Nonlinear Dyn DOI 10.1007/s11071-017-3624-9
ORIGINAL PAPER
Robust trajectory tracking control of cable-driven parallel robots Hamed Jabbari Asl · Jungwon Yoon
Received: 7 May 2016 / Accepted: 16 June 2017 © Springer Science+Business Media B.V. 2017
Abstract In this paper, a robust tracking controller is designed for fully constrained cable-driven parallel robots (CDPRs). One of the main challenges of controller design for this type of robotic systems is that the cables should always be in tension, where this tension is generally generated through an actuator mechanism coupled with gearboxes. On the other hand, the presence of parametric and nonparametric modeling uncertainties is a common problem in designing a precise nonlinear tracking controller for these manipulators. To deal with these problems, in this paper two separate controllers are designed for the subsystems of the robot. First, an adaptive robust feedback controller with an adaptive feedforward term is designed for the dynamics of the CDPR, constituting the outerloop dynamics. This controller is robust with respect to the modeling uncertainties of the system. FurtherH. Jabbari Asl Control System Laboratory, Department of Advanced Science and Technology, Toyota Technological Institute, Nagoya, Japan H. Jabbari Asl Department of Intelligent Mechatronics Engineering, Sejong University, Seoul, South Korea J. Yoon (B) School of Mechanical and Aerospace Engineering & ReCAPT, Gyeongsang National University, Jinju, South Korea e-mail:
[email protected] J. Yoon School of Integrated Technology, Gwangju Institute of Science and Technology, Gwangju 61005, Korea
more, the output of this controller is bounded, which guarantees a saturated desired input for the inner-loop dynamics. Next, a high-gain robust controller is developed for the inner-loop dynamics, which include the actuator–gearbox model. The stability of the overall system is analyzed through a theory of cascaded systems, and it is shown that the system is uniformly practically asymptotically stable. Finally, the effectiveness of the proposed control scheme is validated through simulations on a 4-cable planar robot in both nominal and perturbed conditions. Keywords Cable-driven parallel robot · Robust control · Adaptive control · RISE controller · Bounded-input control
1 Introduction Cable-driven parallel robots (CDPRs) are a class of parallel robotic manipulators, in which the end-effector is moved by means of cables. In these systems, the cables are connected to fixed actuators, which can change the length of the cables in a way to provide a desired position and orientation for the end-effector. This structure reduces the inertia of the moving parts of the robot, which can significantly improve the dynamic performance of the manipulator. In addition, the use of cables, instead of rigid links, can increase the workspace of the robot, reduce the manufacturing and maintenance costs, improve transportability, and simplify the
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assembling of the parts [1,2]. Many applications have recently been developed for the CDPRs, including gait rehabilitation [2], high-speed manipulation [3,4], handling heavy components [5]. One main difference of the CDPRs from the conventional rigid-link manipulators is that the cables can only exert tension and are not able to push on the endeffector. This can be considered as the disadvantage of these mechanisms, since it introduces new challenges in terms of structure and controller design. From the structural point of view, these manipulators can be classified as fully constrained or under-constrained CDPRs [6]. The gravity acts as a passive force in the underconstrained type in order to keep all the cables under tension. In the fully constrained type, the cables can create all degrees of freedom (DOFs) of the robot under the condition that the number of cables is more than that of DOFs of the robot. For these mechanisms, there is a wrench-closure workspace which is defined as a set of feasible poses for the robot, which can be balanced by the tension force of the cables [7]. There is a large literature in extracting the wrench-closure workspace of the CDPRs; see, e.g., [8] and the references therein. In this paper, in the study of the controller design for the CDPRs, it is assumed that the robot trajectory is in the wrench-closure workspace and the robot is fully constrained. Recently, many researches have been devoted to design a controller for the fully constrained CDPRs. In some works, the available control approaches for classical manipulators are directly adapted for the CDPRs. In [9] and [10], respectively, PD and H∞ controllers are designed in cable length coordinates. Linear visionbased controllers are also implemented in [11]. Nonlinear feedforward controller in cable length coordinates is also applied in [3]. Controller design in these coordinates, apart from dynamic modeling accuracy, also requires a precise knowledge of the cable and pulley geometry. Task space controllers are also studied in several works. In [12], backstepping controller is designed for a 3-DOF robot. Also, [13] proposes another modelbased nonlinear controller derived based on the Lyapunov theory. Fuzzy controller is also implemented in [14]. To deal with the requirement of positive tension for CDPRs and also dynamic modeling uncertainty, recently some approaches have been presented. In [15], a positive controller is proposed using control Lyapunov functions. A robust PID controller is developed
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in [1], and to overcome the required knowledge about the uncertainty bound of the dynamic model, adaptive robust controllers are designed in [16,17]. In all of the mentioned studies, the tension of the cables is considered as the control input to the robot dynamics, and the stability of the system is analyzed based on this assumption that the dynamics of the main actuators are negligible. Accordingly, high-gain controllers are implemented for the inner loop of the system, where the electrical actuators generate the desired tensions for the cables. For example, in [1], a lag controller is utilized for the inner loop to track the desired tension of the cables, generated through the outer-loop robot dynamics controller. This analysis, however, is valid only for the cases that the robot handles lightweight components. In other cases, usually the actuators are coupled with high-viscosity gearboxes, which do not allow to arbitrarily increase the bandwidth of the inner-loop dynamics. Therefore, more effort is required in these conditions in terms of controller design and stability analysis. In this paper, a robust controller has been designed for the trajectory tracking of the CDPRs. The main objective is to develop a theoretical framework for stable tracking control of CDPRs considering the whole dynamics of the system and assuming both parametric and nonparametric uncertainties in the system model. The controller design has been performed in two steps. First, a controller is developed for the dynamics of the outer-loop subsystem including the robot dynamics. This controller consists of bounded adaptive feedforward, feedback, and robust terms, which can deal with the uncertainties of the system dynamics. For the outerloop subsystem, the internal force concept is utilized to provide positive tensions for the cables. Next, a robust controller, based on the integral of the sign of the error (RISE) approach, is designed for the inner-loop subsystem, which includes the actuator–gearbox model. Stability analysis is performed for the overall system using a theory of cascaded systems. Finally, simulations are conducted to evaluate the performance of the proposed control scheme. Comparing to the existing works, the proposed approach provides a better stability condition for the system due to considering the dynamics of the whole system, and also the bounded nature of the input of the outer-loop subsystem could improve the transient performance of the system for large initial errors and take into account the power limit of the actuators.
Robust trajectory tracking control of cable-driven parallel robots
The rest of the paper is organized as follows. Section 2 presents the kinematic and dynamic models of the CDPRs. In Sect. 3, a bounded-input adaptive robust controller with an adaptive feedforward term is designed for the dynamics of the manipulator, and then a controller is developed for the actuator–gearbox dynamics. The stability of the overall system is analyzed at the end of this section. Simulation results are presented in Sect. 4, where the proposed control scheme is tested on a 4-cable planar robot. Finally, conclusions are given in Sect. 5.
2 Equations of motion of the CDPRs In this section, equations of motion of the CDPRs are presented, which are used in the next section to design a controller. A CDPR consists of an end-effector, which is supported by n actuated cables. To describe the equations of motion, as shown in Fig. 1, two coordinate frames are considered; i.e., a base frame A {Oa , X a , Ya , Z a }, and an end-effector frame B {Ob , X b , Yb , Z b }. The position and orientation of B with respect to A are, respectively, expressed by the vector q p [x y z] , and the rotation matrix R : B → A, associated with the three Euler angles φ, θ , and ψ, denoting, respectively, the roll, pitch and yaw angle, which is given by ⎡
cψ cθ R ⎣ sψ cθ −sθ
cψ sθ sφ − sψ cφ sψ sθ sφ + cψ cφ cθ sφ
⎤ cψ sθ cφ + sψ sφ sψ sθ cφ − cψ sφ ⎦ cθ cφ (1)
where sa ≡ sin(a) and ca ≡ cos(a). The ith cable is attached to its base at the point Ai [Ai x Ai y Ai z ] and to the end-effector at the point Bi [Bi x Bi y Bi z ] , both with respect to the A frame. Based on this notation, the cable length, li , is defined as the distance between Ai and Bi . By defining [ωx ω y ωz ] as the vector of angular velocity, the following relation can be obtained for the time derivatives of the cable lengths [18]: ˙ = J¯ q˙ p
(2)
where ˙ l˙1 · · · l˙n , and the Jacobian matrix J¯ ∈ n×6 is given by
Fig. 1 A cable-driven parallel robot with attached end-effector and base frames
J¯ [L1 /l1 · · · Ln /ln ]
(3)
where Li [si ; bi × si ] ∈ 6×1 for i = 1, . . . , n, with si Bi −Ai , and bi Bi −q p . Also, the relation between the angular velocities and the time derivatives of the Euler angles is given by = φ˙ θ˙ ψ˙ , where the matrix ∈ 3×3 is defined as [13] ⎡
⎤ 1 0 −sθ ⎣ 0 cφ cθ sφ ⎦ . 0 −sφ cθ cφ
(4)
Considering q [x y z φ θ ψ] ∈ 6 as the generalized coordinates vector for the pose of the endeffector, the dynamic model of a general cable-driven parallel robot with n cables can be written as [1] ˙ + Td = Jτ ˙ q˙ + G(q) + Fd q˙ + Fs (q) M(q)q¨ + C(q, q) (5) where τ ∈ n denotes the vector of tension force, ˙ ∈ M(q) ∈ 6×6 represents the inertia matrix, C(q, q) 6×6 is the centrifugal and Coriolis matrix, G(q) ∈ 6 denotes the gravity vector, Fd ∈ 6×6 is the constant ˙ ∈ 6 is the viscous friction coefficient matrix, Fs (q) 6 Coulomb friction, and Td ∈ is a time-varying dis6×n is defined as turbance
term. Also the matrix J ∈ ¯ J − Jϒ , with ϒ
I3×3 03×3 , 03×3
(6)
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where I3×3 denotes a 3 × 3 identity matrix. Dynamics (5) have the following properties:
Gearbox mi
i
Property 1 The inertia matrix M(q) is positive-definite and symmetric and satisfies the following inequalities:
Motor with servo loop
2rp gi
m x ≤ x M(q)x ≤ m x 2
2
∀x ∈
6
(7)
i
where m and m ∈ + are some positive constants, and · denotes the Euclidean norm.
Fig. 2 actuator–gearbox model
˙ matrices satisfy the folProperty 2 M(q) and C(q, q) lowing relation:
The cables are driven by rotary motors whose dynamics are usually compensated through a high-gain servo loop to follow the desired torque, having voltage or current as their input. In order to have the manipulator to handle heavy loads, the motors are usually coupled with gearboxes, which have friction opposing with the torque of motors (see Fig. 2). Therefore, neglecting the friction in the pulley, the applied tension to the cables can be expressed as the scaled difference of vector of motor torque τ m = [τm1 . . . τmn ] ∈ n and the viscous and Coulomb friction torque of gear boxes τ g = τg1 . . . τgn ∈ n as follows [19]:
˙ ˙ x=0 x M(q) − 2C(q, q)
∀x ∈ 6 .
(8)
Property 3 The following linearly parameterized equality is valid for the dynamic model of (5): Yd (qd , q˙ d , q¨ d )ϑ = M(qd )q¨ d + C(qd , q˙ d )q˙ d +G(qd ) + Fd q˙ d + Fs (q˙ d )
(9)
where ϑ ∈ p is the vector of constant system parameters with p defining the number of parameters, and Yd (qd , q˙ d , q¨ d ) ∈ 6× p is the measurable desired regression matrix, which depends on the desired value of q and its first two derivatives, denoted by qd , q˙ d , q¨ d ∈ 6 . Property 4 The Coriolis and centrifugal, friction and gravity terms can be bounded as C(q, q) ˙ ≤ ζc q ˙ , G(q) ≤ ζg , Fd q˙ + Fs (q) ˙ + ζ fs ˙ ≤ ζ fd q
(10)
where ζc , ζg , ζ fd , ζ fs ∈ + are positive constants. Property 5 The following relation is satisfied for ˙ C(q, q): C(q, x1 )x2 = C(q, x2 )x1 ∀x1 , x2 ∈ 6 .
(11)
Also the following assumption is considered for the dynamic equation of (5). Assumption 1 The time-varying disturbance term, which represents unstructured uncertainties in the model, is bounded such that Td (t) ≤ ζt where ζt ∈ + is a positive constant.
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(12)
r p τ = τ m − Fdg ω + Fsg (ω)
(13)
τg
where r p is the radius of pulley, ω ∈ n denotes the vector of shaft speed, Fdg ∈ n×n is a diagonal constant matrix, and Fsg (ω) ∈ n denotes the Coulomb friction. The effect of friction τ g can be approximated as τˆ g ∈ n , and hence partially be compensated by feedforward terms in the controller, provided the shaft velocities are measured or estimated with high accuracy and small time delay. Based on this consideration, the following assumption is stated for the friction model. Assumption
2 The compensated friction model, stated by τ g − τˆ g , and its first three time derivatives are bounded.
3 Trajectory following control of the CDPRs In this section, a trajectory tracking controller is designed for the CDPRs. The controller design objective is to find a positive tension vector τ such that the robot end-effector could follow a predefined desired trajectory. The proposed controller compensates for the
Robust trajectory tracking control of cable-driven parallel robots
parametric uncertainties of the robot through an adaptive scheme and deals with unstructured (nonparametric) uncertainties of the robot and the actuation system using robust control terms. The controller design and stability analysis are performed in three steps. First, a bounded-input adaptive robust controller is designed for the robot dynamics. This step generates desired tensions which have to be tracked to complete the control objective. Then, a robust controller is designed for the actuator–gearbox dynamics to follow the desired tensions. Finally, the stability of the overall system is analyzed based on a theory of cascaded systems.
3.1 Adaptive robust control of cable part dynamics To design a trajectory tracking controller for the robot dynamics, the error signal e ∈ 6 is first defined as e q − qd ,
(14)
where qd is defined in Property 3, which denotes the vector of desired trajectory. The following assumption is considered for the desired trajectory. Assumption 3 The time-varying reference trajectory of the end-effector is selected such that qd (t), q˙ d (t) and q¨ d (t) are bounded; i.e., qd (t), q˙ d (t), q¨ d (t) ∈ L∞ . In order to facilitate the subsequent design and analysis, the vector Tanh(·) ∈ n and the matrix Cosh(·) ∈ n×n are defined as in the following: Tanh(x) [tanh(x1 ) . . . tanh(xn )]
(15)
Cosh(x) diag{cosh(x1 ), . . . , cosh(xn )}
(16)
for x = [x1 . . . xn ] ∈ n . To aid the controller design, the filtered error r ∈ 6 is defined as follows: r e˙ + α1 Tanh(e)
(17)
where α1 ∈ + is a constant adjustable gain. Now, the open-loop tracking error system can be developed by differentiating (17), premultiplying by M(q), and using (5) to obtain the following: ˙ − Y(e, r)ϑ − Td + Jτ M(q)˙r = −C(q, q)r
(18)
where Yϑ ∈ 6 is defined as Y(e, r)ϑ − M(q) α1 Cosh−2 (e)˙e − q¨ d ˙ (α1 Tanh(e) − q˙ d ) − C(q, q) ˙ + G(q). + Fd q˙ + Fs (q)
(19)
The dynamics of r can be rewritten as ˜ − Td + Jτ ˙ − Yd ϑ + Y M(q)˙r = −C(q, q)r
(20)
˜ (e, r) ∈ 6 is defined as where Y ˜ Yd ϑ − Yϑ. Y
(21)
and Yd ϑ is given in (9). From the properties of the system dynamics (Properties 1–5), and knowing the fact that cosh−1 (·) ≤ 1, it is possible to show that ˜ Y ≤ ζ0 + ζ1 e + ζ2 r
(22)
where ζ0 , ζ1 , ζ2 ∈ + are some bounded positive constants. This relation can be verified from the results of [20,21], and considering a constant value for the differ˙ and Fs (q˙ d ). Furthermore, the following ence of Fs (q) bound can be assumed for the parameter vector: ϑ ≤ ϑ M .
(23)
The tension τ in (20) is applied through actuators, which generally include electrical motors and gearboxes. Dynamics of motors are usually fast enough comparing to the dynamics of the robot; however, the dynamics of the combined actuator–gearbox system still have comparable response time to the robot dynamics. Accordingly, in the sequel, first a desired tension τ , denoted by τ d ∈ n , will be designed for the robot dynamics (outer-loop subsystem) and then a controller for the actuator–gearbox dynamics (inner-loop subsystem) will be introduced in the next subsection to track τ d . Considering the error between the true and desired values of the tension, (20) can be written as ˜ ˙ − Yd ϑ − Td + Y M(q)˙r = − C(q, q)r + Jτ d + J (τ − τ d ) .
(24)
In the design of the controller for the robot dynamics, the interconnection term J (τ − τ d ) will not be consid-
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ered. However, stability of the overall system is studied at the end of the section considering the effect of this term. The objective of the controller design is to find the appropriate tension of cables to regulate the dynamics in (20). For these dynamics, it is required to have a positive tension in all cables. Therefore, to find the tension vector, the following general solution is applied, which follows the internal forces concept [4]: ˆ†
τ d J τ 0 + κ n¯
(25)
where, assuming an uncertainty in the measurement of J, Jˆ ∈ 6×n denotes its estimate, κ ∈ is an arbitrary value which is selected in a way to yield a positive tension for the cables, n¯ ∈ n is the null space of Jˆ such that Jˆ n¯ = 0, and Jˆ † τ 0 is the particular solution for the tension, in which Jˆ † ∈ n×6 denotes the pseudoinverse of Jˆ and, τ 0 ∈ 6 is the applied wrench on the end-effector, which is proposed as τ 0 Yd ϑˆ − K p Tanh(e) − Kd Tanh(r) + u
(26)
where K p , Kd ∈ 6×6 are diagonal positive-definite gain matrices, and ϑˆ ∈ p is an estimated value of ϑ through the following robust adaptation law: ˆ ϑ˙ˆ Proj(− Yd r − σ1 ϑ)
Remark 1 Equations (27), (29) are leakage-like adaptation laws. These laws can improve the robustness of the adaptive scheme in the presence of disturbance or unmodeled dynamics [23,24]. Now, substituting (25) with (26) in (20), and adding and subtracting τ 0 , the closed-loop dynamics of the tracking error can be written as ˜ ˙ − Yd ϑ˜ + Y M(q)˙r = − C(q, q)r − K p Tanh(e) − Kd Tanh(r) + u − Td + JJˆ † − I τ 0 + κ J − Jˆ n¯ (30) where ϑ˜ ϑ − ϑˆ ∈ p , and I ∈ 6×6 denotes a 6 × 6 identity matrix. Remark 2 Considering the boundedness property of (26), and assuming that Jˆ is bounded, it is easy to verify that ˆ† (31) J J − I τ 0 ≤ ζ3 (32) κ J − Jˆ n¯ ≤ ζ4 for some positive values of ζ3 , ζ4 ∈ + .
(27)
Now the following theorem is stated for the stability analysis of the controller.
where σ1 ∈ + is an adjustable gain, ∈ p× p is a diagonal positive-definite adaptation gain matrix, and Proj(·) is a continuous projection operator, similar to what is used in [22], which is applied to ensure that ϑˆ remains within a bounded value. The bounded term u ∈ in (26) is a robustifying term, which is defined as
Theorem 1 Consider the open-loop dynamics of the system defined in (20) with Assumptions 1 and 3. Let the controller be defined through (26), (27), (28) and (29). Then the tracking error signal is uniformly ultimately bounded provided the control gains satisfy the following sufficient conditions:
u−
rδˆ2 r δˆ +
(28)
λmin (Kd ) > ζ2
in which ∈ + is a positive constant gain, and δˆ ∈ is an estimated value of δ ∈ + through the following robust adaptation law: δ˙ˆ Proj γ r − σ2 γ δˆ
(29)
where σ2 , γ ∈ + are adjustable gains, and δ ζ0 + ζ3 + ζ4 + ζt , in which ζ3 and ζ4 are positive constants and subsequently defined in (31) and (32).
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λmin (K p ) > 0
+
2 2 2 −1 1 + cosh exp μ 1
(33)
ζ12
4α1 λmin (K p ) 4 2 2 −1 exp 1 + cosh μ , 1
(34)
where λmin (·) denotes the minimum eigenvalue of a matrix, μ ∈ + is defined as μ max {d, h1 (0)}, and d, 1 , 2 ∈ + and h1 ∈ 13+ p are subsequently defined parameters.
Robust trajectory tracking control of cable-driven parallel robots
Proof The following continuously differentiable positive-definite and radially unbounded function is considered to prove the theorem: 1 k pi ln (cosh (ei )) V1 r Mr + 2 n
Using the following inequality −
r2 δˆ2 ≤ − r δˆ + , r δˆ +
(40)
V˙1 can be bounded as
i=1
1 1 2 + ϑ˜ −1 ϑ˜ + δ˜ 2 2γ
(35)
where k pi denotes the ith diagonal element of K p , and δ˜ δ − δˆ ∈ . This function can be bounded as 1 1 tanh2 (h1 ) ≤ ψ1 (h1 ) ≤ V1 ≤ ψ2 (h1 ) 2
V˙1 ≤ − α1 λmin (K p ) Tanh(e)2 − λmin (Kd ) Tanh(r)2 + ζ1 e r 1 + ζ2 r2 − σ1 ϑ˜ ϑˆ + δ˜ r − δ˙ˆ + . γ (41)
(36) Substituting (29) in (41), knowing the fact that where ψ1 (·) : → and ψ2 (·) : → are, respectively, defined as ψ1 1 ln(cosh (h1 )) and ψ2 2 h1 2 , in which 13+ p
13+ p
1 1 1 1 min λmin (K p ), m, λmin ( −1 ), , 2 2 2γ 1 1 1 , 2 max λmax (K p ), m, λmax ( −1 ), 2 2 2γ
V˙1 ≤ − α1 λmin (K p ) Tanh(e)2
and h1 e r ϑ˜ δ˜ ∈ 13+ p , with λmax (·) denoting the maximum eigenvalue of a matrix. Using (17) and (30), the time derivative of (35) can be obtained as V˙1 = − α1 Tanh (e)K p Tanh(e) − r Kd Tanh(r) ˜ − r Td + r u + r JJˆ † − I τ 0 + r Y 1 ˙ˆ −1 ϑ˙ˆ + Yd r − δ˜δ. + κr J − Jˆ n¯ − ϑ˜ γ (38) Substituting (27) and (28) in (38), exploiting the fact that Tanh(r) ≤ r, and utilizing (22), (31), and (32), equation (38) can be upper bounded as V˙1 ≤ − α1 λmin (K p ) Tanh(e)2 − λmin (Kd ) Tanh(r)2 + ζ1 e r + ζ2 r2 − −
r δˆ +
1 ˙ ˆ δ˜δˆ + σ1 ϑ˜ ϑ. γ
(42)
˜ one can and using the same inequality as (42) for δδ, bound V˙1 as follows:
(37)
r2 δˆ2
σ1 2 σ1 ϑ2 , σ1 ϑ˜ ϑˆ ≤ − ϑ˜ + 2 2
+ δ r (39)
− λmin (Kd ) Tanh(r)2 −
σ1 ˜ 2 ϑ 2
σ2 2 δ˜ + ζ1 e r + ζ2 r2 2 σ1 σ2 ϑ2 + δ 2 + . + 2 2 −
(43)
Also, knowing the fact that ∀x ∈ n tanh2 (x) ≤ Tanh(x)2 ≤ x2 ,
(44)
and using (23), V˙1 can be further bounded as V˙1 ≤ − α1 λmin (K p ) tanh2 (e)
σ1 ˜ 2 ϑ 2 e r tanh(e) tanh(r) σ2 2 − δ˜ + ζ1 2 tanh(e) tanh(r) 2 2 r tanh (r) + ζ2 + β1 tanh2 (r) − λmin (Kd ) tanh2 (r) −
(45)
2 + σ2 δ 2 + is a positive constant, where β1 σ21 ϑ M 2 which is adjustable through the parameters σ1 , σ2 , and
. Equation (45) can be written as
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σ1 ˜ 2 σ2 ˜2 V˙1 ≤ −y Py − ϑ − δ + β1 2 2
(46)
where y = [tanh(e) tanh(r)] ∈ 2 , and P is given by P=
α1 λmin (K p ) er − ζ21 tanh(e) tanh(r)
er − ζ21 tanh(e) tanh(r)
λmin (Kd ) −
ζ2 r2 tanh2 (r)
.
(47)
The lower bound on V1 can be utilized from (36) to state that 1 h1 ≤ cosh−1 exp . (53) V1 1 Therefore, a sufficient condition for (52) can be written as 2 1 −1 λmin (Kd ) > ζ2 1 + cosh V1 exp 1
The following conditions are required for P to be positive-definite:
+
4α1 λmin (K p ) 4 1 −1 exp 1 + cosh V1 . 1
α1 λmin (K p ) > 0 ζ2 r2 tanh2 (r) ζ12 e2 r2 + . 4α1 λmin (K p ) tanh2 (e) tanh2 (r) (48)
λmin (Kd ) >
ζ12
(54)
Now, the same analysis as in [25] can be conducted here; given (50) and (36), provided the condition (54) is satisfied, h1 is uniformly ultimately bounded, according to the Lyapunov theorem extension [26], in the sense that
If these conditions are satisfied, (46) can be bounded as
e ≤ h1 < d
σ1 ˜ 2 σ2 ˜2 V˙1 ≤ −β2 y2 − ϑ − δ + β1 2 2
where d ∈ + defines the radius of a ball containing the position tracking errors, which can be selected according to [27] as
(49)
∀t ≥ T (d, h1 (0)) ,
(55)
where β2 = λmin (P) ∈ + is a positive constant. Finally, using (44), expression (49) can be written as
d > ψ1−1 ◦ ψ2 ψt−1 (β1 )
V˙1 ≤ −ψt (h1 ) + β1
and T ∈ + is a constant that states the time to reach the ball, which is given by [27]
(50)
where the function ψt : 13+ p → is a strictly increasing nonnegative function, which is defined as ψt β¯2 tanh2 (h1 ), with β¯2 min{β2 , σ1 /2, σ2 /2}. Using the fact that ∀x ∈ n (51)
the sufficient condition for Kd can be written as follows: λmin (Kd ) > ζ2 (1 + h1 )2
123
ζ12 4α1 λmin (K p )
⎧ ⎪ ⎪ ⎨0
h1 (0) ≤ ψ2−1 ◦ ψ1 (d) . T ψ2 (h1 (0))−ψ1 ψ2−1 ◦ψ1 (d) ⎪ h1 (0) > ψ2−1 ◦ ψ1 (d) ⎪ ⎩ −1 ψt ψ ◦ψ1 (d)−β1 2
(57)
x ≤ 1 + x , tanh(x)
+
(56)
(1 + h1 )4 .
(52)
Therefore, from (36) and (55), the final condition for Kd , which is given in (34), can be expressed in terms of either the ultimate bound or the initial conditions of the system.
The region of attraction can be made large, and also the size of the ultimate bound d can be made small by selecting sufficiently large control gains. From this, and since Kd depends on initial conditions of the states, it
Robust trajectory tracking control of cable-driven parallel robots
can be concluded that the system is semi-globally uniformly ultimately bounded (see Remark 40. The ultimate bound of the tracking error also depends on the values of σ1 , σ2 and which are adjustable parameters. Small values of these parameters will decrease the final tracking error, while on the other hand, will reduce the robustness of the closed-loop system in the presence of noise and external disturbances. Therefore, a tradeoff between the precision and reliability of the system should be considered in selecting these parameters.
To facilitate the controller design and subsequent analysis, the following filtered tracking error is defined:
Remark 3 Since the condition for Kd , given in (34), depends on the ultimate bound d, and β1 accordingly, the terms ϑ M and δ, which depend on the upper bounds of uncertainties of the system, are required to be known. However, it is worth mentioning that these terms in β1 are scaled by the control gains σ1 and σ2 . Setting these gains at zero can release the requirement of knowing the upper bound of the uncertainties, but as discussed before, this may reduce the robustness of the adaptive laws (27) and (29).
r p δ τ = τ˙ m − τ˙ g − r p τ˙ d + α2 r p eτ .
δ τ e˙ τ + α2 eτ
(60)
where α2 ∈ + . It is worth mentioning that δ τ is not measurable since it depends on e˙ τ . Now, the open-loop tracking error system can be developed by using (59) to obtain the following: (61)
For these dynamics, the following controller is proposed: τ˙ m = r p τ˙ d + τ˙ˆ g − χ
(62)
where χ(t) ∈ n is the robust integral of the sign of the error feedback control term, which is defined as [29] χ (t) (ks + 1) eτ (t) − (ks + 1) eτ (0) ! t
+ (ks + 1) α2 eτ t¯ + 2kr sgn eτ (t¯) dt¯
3.2 Robust control of actuator–gearbox model
0
The control objective for the actuator–gearbox system is to design the motor torque τ m such that the output tension τ follows the desired tension τ d , defined by the outer-loop controller. Note that to implement the tension tracking controller, we need to be ensured that the desired tension and its time derivative are bounded, which can be easily verified from (25) and (26). To design the tension tracking controller, inspired from the dynamic extension method applied in [28], the following dynamics can be considered for (13): r p τ˙ = τ˙ m − τ˙ g .
(58)
Now, by defining eτ τ − τ d ∈ n as the tension tracking error, its dynamics can be written as r p e˙ τ = τ˙ m − τ˙ g − r p τ˙ d
(59)
where τ˙ m can be considered as the input for these dynamics. Therefore, in the sequel, a controller will be designed for (59) with τ˙ m as its input, and the actual input can then be recovered through the time integrat tion as τ m = 0 τ˙ m (t¯)dt¯.
(63) where ks , kr ∈ + are adjustable gains, and sgn(·) denotes the signum function. It should be noted that the second term in right-hand side of (63) is used to ensure that χ (0) = 0. Substituting controller (62) into (61) gives the following closed-loop dynamics of the system: r p δ τ = τ˙ˆ g − τ˙ g + α2 r p eτ − χ.
(64)
Using (60), the time derivative of (64) can be written as ˜ + Nd r p δ˙ τ = − (ks + 1) δ τ − kr sgn(eτ ) + N
(65)
˜ α2 r p (δ τ − α2 eτ ) ∈ n , and Nd τ¨ˆ g − where N τ¨ g ∈ n . From Assumption 2, it can be verified that ˙ d (t) ∈ L∞ . Furthermore, N ˜ can be bounded Nd (t), N as ˜ N ≤ ρ h2
(66)
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where ρ ∈ + is a known positive constant, and h2 ∈ 2n is defined as follows: h2 eτ δ . τ
(67)
Lemma 1 Let the auxiliary function Q(t) ∈ be defined as follows: Q(t)
2˙eτ kr sgn(eτ )dt¯+2
n
kr |eτ i (0)|−Nd eτ (68)
i=1
0
where the subscript i = 1, . . . , n denotes the ith element of the vector. If the control gain satisfies the following condition: kr > Nd ∞ +
1 N ˙ d ∞ α2
(69)
where ·∞ denotes the infinity norm, then Q(t) ≥ 0 α2 eτ Nd −
(70) n
˙d ≤ 0 . kr α2 |eτ i | − eτ N
(71)
1 3 h2 2 + (2kr + Ndi ∞ ) |eτ i | 2 n
V2 ≤
Now, the following lemma is presented which is used in the subsequent stability analysis.
!t
with Q defined in (68). This function satisfies the following inequality:
with 3 max{ 21 , r p }. Using (60) and (65), and computing the derivative of Q(t), the time derivative of (73) can be obtained as V˙2 = − α2 eτ eτ − (ks + 1) δ τ δ τ + eτ δ τ ˙ ˜ + δ τ N − eτ Nd + α2 eτ Nd
−2
n
kr α2 |eτ i | .
Using (71) in Lemma 1, and the property of (42), the following upper bound can be obtained for V˙2 : 1 1 ˙ eτ 2 − δ τ 2 − ks δ τ 2 V2 ≤ − α2 − 2 2 n ˜ (76) + δ τ N kr α2 |eτ i | . − i=1
Now, the following theorem for the stability result of the controller is stated: Theorem 2 Consider the error system defined by (61). The controller τ˙ m given by (62) and (63) ensures that the system signals are bounded and the tracking errors, eτ and δ τ , are exponentially regulated, provided kr satisfies (69), and the control gains are selected according to the following sufficient conditions: α2 >
1 ρ2 , β3 > 2 4ks
(72)
where β3 ∈ + is a subsequently defined adjustable constant. Proof The following positive-definite function is considered to prove the theorem: V2 =
1 1 eτ eτ + r p δ τ δτ + Q 2 2
123
(75)
i=1
i=1
Proof The proof is given in [30].
(74)
i=1
(73)
Considering (66) and (67), equation (76) can be written as V˙2 ≤ − β3 h2 2 − ks δ τ 2 + ρ δ τ h2 −
n
kr α2 |eτ i |
(77)
i=1
where β3 min{(α2 − 21 ), 21 }. Now, by completing the squares for the second and third terms in the right-hand side of (77), the following expression can be obtained:
V˙2 ≤ −β4 h2 2 −
n
kr α2 |eτ i |
(78)
i=1
ρ where β4 β3 − 4k ∈ is a positive value provided s gain condition (72) is satisfied. The inequalities in (74) and (78) can be used to verify that 2
Robust trajectory tracking control of cable-driven parallel robots
V˙2 (t) ≤ −β5 V2 (t)
(79)
with β5 ∈ + satisfying 2β4 kr α2 . , 0 < β5 ≤ min 3 2kr + Nd ∞
(80)
Therefore, it can be concluded that the error signals eτ and δ τ are exponentially stable.
As mentioned earlier, the actual control input can be obtained from the time integration of (62) as follows: !
t
τ m = r p τ d + τˆ g −
χ (t¯)dt¯.
(81)
0
(i) Subsystem (84) is USPAS. (ii) The subsystem x˙ 1 = g1 (x1 , θ¯ 1 ) is USPAS with an additional condition that the convergence of the solutions of the driven subsystem does not depend on the size of the attractive neighborhood of the origin. (iii) The interconnection term is bounded as (x1 , x2 , θ¯ 1 , θ¯ 2 ) ≤ G 0 (x)g0 (x2 )
(85)
for x = [x1 x2 ] ∈ n1+n2 , all θ¯ 1 ∈ 1 , and all θ¯ 2 ∈ 2 , where G 0 (·) : ≥0 → ≥0 is a nondecreasing function, and g0 (·) is a class K function. (iv) Solutions of the system under the cascaded interaction are all uniformly bounded.
3.3 Stability of the overall system Proof The proof is given in [31]. Until now, stability properties of the outer-loop and inner-loop subsystems are analyzed without considering the interaction term J (τ − τ d ) in (24). To show the stability properties of the overall system, the following definition and lemma are invoked: Definition 1 Uniformly semi-globally practically asymptotically stable (USPAS) [31]: Consider the following nonlinear system: x˙ = g(x, θ¯ , t),
(82)
where t ∈ ≥0 , x ∈ n , θ¯ ∈ m is a constant parameter, and g : n × m × ≥0 → n is locally Lipschitz in x and piecewise continuous in t for all θ¯ . Let ⊂ m be a set of parameters. System (82) is said to be USPAS on if the domain of attraction and the ball which is uniformly asymptotically stable (see the definition in [31]) can be arbitrarily enlarged and decreased, respectively, by a proper adjustment of θ¯ . Lemma 2 Consider the following cascaded system x˙ 1 = g1 (x1 , θ¯ 1 ) + (x1 , x2 , θ¯ 1 , θ¯ 2 ) x˙ 2 = g2 (x2 , θ¯ 2 )
The outer-loop and inner-loop subsystems of the CDPR can be expressed in forms of (83) and (84). Therefore, to prove that the system is USPAS, it is sufficient to show that the conditions (i)–(iv) of Lemma 2 are satisfied. According to the results of Theorems 1 and 2, it is easy to verify that the conditions (i) and (ii) are satisfied. It should be noted that the inner-loop subsystem is exponentially stable, which is a stronger stability result comparing to the USPAS. The interconnection term of the cascaded system is J (τ − τ d ). In this expression, (τ − τ d ) exponentially converges to zero and J is a bounded matrix, which lead to the conclusion that the condition (iii) is satisfied. One approach to show that the condition (iv) is satisfied, is to check the criteria mentioned in [32]. Based on the results in [32], all solutions of the cascaded system are uniformly bounded provided that (a) The interconnection term satisfies the linear growth restriction, such that
(83) (84)
where g1 : n1 ×m1 ×≥0 → n1 , g2 : n2 ×m2 × ≥0 → n2 and : n1+n2 ×m1+m2 ×≥0 → n1 are locally Lipschitz and piecewise continuous in time. Let 1 ⊂ m1 and 2 ⊂ m2 be the sets of parameters. Cascaded system (83-84) is USPAS on 1 × 2 if:
(x1 , x2 , θ¯ 1 , θ¯ 2 ) ≤ g01 (x2 ) x1 + g02 (x2 ) (86) where g01 and g02 are class K functions. (b) There exist positive constants c1 and c2 such that the Lyapunov function V (x1 ), which establishes stability of x˙ 1 = g1 (x1 , θ¯ 1 ), also satisfies
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H. Jabbari Asl, J. Yoon
Bounded adaptive robust outer-loop controller
τd τd
Jˆ † τ 0
CDPR model
Inner-loop controller
e
τm
RISE controller
τ
n
ω
τ0
Feedforward controller
τ Adaptive Feedforward
Bounded PD term
Adaptive robust term
Reference trajectory
qd
r
e
Tanh( h e)
q
e
1
Fig. 3 Block diagram of the proposed control approach
∂V ∂x x1 ≤ c1 V for x1 ≥ c2 . 1
(87)
(c) Subsystem (84) is exponentially stable. In a similar way of item (iii) of the lemma, it can be concluded that the condition (a) is satisfied. It is wellknown that the condition (b) is satisfied for all polynomial Lyapunov functions [32]. V1 in (35) includes the "n term i=1 k pi ln(cosh (ei )), which can be shown that " ∂ n k ln (cosh (e )) i i=1 pi e ≤ K p F e ∂e ≤ c1
n
In this section, simulation studies have been performed to evaluate the effectiveness of the proposed robust controller. In these studies, a planar CDPR is considered, which has four cables with three degrees of freedom. The geometrical structure and configuration of this manipulator is shown in Fig. 4. The generalized coordinates vector for this robot is q = [x y φ] ∈ 3 , where its dynamics, considering (5) and the planar motion of the robot, can be written as Mq¨ + G + Td = Jτ
(88)
k pi ln(cosh(ei ))
i=1
where · F denotes the Frobenius norm. Therefore, the condition (b) can be verified. Also the condition (c) is satisfied from the result of Theorem 2, and this ends the proof. The block diagram of the proposed controller is shown in Fig. 3. Remark 4 Although the stability analysis presented above shows the semi-global stability of the system, it is worth mentioning that the result does not satisfy the standard semi-global result because of the limited wrench-closure workspace of the robot and also the ˆ In view possible singularity in the inverse of matrix J. of the above stability result, the global region of the robot is limited to wrench-closure and singularity-free workspace.
123
4 Simulation results
(89)
where M ∈ 3×3 and G ∈ 3 are given by ⎡
⎤ ⎡ ⎤ m 0 0 0 M = ⎣ 0 m 0 ⎦ , G = ⎣ mg ⎦ 0 0 I 0
(90)
in which m and I are, respectively, the mass and inertia of the end-effector, and g is the gravity acceleration. Therefore, for this robot, one has ϑ = [m I ] , and ⎤ q¨d x 0 Yd = ⎣ g + q¨dy 0 ⎦ . 0 q¨dφ ⎡
(91)
The disturbance term is also modeled as Td = sin(t)[0.7 0.7 0.05] . The parameters of these dynamics and the actuator–gearbox model, together with the controller gains, are all given in Table 1. In the simulations, the initial position of the robot is set at q(0) =
Robust trajectory tracking control of cable-driven parallel robots 0.35
Norm of signals
0.3 0.25 0.2 0.15 0.1 0.05 0
0
5
10
15
20
25
30
Time [s]
Fig. 4 Configuration of the studied 4-cable planar parallel robot
Fig. 5 Simulation 1: time evolution of the norm of tracking errors 10
Parameter
Value
Unit
m
8
kg
I
0.3
kg m2
g
9.81
m s−2
rp
0.04
m
Kp
diag{5, 5, 0.1}
–
Kd
diag{10, 10, 0.5}
–
α1
0.4
–
diag{0.5, 5}
–
γ
2
–
σ1 , σ2
0.01
–
0.05 [0.9m 1.5I ] 0.5
Nm
ks
5
–
kr
1
–
α2
3
–
δˆ ϑˆ1 ϑˆ2
6 4 2
0
5
10
15
20
25
30
Time [s]
Fig. 6 Simulation 1: time evolution of estimates of δ and ϑ = [ϑ1 ϑ2 ]
[0.1 0.55 0] , and the following trajectory is considered as the desired reference input for the manipulator:
π ⎧ ⎨ xd = 0.25 cos 15 t π t + 0.5 . y = 0.25 sin 15 ⎩ d φd = 0
8
0
– T kg kg m2
ˆ ϑ(0) ˆ δ(0)
T
Parameter estimates
Table 1 Simulation parameters
(92)
Considering the practical results obtained for the compensation of the gearbox dynamics in [33], the compensation error of the feedforward controller is considered as τ g − τˆ g = 0.5ω + 0.5 tanh(10ω). Also pose information of the end-effector and the tension forces are assumed to be, respectively, measured through a visual sensor and load cells, as utilized in [16]. In the first simulation, ideal measurements are assumed for the robot task space information and load
cell data. The results of this simulation are illustrated in Fig. 5 through Fig. 8. The norm of error signals is shown in Fig. 5. The time evolution of the parameter updates for ϑˆ and δˆ is shown in Fig. 6. The desired and manipulator trajectories in X a Ya plane are illustrated in Fig. 7, and Fig. 8 demonstrates the tension of the cables. As it is observed from this simulation, the tracking error of the system is remained in a small bounded area while the cables are all in tension, which verify the results of the theorem. Remark 5 Note that the tanh(·) function used in controller (26) is a slow saturation function. It means that hard saturation effects only appear for large initial errors and a mechanism with large wrench-closure workspace [34]. To demonstrate the superiority of the both proposed inner-loop and outer-loop controllers over the classical ones, here some comparison results are presented. First, having the same simulation conditions as the previous study, the inner-loop controller is replaced with a welltuned PID controller, which is commonly used in the literature for the tracking of desired tension. The tra-
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H. Jabbari Asl, J. Yoon
0.7
0.7 Desired trajectory Robot trajectory
0.6
y [m]
y [m]
0.6
0.5
0.4
0.5 Desired trajectory Adaptive FF Non-adaptive FF
0.4 0.3 0.2
0.3 -0.3
-0.2
-0.1
0
0.1
0.2
-0.4
0.3
-0.2
0
0.2
0.4
x [m]
x [m] Fig. 7 Simulation 1: desired path and robot’s trajectories in X a Ya plane
Fig. 10 Simulation 3: desired path and robot’s trajectories in X a Ya plane using a non-adaptive feedforward term 150
120
τ1 τ2 τ3 τ4
100
100 τ1 τ2 τ3 τ4
60 40 20 0
τ [N]
τ [N]
80
50
0 0
5
10
15
20
25
30
0
5
10
15
20
25
30
Time [s]
Time [s]
Fig. 8 Simulation 1: time evolution of the cables’ tensions
Fig. 11 Simulation 3: time evolution of the cables’ tensions applying the non-adaptive controller
0.7 0.7
0.5 Desired trajectory Proposed controller PID controller
0.4
0.3
-0.2
-0.1
0
0.1
0.2
0.3
x [m] Fig. 9 Simulation 2: desired path and robot’s trajectories in X a Ya plane using PID controller for the inner loop
jectories of the robot in X a Ya plane, using the two controllers, are illustrated and compared in Fig. 9, which shows a significant improvement obtained by the proposed controller comparing to the classical PID controller. To show the advantage of the outer-loop controller, the parameter vector in the adaptive feedforward ˆ is replaced with a constant estimated value term, Yd ϑ, with 5% deviation from the nominal values. The robot
123
0.5 0.4
0.3 -0.3
Desired trajectory Robot trajectory
0.6
y [m]
y [m]
0.6
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
x [m]
Fig. 12 Simulation 4: desired path and robot’s trajectories in X a Ya plane having noise in measurements of position and velocity of the end-effector, also in the load cell data
trajectory in X a Ya plane for this case is shown in Fig. 10 and compared with the result of the adaptive feedforward term. The tensions of cables for this simulation are also illustrated in Fig. 11. The results demonstrate a clear improvement in performance through applying the proposed controller, as satisfactory trajectory fol-
Robust trajectory tracking control of cable-driven parallel robots
lowing is obtained with applying less control energy than the non-adaptive controller. In order to evaluate the performance of the proposed controller in a more realistic condition, the position and velocity information of the end-effector together with the load cell measurements are augmented with white noise. The covariance of noise for these three quantities is, respectively, set at 2 × 10−3 , 2 × 10−4 , and 20. The trajectory of the robot in X a Ya plane for this simulation is illustrated in Fig. 12, which shows the robustness of the controller in the presence of noise in the measurements. 5 Conclusion This paper addressed the problem of stable trajectory tracking control of fully constrained cable-driven robots when the actuation system was coupled with high-viscosity gearboxes to handle heavy loads. The controller design is performed in two steps. First, an adaptive robust feedback controller with an adaptive feedforward compensation term is designed for the robot dynamics. This controller deals with the unstructured and parametric uncertainties of the dynamic model of the robot. An important advantage of this controller is that it generates bounded desired tension forces for the cables, which may saturate the control effort when there is a large tracking error. A corrective term is utilized to ensure positive tensions for the cables. Second, a robust controller is designed for the inner-loop dynamics, which compensates for the dynamics of the gearbox, including friction. The stability of the overall system is analyzed based on the theory of cascaded systems, and it is guaranteed that the system is USPAS. The controller performance is validated through simulations on a 4-cable planar cable robot, and satisfactory results are obtained in both nominal and perturbed conditions. Acknowledgements This work was supported by the National Research Foundation Korea (NRF) (2014R1A2A1A11053989) and Dual Use Technology Program of Civil and Military.
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