Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference Shanghai, P.R. China, December 16-18, 2009
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Coordination control of parallel manipulators with actuation redundancy Weiwei Shang, Shuang Cong, and Shilong Jiang
Abstract—A coordination controller is proposed to improve the coordination relation of multiple kinematic chains in the parallel manipulators with actuation redundancy. The coordination errors are defined using the tracking errors of the neighboring active joints, and the coordination controller is designed in the task space. And the coordination controller is proved to guarantee asymptotic convergence to zero of both tracking and coordination errors by the Barbalat's Lemma. The trajectory tracking experiments of an actual parallel manipulator with actuation redundancy are carried out, and then the control performances between the coordination controller and the augmented PD (APD) controller are compared.
I. INTRODUCTION
P
arallel manipulators have many advantages over serial manipulators such as high stiffness, high payload, high precision, high speed and high acceleration [1]. This has opened up broad possibilities for the use of parallel manipulators in many fields. However, the relatively limited workspace, complex torque transmission, and abundant singularities in their workspaces decline parts of above mentioned advantages. Based on the parallel manipulators with normal actuator, when the passive joints of the parallel manipulators are replaced with the active joints, or other chains with active joints are appended, then the parallel manipulators with actuation redundancy can be designed [2]. The parallel manipulator with actuation redundancy can improve the torque transmission ability [3], and eliminate the actuation singularity [4], [5]. In addition, actuation redundancy allows a lighter and faster mechanical structure that can carry the same loads to be built [6]. It also allows greater safety in case of breakdown of individual actuators [7], [8]. If the mechanism is redundantly actuated, it can still be controlled if one or more of the actuators break down. Due to the actuation redundancy, the kinematic equations of the parallel manipulators are tightly coupled, and the dynamic characteristics of the parallel manipulators are usually highly nonlinear [9]. All these difficulties make the dynamic controller design for parallel manipulators with actuation This work was supported by the Anhui Provincial Natural Science Foundation with Grant No.090412040, K. C. Wong Education Foundation of Hong Kong, and the China Postdoctoral Science Foundation funded project with Grant No. 20080440717. W. W. Shang and S. Cong are with the Department of Automation, University of Science and Technology of China, Hefei, Anhui, 230027, P. R. China (e-mail:
[email protected];
[email protected] ). S. L. Jiang is with the PKU-HKUST Shenzhen-HongKong Institution, 518057, P. R. China (email:
[email protected] ).
978-1-4244-3872-3/09/$25.00 ©2009 IEEE
redundancy a work full of challenge, which has aroused the interest of researchers in recent years. The traditional dynamic controllers include the augmented PD (APD) controller and computed-torque (CT) controller [9-11]. In order to overcome the uncertainty in dynamics and disturbance of trajectory, the adaptive controller [12], predictive controller [13], and robust controller [14] are used to the trajectory tracking of the parallel manipulators. However, when these controllers are applied to the parallel manipulators, the mechanical structure characteristics of a number of kinematic chains are not taken into account. Thus the correlation among the kinematic chains is ignored, and the trajectory tracking accuracy of the parallel manipulators is poor at high-speed [15]. So, the parallel manipulators are considered as virtual axis machine tools, and the coordination error between the virtual axes can be defined in the task space, thus the cross-coupled control technology for the motion control systems with multiple axes can be used to the coordination control of the parallel manipulators [16], [17]. The trajectory tracking accuracy can be increased with the cross-coupled control technology, but the coordination behaviors between the kinematic chains of the parallel manipulators are defined ambiguously. Especially for the parallel manipulators with actuation redundancy, considering the number of the actuators is greater than that of the task space coordinates, thus the motion due to the actuation redundancy can not be defined by the coordination error in the task space. So, in this paper, the coordination behavior will be defined in the active joint space, and this coordination behavior will be transformed into the coordination behavior in the task space. Then, the coordination controller can be designed in the task space, and the coordination control of the parallel manipulators with actuation redundancy will be implemented. The main contribution of this paper is the development of coordination controller for an actual parallel manipulator with actuation redundancy. On the assumption that the kinematic chains of the parallel manipulator is cut at the common end-effector, the manipulator can be transformed into an open-chain system including several kinematic chains, each of which contains an active joint and several passive joints. Considering the correlation of the neighboring kinematic chains, and the coordination errors can be defined using the tracking errors of the neighboring active joints. Then these coordination errors of the neighboring active joints will be transformed into the coordination errors in the task space, and
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FrC06.4 the coordination controller can be designed in the task space. The coordination controller includes three terms: dynamics compensation, friction force compensation, and combined error elimination. Then, the stability of the parallel manipulator system controlled by the coordination controller is proved with the Lyapunov theory, and the convergence of the tracking errors and coordination errors are proved by the Barbalat’s Lemma. The trajectory tracking experiments of an actual parallel manipulator with actuation redundancy is implemented, and the experiment results are compared with the traditional APD controller. Our experiment results show that, compared with the APD controller, the tracking error and the coordination error of the active joints decrease greatly using the coordination controller. The paper is organized as follows. In section II, the dynamic model of a parallel manipulator with actuation redundancy is formulated in the task space. In section III, the coordination controller is designed, and the stability of the parallel manipulator system is proven. In section IV, the trajectory tracking experiments of an actual parallel manipulator with redundant actuation are carried out, and the experiment results are compared with the APD controller. Finally, several remarks are concluded.
A3
B1
Y
q a3
q b1
B3
q a1 A1
q b3
O
B2
q b2
q a2 X
A2
Fig. 1. Coordinates of the planar parallel manipulator with actuation redundancy.
end-effector and the three active joints of the parallel manipulator; M e is the inertia matrix in the task space, and Ce is the Coriolis and centrifugal force matrix in the task space. The detailed definition of the above symbols can be found in [18].
III. COORDINATION CONTROLLER DESIGN II. DYNAMIC MODEL The experiment platform is a planar parallel manipulator with actuation redundancy. As shown in Fig. 1, a reference frame is established in the task space of the parallel manipulator. The unit of the frame is meter. The parallel manipulator is actuated by three servo motors located at the base A1, A2, and A3, and the end-effector is mounted at the common joint O, where the three kinematic chains meet. Coordinates of the three bases are A1 (0, 0.25), A2 (0.433, 0), and A3 (0.433, 0.5), and all of the links have the same length l = 0.244m . The definitions of the joint angels are shown in Fig. 1, where qa1 , qa 2 and qa 3 refer to the active joint
angles; qb1 , qb 2 and qb3 refer to the passive joint angles. Cutting the parallel manipulator at the common point O in Fig. 1, one can have an open-chain system including three independent planar 2-DOF serial manipulators, each of which contains an active joint and a passive joint. The dynamic model of the parallel manipulator equals to the model of the open-chain system plus the closed-loop constraints, thus the dynamics of the parallel manipulator can be formulated by combining the dynamics of the three serial manipulators under the constraints [9], [18]. Thus, the dynamic model in the task space of the parallel manipulator can be formulated as [18] T
T
M e qe + Ce qe + S f a = S τ a , T
where qe = [ x y ]
In the coordination control, an additional feedback signal termed the coordination error is introduced. The coordination error represents the coordination relation among the active joints and is not equivalent to the conventional tracking error. For the parallel manipulator, the synchronization error of each active joint contains the information from itself and the adjacent joints [19]. Hence, the motions of all active joints are coordinated and the tracking accuracy of the end-effector is substantially smaller than that without using the synchronization error. d Let q ai be the desired trajectory of the ith active joint, then the tracking error can be defined as e ai = q aid − q ai (2) and the tracking error vector can be written as T
ea = [ ea1 ea 2 ea 3 ] .
With the coordination proposed in [20], we would like to keep the tracking error eai (t ) → 0 as t → ∞ , and regulate the coordination relation among the three active joints to keep following equation satisfied lim e a1 = lim e a 2 = lim e a 3 = 0 . (4) t →∞
t →∞
t →∞
t →∞
lim e a 2 = lim e a 3 ,
t →∞
t →∞
lim e a 3 = lim e a1 .
is the position coordinates of the
joints; f a is the friction torque vector of the active joints [18]; S is the Jacobian matrix between the velocity of the
t →∞
Eq.(4) can be divided into three sub-goals as lim e a1 = lim e a 2 ,
(1)
end-effector; τ a is the actuator torque vector of the active
(3)
t →∞
t →∞
(5)
According to the concept of coordination, the coordination errors between all possible pairs of two neighboring active joints can be defined as ε a1 = ea1 − ea 2 ,
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ε a 2 = ea 2 − ea3 , ε a3 = ea 3 − ea1 ,
to different functions. The first term is the dynamics (6)
compensation defined by the referenced trajectory qer and
where ε ai denotes the coordination error of the active joint i . One can find that, the coordination relation Eq. (5) of the three active joints can be satisfied when ε ai = 0 . Unlike the traditional non- coordinated control considering the position control only, here the coordination control between two neighbor active joints is also required so that Eq. (5) holds. Then the coordination error vector can be written as
qer , which can be written as τ e1 = M e qer + C e qer ; the second term is the friction compensation defined by the Coulomb + viscous friction model [18], which can be written
T
εa = [ε a1 ε a 2 ε a 3 ] .
(7)
Then this coordinated error of the neighboring active joints will be transformed into the task space as follows εe = S T ε a . (8) With the coordination error vector ε e , the cross-coupled error vector ecc can be defined as t
ecc = et + R ∫ εe ( w )dw , 0
where R = diag ( r , r ) is coupled parameter matrix;
(9) the
tracking error vector et can be defined as et = qed − qe , where
qed be the desired trajectory of the end-effector. Then the cross-coupled velocity error vector ecc can be written as ecc = et + R ε e . (10) With the cross-coupled error ecc and cross-coupled velocity error ecc , the combined error vector s can be defined as s = ecc + Pecc , (11) where P = diag ( p , p ) is parameter matrix of combined
error. And the combined velocity error vector s can be expressed as s = ecc + Pecc . (12) The referenced tracking velocity qer and acceleration qer can be defined as qer qer
= qe + s = qed = qe + s = qed
+ R ε e + Pecc ,
(13)
+ Rεe + Pecc .
(14)
Compared with the desired velocity qed and desired acceleration qed , qer and qer not only contain the desired
trajectory information, but also include the actual tracking error and coordination error. Thus qer and qer are more suitable to the trajectory tracking control. Based on the structure of the APD control law [9], [18], with the referenced tracking trajectory qer and qer , and the combined error s , the coordination control law can be designed as τ e = M e qer + C e qer + S T f a + K d s , (15) where K d is a symmetric, positive definite matrix. The control law Eq. (15) can be divided into three terms according
as τ e 2 = S T f a ; the third term is the error elimination defined by the combined error, which can be written as τ e3 = K d s ,with the definition of combined error s , the expression of τ e3 can be rewritten as τ e3 = K d s = K d ( ecc + Pecc ) t = K d ⎛⎜ et + Pet + R ε e + R ∫ εe ( w )dw ⎞⎟ . (16) 0 ⎝ ⎠ From Eq. (16), one can see that the error elimination term contains the information of tracking error and coordination error, where the tracking error is eliminated by the PD control and the coordination error is eliminated by the PI control. In the following, we will prove the asymptotic stability of the parallel manipulator system controller by the coordination controller. Theorem: The proposed synchronization controller Eq. (15) guarantees asymptotic convergence to zero both of the tracking errors and coordination errors of the parallel manipulator system Eq. (1), i.e, et (t ) → 0 and ε a (t ) → 0 as t →∞. Proof: Choose the Lyapunov function candidate as 1 V ( t ) = sT M e s . (17) 2 Differentiating V (t ) with respect to time yields
1 s. V ( t ) = sT M e s + sT M (18) e 2 Considering the structural properties, then one can have s = 2 sT C s into − 2C ) s = 0 . Substitute sT M sT ( M e e e e
Eq.(18), one can get V (t ) = sT ( M e s + C e s ) . The closed-loop system equation can be written as M e s + C e s + K d s = 0 .
(19) (20)
T
Multiplying both sides of Eq. (20) by s , and then substituting the resulting equation into Eq. (19) yields V (t ) = − sT K d s . (21) Thus, s is bounded, and one knows that ecc and ecc are both bounded from Eq. (11), further et is bounded from Eq. (10). Considering the closed-loop system Eq. (20), one obtains s = − M e−1 (C e s + K d s ) . (22) Thus, s is bounded. From Eq. (21), one gets V(t ) = − sT K d s − sT K d s = −2 sT K d s . (23) Thus V(t ) is bounded, and V (t ) is uniformly continuous.
With the Barbalat’ Lemma, s → 0 as t → ∞ . Since s → 0
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as t → ∞ , ecc → 0 and ecc → 0 as t → ∞ from Eq. (11). Thus, one knows et → 0 and εa → 0 as t → ∞ from Eqs. (8), (9) and (10). IV. ACTUAL EXPERIMENT AND PERFORMANCES COMPARISON
Fig.2. The prototype of the parallel manipulator with redundant actuation.
0.30 0.28
Y-axis(m)
As shown in Fig.2, the actual experiment platform is a planar parallel manipulator with redundant actuation designed and produced by Googol Tech. Ltd. in Shenzhen, China. It is equipped with three permanent magnet synchronous motors with harmonic gear drives. The active joint angles are measured with the absolute optical-electrical encoders. The synchronization controller and APD controller are programmed with the Visual C++, and the algorithms ran on a Pentium III CPU at 733MHz. with the sampling period 2ms. In the trajectory tracking experiments, the desired trajectory of the end-effector of our parallel manipulator is a circle with constant speed 0.2m/s. As shown in Fig. 3, the center is (0.29, 0.25) and the starting point is (0.29, 0.29), thus the radius is 0.04m. In order to implement the coordination controller, the dynamic and friction parameters in the dynamic model Eq. (1) must be known. In the experiments, the nominal values of the dynamic parameters are selected as the actual values and the nominal values can be found in [18]. Then, with the known dynamic parameters, the friction parameters can be identified by the Least Squares method [18]. Using the trial-and-error method, the coordination controller parameters are tuned as follows: the coupled parameter matrix R = diag ( 30, 30 ) ,
0.2 0.24
and the gain matrix K d = diag ( 210, 210 ) .
( ) ( M q
τ a = ST
where
d e
+
e = q − qe
)
d e e
+ Ce qed + K p e + K v e + f a , (24)
is
the
position
error
of
the
end-effector; K p and K v are both symmetric, positive definite matrices with constant gains. In fact, K p and K v are tuned by the actual experiments as follows: K p = diag ( 35000, 35000 ) , K v = diag ( 300, 300 ) . In order to make a fair comparison between the two controllers, we ensure the control input vector of the active joints from our coordination controller is not larger than that
0.24
0.22
the combined error parameter matrix P = diag (100, 100 ) , Moreover, to demonstrate that the coordination controller can improve the tracking accuracy of the parallel manipulator with redundant actuation, as comparison, experiments of using APD controller have been implemented [18]. This is because that the APD controller has similar nonlinear dynamics compensation and friction compensation. In the APD controller, the control input vector of the active joints can be calculated as
0.26
0.26
0.28 0.3 X-axis(m)
0.32
0.34
Fig.3. The desired trajectory of the end-effector.
of the APD controller and tune the controller parameters to get their best tracking accuracy. That is to say the cost of the control energy of the coordination controller is less than that of the APD controller. Under the above condition, we get the error curves of the trajectory tracking experiments. The tracking errors of the three active joints are shown in Fig.4a, Fig.4b,and Fig.4c, respectively. From the experiment curves, one can see that the coordination controller can decrease the tracking errors during the whole motion process obviously, and the maximum error in the motion is smaller. Thus, the planar parallel manipulator with actuation redundancy can get better tracking accuracy using the coordination controller. Fig.5a, Fig.5b, and Fig.5c are the coordination errors of the three active joints. One can find that the coordination errors are smaller with the coordination controller, thus the coordination relation of the three active joints are better using the coordination controller than using the APD controller.
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0
-1
-2
0
0.4
0.8 time(s)
1.2
x 10 -3
2
coordination error(rad)
tracking error(rad)
1
APD Coordination
0
-2
1.4
0
0.4
(a)
coordination error(rad)
tracking error(rad)
2
0
-1
-2
APD Coordination
0
0.4
0.8 time(s)
1.2
1.4
tracking error(rad)
APD Coordination
0
0.4
0.8 time(s)
1.2
1.4
Fig.5. The coordination errors among the three active joints: (a) between joint 1 and joint 2; (b) between joint 2 and joint 3; (c) between joint 3 and joint 1.
x 10 -3
V. CONCLUSION
-1 -2 -3 0
APD Coordination
0.4
0.8 time(s)
1.2
1.4
(c) Fig. 4 The tracking errors of the three active joints: (a) joint1; (b) joint 2; (c) joint 3.
coordination error(rad)
x 10
-3
(c)
0
2
1.4
0
-2
(b)
1
1.2
(b)
x 10 -3
1
0.8 time(s)
A coordination controller is proposed for an actual parallel manipulator with redundant actuation. Our theory analysis implies that, the proposed controller can improve the coordination relation of the active joints in the trajectory tracking control. Our experiment results indicate that, compared with the traditional APD controller, the tracking errors and the coordination errors of the parallel manipulator decrease greatly with the coordination controller. Also the proposed controller can be used to other manipulators, such as parallel manipulators with multiple DOFs, to realize high-speed and high accuracy motion.
x 10 -3 APD Coordination
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-2 0
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