.79 9.11 5.30
–14.99 –11.68 –9.52 –12.13 –11.90 –11.68 –11.54 –15.28 –11.68 –9.32
there is no tracking error for the master motion. Only the slave motion tracking errors will affect the final contour tracking performance. The guideline for determining control gains is set up through the stability analysis. It is demonstrated that PID PDC is stable and can obtain very good contour tracking performance. The new developed PD PDC is applied to a robotic system for improving the contour tracking performance. Linear and nonlinear motion contours are used to verify the effectiveness of the new controller through comparison with PID TDC and CCC. Also, it is demonstrated that PID PDC is better than its PD counterpart. More advanced control in position domain for multiple DOF system control would be a future research direction.
Figure 10. Comparison of contour errors between PID and PD control for an ellipse motion.
error controlled by PDC is about half of that by CCC and one-third of that by TDC. Figure 10 shows the comparison results of contour tracking errors controlled by PD and PID in position domain. Once again, it demonstrates that PID PDC is better than PD PDC in terms of reduction in contour errors.
5. Conclusions Many machining applications involve the generation of complex profiles and need to synchronise the motions of different axes in order to obtain good contour tracking performance. In this paper, a PID PDC is proposed for contour tracking control as an alternative to traditional PID TDC. The proposed PDC adopts the master–slave synchronised control concept. In the developed PDC, the master motion is sampled equidistantly and used as a reference; therefore,
Acknowledgement This research is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) through a Discovery Grant.
Notes on contributors Puren Ouyang is an assistant professor in the Department of Aerospace Engineering at Ryerson University, Toronto, Canada. He has been working in the areas of robotics and control, mechatronics, and precision manipulator and devices for more than 20 years. Dr Ouyang has also specialised in robotic systems with compliant mechanisms. He has developed a novel position domain control method that can significantly improve the contour tracking performance as compared with time domain control methods. This research was recognised in an IEEE conference in 2011 through the best paper award. In the robotics and control area, he has developed a series of robust and adaptive online learning control methods. Dr Ouyang has published more than 40 refereed journal papers and 30 conference articles.
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P.R. Ouyang et al. Vangjel Pano received his Bachelor’s degree in Aerospace Engineering from Ryerson University in Canada in 2011. Currently, he is studying as an MASc student in the Aerospace Engineering Department of Ryerson University. His research interests include robotics and advanced control.
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Truong Dam graduated from Ryerson University in Toronto, Canada. He received his BEng and MSc for aerospace engineering in 2010 and 2012, respectively. He is currently employed in the aerospace sector.
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