Practical adaptive position feedback regulator for parallel robots with bounded inputs

Abstract

This paper addresses the regulation problem of parallel robots by a proportional derivative plus desired gravity compensator (PD-DGC) controller. Due to inaccurate measurements, unmodeled dynamics, and vibrations specially in cable-driven robots and external disturbance in practice, the model of the robot is often plagued with kinematic and dynamic uncertainties. In this paper, two new generations of PD-DGC controller, namely adaptive with respect to the parameters in gravity term, and time-varying PD-DGC in the presence of bounded disturbance, are proposed. Toward not requiring accurate velocity measurement, PD-DGC with merely position feedback in complement to the time-varying controller is designed in the presence of bounded control efforts. Incorporating both methods to establish a simple but strong robust adaptive controller is also investigated by adding an extra assumption on adapted parameters. The asymptotic stability of the closed-loop system is analyzed by the Lyapunov direct method. Experimental results on 2-DOF eye surgery and 3-DOF flexible link ARAS cable-driven robot demonstrate the effectiveness of the proposed approaches in practice.

Keywords

parallel robots regulation adaptive control bounded input cable-driven robots

1. Introduction

Despite being vastly utilized in many industrial applications, robotics control methods suffer from challenging issues such as unmodeled dynamics/external disturbance, input constraint, inaccurate velocity for feedback, etc. that make it difficult to achieve a promising efficiency. For this purpose, a considerable amount of literature has been published on control strategies addressing these problems, see for example, Chen et al. (2018); Harandi et al. (2021b); Moghaddam et al. (2020); Sun et al. (2016). Nevertheless, the major difficulty of most methods is their complexity that leads to a gap between industrial exposure and the current state of the art control science. It is well-known that proportional derivative with gravity compensation (PD-GC) is a common controller for regulation objectives in the industry Yang et al. (2010). A particular form of this controller, which is PD with desired gravity compensator (PD-DGC), is basically developed to stabilize a serial robot with revolute joints in joint space by merely compensation of gravity torque in the desired pose Kelly (1997); Niu et al. (2018); Tummalapalli (2019); Tzafestas (2012); Zavala-Rio and Santibanez (2007); Su and Parra-Vega (2008); Zavala-Río and Santibáñez (2006). The main advantage of this controller compared to the simple PD-GC is that it merely needs the exact value of gravity torque in the desired pose, instead of gravity in the actual pose. Therefore, PD-DGC is simpler and more suitable in practice. It has been shown in Kelly (1997) that global asymptotic stability is achievable if the proportional gain is higher than the maximum value of the partial derivative of gravity torque with respect to configuration variables. In the mentioned papers, it has been assumed that the robot’s model is precisely known without any uncertainties, the actuators are not limited, and all the states are available for feedback. Since these assumptions fail in practical applications, design of new PD-based controllers improving the performance of the system, ensuring asymptotic stability of the closed-loop system, and ease of implementation in the industry on a wide class of robots, is a stringent requirement.

Some researchers have tried to address these shortcomings, and as an extension, PD-GC for serial robots with bounded inputs has been developed using analytical methods López-Araujo et al. (2013a,b). Based on the structure of these controllers, they are called SP-SD Santibáñez and Kelly (1996). The main idea in these papers is the representation of gravity torque in regressor form and using SP-SD functions to generate a bounded input control law. Furthermore, in Yang et al. (2021), a modified version of PD-GC for regulation control of serial robots in joint space has been proposed such that asymptotic stability of desired pose together with correct estimation of gravity vector is ensured by merely position feedback. A robust PD controller based on linear matrix inequality for a 2-DOF flexible link robot has been reported in Mohamed et al. (2016). Curious readers are referred to Li and Li (2020, 2021) to see recent developments of bounded input controllers.

In contrast to the above-mentioned PD-GC controllers, less attention has been paid to the further development of PD-DGC. In Zavala-Rio and Santibanez (2007), a bounded input PD-DGC for serial robots has been proposed and asymptotic stability has been proved using Lyapunov direct method. Furthermore, position feedback with bounded input version of the PD-DGC has been designed in Su and Parra-Vega (2008) in which the stability of the closed-loop system depends on an unwarranted condition on damping friction coefficients. The main disadvantage of the last two mentioned works is their need to access to exact dynamic information of the system. Furthermore, these research papers do not study the effect of external disturbance on the performance of the PD-DGC controller. Moreover, to the best of authors’ knowledge, still PD-DGC is not developed and examined on parallel robots.

Parallel robots are closed-loop mechanisms such that the end-effector is attached to the base through some kinematic chains Tajdari and Ebrahimi Toulkani (2021). These robots outperform serial manipulators in terms of agility and precision Harandi and Taghirad (2017). However, a smaller workspace is the most crucial deficiency of them. Cable-driven robots are a particular well-developed type of parallel robots in which the rigid links are replaced by flexible cables Lin et al. (2016). They inherit most of the positive features of parallel robots while enhancing the reachable workspace Bayani et al. (2016). Nevertheless, due to the admissible unidirectional force in the cables and flexibility of them that results in vibrations, control of cable-driven robots is a challenging problem Sun et al. (2020), see also Jamshidifar et al. (2020) and references therein for more details about the effects of vibrations of the cables. In order to stabilize a parallel mechanism, PD-GC is a common controller due to its simplicity. A PD-GC based on passivity notion had been proposed in Harandi et al. (2019) for cable-driven robots while the authors tried to ensure positive tension in cables using the interconnection matrix. However, in contrast to serial robots, less attention has been paid to develop the PD-GC for parallel robots.

In this paper, we design PD-DGC in task space for a class of parallel robots with dynamic uncertainties. This class includes robotic manipulators with revolute joints as well as cable-driven robots if the mass of cables is negligible. In this paper, a new adaptive PD-DGC (APD-DGC) controller is first proposed with respect to unknown parameters in gravity torque/force, and asymptotic stability is ensured considering a suitable strict Lyapunov function. Since the robot’s velocity is not precisely available, and the actuators have limited power, a novel output feedback time-varying PD-DGC with bounded inputs (OFTVPD-DGC-BI) is developed, to address the unmodeled dynamics and external disturbance while considering the input constraints by using a particular case of SP-SD functions. For this purpose, an observer for velocity is designed, and the stability of the closed-loop system is analyzed using the Lyapunov direct method. Since the combination of adaptive and robust methods results in a vigorous controller, incorporation of both approaches to build an adaptive OFTVPD-DGC-BI (AOFTVPD-DGC-BI) controller is also considered. It is shown that asymptotic stability is ensured under the assumption that the unknown parameters in gravity torque/force are correctly estimated. Note that although it is a well-known fact that increasing the damping term will lead to the reduction of the ultimate bound of the error Qu (1994), in this work, only the proportional gain is increased to have less dependency on the inaccurate velocity. Notice that the proposed controllers can significantly improve the performance of simple PD-GC and PD-DGC, while they are simply implementable in practice.

To the best of the authors’ knowledge, these types of PD-DGC have not been developed in the literature, even for serial robots. Furthermore, the design of a general position feedback regulator with bounded inputs has not been reported for parallel robots in literature, see discussion section for an analysis about the proposed controllers. Note that although the proposed controllers are designed for parallel robots, they can also be applied to serial robots with minor modifications. The proposed controllers are implemented on two different structure parallel robots to verify the performance of the methods in practice. The first manipulator is a complex 2-DOF eye surgery robot called ARAS-Diamond and the second one is a 3-DOF fully actuated cable-driven robot named ARAS-CAM. These systems provide an opportunity to analyze the efficiency of the proposed controllers on rigid manipulators as well as flexible link robots that suffer from input constraints and natural vibrations. Furthermore, the adaptive controller proposed in Harandi et al. (2021a) is implemented on ARAS-CAM to compare the results.

2. Preliminaries

Dynamic equation of a general n-DOF parallel robot in task space is given as (Taghirad 2013, Ch.5)

M 〈 X 〉 \ddot{X} + C 〈 X, \dot{X} 〉 \dot{X} + g 〈 X 〉 = J^{T} 〈 X 〉 τ,

(1)

where

X, \dot{X}, \ddot{X} \in ℝ^{n}

are generalized position, velocity, and acceleration vectors, respectively,

M 〈 X 〉 \in ℝ^{n \times n}

is inertia matrix,

C 〈 X, \dot{X} 〉 \in ℝ^{n \times n}

is the matrix of Coriolis and centrifugal terms,

g 〈 X 〉 \in ℝ^{n}, J 〈 X 〉 \in ℝ^{m \times n}

and

τ \in ℝ^{m}

denote gravity torque/force, Jacobian matrix and the control inputs, respectively, and m denotes the number of actuators. Note that to prevent ambiguity of readers, in this paper the elements that a matrix or a vector is a function of them, are represented by ⟨⋅⟩, and this dependency may be omitted when it is obvious. The terms in dynamical model (1) satisfy some important properties (Taghirad 2013, ch. 5).

property 1

The inertia matrix M⟨X⟩ is symmetric positive definite.

property 2

The matrix $\dot{M} - 2 C$ is skew-symmetric, that is, for every vector $ξ \in ℝ^{n}$ , $ξ^{T} (\dot{M} - 2 C) ξ = 0$ .

property 3

The gravity term in (1) is linear with respect to parameters of the system. That is, g⟨X⟩ = Y_g⟨X⟩θ_g.

property 4

The terms in matrix $C 〈 X, \dot{X} 〉$ are linear respect to $\dot{X}$ . Hence, there exists a positive constant κ_c such that ‖C⟨α, β⟩γ‖ ≤ κ_c‖β‖‖γ‖.

property 5

In the robots with actuated revolute joints, g⟨X⟩ is bounded. In other words, every element of gravity torque has a lower and upper bound ${\underline{γ}}_{i} \leq g_{i} 〈 X 〉 \leq {\bar{γ}}_{i}$ . ∂g⟨X⟩/∂X is also bounded and has the upper bound ‖∂g⟨X⟩/∂X‖ ≤ κ_g, and therefore

‖ g 〈 X 〉 - g 〈 Y 〉 ‖ \leq κ_{g} ‖ X - Y ‖ .

(2)

property 6

For every X, Y in the workspace of the robot, we have the following inequality

\frac{k_{g}}{2} ‖ Y - X ‖^{2} + U 〈 X 〉 - U 〈 Y 〉 + g^{T} 〈 Y 〉 (Y - X) \geq 0.

Notice that as described in Taghirad (2013) and also the articles with this topic on the serial robots, for example, Kelly (1997), the last two properties holds for the robots with merely actuated revolute joints. However, cable-driven robots also satisfy these properties if the mass of cables is negligible.

3. Main results

In this section, PD with desired gravity compensation is designed for parallel robots that satisfy the above properties. A necessary condition for the controller’s gains is derived, and APD-DGC with respect to parameters in g⟨X⟩ will be proposed. Then, OFTVPD-DGC-BI is designed such that merely position feedback is required and stability is ensured in the presence of uncertainties and disturbance. Finally, the incorporation of both methods is analyzed. Notice that although the proposed controllers are designed for parallel robots, they can be used for any other mechanical systems if they satisfy the above properties. Therefore, the controllers are also applicable to serial robots with actuated revolute joints with slight modifications. Note that similar to previous articles that design a controller in task space for serial or parallel robots Dixon (2007); Harandi et al. (2019), it is assumed that the initial and desired pose of the robot are within the reachable workspace of the robot and far from its singular points.

3.1 Adaptive PD-DGC

In contrast to a simple PD-GC which compensate gravity torque in the actual configuration of the robot, PD-DGC only compensates the gravity in the desired configuration. Hence, the control law is as follows

τ = J^{- T} 〈 X 〉 (g 〈 X_{d} 〉 - K_{p} \tilde{X} - K_{d} \dot{X}),

(3)

in which X_d denotes desired position,

\tilde{X} = X - X_{d}

, and K_p and K_d are positive definite symmetric matrices¹. First, let us analyze the equilibrium point of the closed-loop system denoted by X*. Replace the control law (3) into (1), the closed-loop equation is

M \ddot{X} + C \dot{X} + g 〈 X 〉 - g 〈 X_{d} 〉 + K_{p} \tilde{X} + K_{d} \dot{X} = 0.

(4)

In X = X*, velocity and acceleration are zero. Thus, the resulting dynamic is

g 〈 X^{*} 〉 - g 〈 X_{d} 〉 + K_{p} (X^{*} - X_{d}) = 0.

(5)

Invoking Kelly (1997), in order to derive the condition that X* = X_d is the only equilibrium point of the system, let us define

ψ : = X^{*} - X_{d}, ρ 〈 ψ, X_{d} 〉 : = K_{p}^{- 1} (g 〈 X_{d} 〉 - g 〈 X_{d} + ψ 〉) .

Note that considering (5), it is clear that ψ = 0 is an equilibrium point of the system. Using property 5, we obtain

‖ ρ 〈 α, X_{d} 〉 - ρ 〈 β, X_{d} 〉 ‖ \leq \frac{κ_{g}}{λ_{m} {K_{p}}} ‖ α - β ‖,

in which λ_m{K_p} denotes the minimum eigenvalue of K_p, and κ_g was introduced in (2). Finally, by applying contraction-mapping theorem, it is inferred that K_p should satisfy the following inequality

λ_{m} {K_{p}} > κ_{g},

(6)

so that ψ = 0 is the unique equilibrium point of the robot. Now consider the following Lyapunov candidate

V = \frac{1}{2} {\dot{X}}^{T} M 〈 X 〉 \dot{X} + U 〈 X 〉 + \frac{1}{2} {\tilde{X}}^{T} K_{p} \tilde{X} - {\tilde{X}}^{T} g 〈 X_{d} 〉,

(7)

where U⟨X⟩ denotes the potential energy of the robot. Note that it is a suitable Lyapunov candidate because it is positive definite respect to

\dot{X}

. Furthermore, it is also positive definite respect to

\tilde{X}

since

{\frac{\partial V_{n}}{\partial \tilde{X}} |}_{\tilde{X} = 0} = (g 〈 X 〉 + K_{p} \tilde{X} - g 〈 X_{d} 〉) |_{\tilde{X} = 0} = 0,

and

‖ {\frac{\partial^{2} V_{n}}{\partial {\tilde{X}}^{2}} |}_{\tilde{X} = 0} ‖ \geq λ_{m} {K_{p}} - ‖ \frac{\partial g 〈 X 〉}{\partial \tilde{X}} ‖

is positive definite due to property 5 and condition (6) in which

V_{n} : = U 〈 X 〉 + \frac{1}{2} {\tilde{X}}^{T} K_{p} \tilde{X} - {\tilde{X}}^{T} g 〈 X_{d} 〉 .

Time derivative of (7) is

\dot{V} = {\dot{X}}^{T} M \ddot{X} + \frac{1}{2} {\dot{X}}^{T} \dot{M} \dot{X} + {\dot{X}}^{T} (g 〈 X 〉 + K_{p} \tilde{X} - g 〈 X_{d} 〉),

in which we used the relation that $\frac{\partial U}{\partial X} = g 〈 X 〉$ . Replacing $M \ddot{X}$ from (4) and considering property 2, one can easily verify that $\dot{V} = - {\dot{X}}^{T} K_{d} \dot{X}$ which yields $\dot{X}$ converges to zero. Asymptotic stability is ensured simply by Krasovskii–Lasalle theorem Lee et al. (2018).

In control law (3), it is assumed that gravity torque in the desired position is precisely known. However, this is not practically satisfied when the parameters in gravitational torques/forces are unknown. This situation leads to a constant error since X* ≠ X_d is the solution of (5). With the purpose of rectifying this issue, an adaptive PD controller with desired gravity compensation is proposed in the following theorem to ensure asymptotic stability within the workspace.

Theorem 1

Consider a parallel robot with properties 1–5, dynamic equation (1), control law

τ = J^{- T} 〈 X 〉 (Y_{g} 〈 X_{d} 〉 {\hat{θ}}_{g} - K_{p} \tilde{X} - K_{d} \dot{X}),

(8)

and the following adaptation law

{\dot{\hat{θ}}}_{g} = - Γ^{- 1} Y_{g}^{T} 〈 X_{d} 〉 (\dot{X} + μ f 〈 \tilde{X} 〉),

(9)

in which Γ is a positive definite constant matrix and μ is a constant satisfying

\begin{array}{l} 0 < μ < \min {\sqrt{\frac{(λ_{m} {K_{p}} - κ_{g})}{λ_{M} {M}}}, \\ \frac{λ_{m} {K_{d}} (λ_{m} {K_{p}} - κ_{g})}{\begin{matrix} \frac{λ_{M}^{2} {K_{d}}}{4} + (λ_{m} {K_{p}} - κ_{g}) (δ λ_{M} {M} + κ_{c}) \end{matrix}}} \end{array}

(10)

with λ_M{M} means the maximum eigenvalue of M and

f 〈 \tilde{X} 〉 = {[f 〈 {\tilde{X}}_{1} 〉, \dots, f 〈 {\tilde{X}}_{n} 〉]}^{T} \in ℝ^{n}

is a smooth function with the following properties:

p1: $f 〈 \tilde{X} 〉$ is a strictly increasing function whose elements are only in the first and third quarters of the coordinate and f(0) = 0 (i.e. ${\tilde{X}}^{T} f 〈 \tilde{X} 〉 \geq 0$ ),

p2: $‖ f 〈 \tilde{X} 〉 ‖ \leq 1$ ,

p3: $‖ \frac{\partial f}{\partial X} ‖ \leq δ$ .

Then the equilibrium point X_d is asymptotically stable. □

Note that $tanh 〈 \tilde{X} 〉, \frac{2}{π} arctan 〈 \tilde{X} 〉$ and $\tilde{X} / (η + ‖ \tilde{X} ‖), η \in ℝ^{+}$ are some typical examples of $f 〈 \tilde{X} 〉$ . Furthermore, the controller gains shall be chosen such that inequality (6) is satisfied which leads to satisfaction of condition (10) with a positive value for μ.

proof

Referring to property 3, gravity term may be written in regressor form as g⟨X⟩ = Y_g⟨X⟩θ_g. Hence, g⟨X_d⟩ = Y_g⟨X_d⟩θ_g. The equation of closed-loop system is

M \ddot{X} + C \dot{X} + g 〈 X 〉 - Y_{g} 〈 X_{d} 〉 {\hat{θ}}_{g} + K_{p} \tilde{X} + K_{d} \dot{X} = 0

(11)

Consider the following Lyapunov candidate

\begin{array}{l} V = \frac{1}{2} {\dot{X}}^{T} M 〈 X 〉 \dot{X} + μ {\dot{X}}^{T} M 〈 X 〉 f 〈 \tilde{X} 〉 + U 〈 X 〉 \\ - U 〈 X_{d} 〉 + \frac{1}{2} {\tilde{X}}^{T} K_{p} \tilde{X} - {\tilde{X}}^{T} g 〈 X_{d} 〉 + \frac{1}{2} {\tilde{θ}}_{g}^{T} Γ {\tilde{θ}}_{g}, \end{array}

(12)

with

{\tilde{θ}}_{g} = {\hat{θ}}_{g} - θ_{g}

. It is clear that the last term is positive definite with respect to

\tilde{θ}

. In order to show that the other terms are positive definite, we rewrite them in the following form

\frac{1}{2} {(\dot{X} - μ f 〈 \tilde{X} 〉)}^{T} M 〈 X 〉 (\dot{X} - μ f 〈 \tilde{X} 〉) + \frac{1}{2} {\tilde{X}}^{T} K_{p} \tilde{X} - \frac{μ^{2}}{2} f^{T} 〈 \tilde{X} 〉 M 〈 X 〉 f 〈 \tilde{X} 〉 + U 〈 X 〉 - U 〈 X_{d} 〉 - {\tilde{X}}^{T} g 〈 X_{d} 〉 .

The first term is non-negative and disappear in $\dot{X} = μ f 〈 \tilde{X} 〉$ . Now, it is sufficient to show that V is positive definite with respect to $\tilde{X}$ . For this purpose, we have the following inequality

\begin{array}{l} \frac{1}{2} {\tilde{X}}^{T} K_{p} \tilde{X} - \frac{μ^{2}}{2} f^{T} 〈 \tilde{X} 〉 M 〈 X 〉 f 〈 \tilde{X} 〉 + U 〈 X 〉 - U 〈 X_{d} 〉 \\ - {\tilde{X}}^{T} g 〈 X_{d} 〉 \geq \frac{1}{2} λ_{m} {K_{p}} ‖ \tilde{X} ‖^{2} - \frac{μ^{2}}{2} λ_{M} {M} ‖ \tilde{X} ‖^{2} \\ + U 〈 X 〉 - U 〈 X_{d} 〉 - {\tilde{X}}^{T} g 〈 X_{d} 〉, \end{array}

(13)

where the following terms were used in (13)

\frac{1}{2} {\tilde{X}}^{T} K_{p} \tilde{X} \geq \frac{1}{2} λ_{m} {K_{p}} ‖ \tilde{X} ‖^{2},

\frac{μ^{2}}{2} f^{T} 〈 \tilde{X} 〉 M 〈 X 〉 f 〈 \tilde{X} 〉 \leq \frac{μ^{2}}{2} λ_{M} {M} ‖ \tilde{X} ‖^{2} .

Invoking property 6 and the first term in (10), it is clear that (13) is positive definite. Differentiate (12) with respect to time and substitute $M 〈 X 〉 \ddot{X}$ from (11) in it, yields

\begin{array}{l} \dot{V} = - {\dot{X}}^{T} K_{d} \dot{X} + μ {\dot{X}}^{T} M 〈 X 〉 \dot{f} 〈 \tilde{X} 〉 + {\dot{X}}^{T} Y_{g} 〈 X_{d} 〉 {\tilde{θ}}_{g} \\ + μ {\dot{X}}^{T} C 〈 X, \dot{X} 〉 f 〈 \tilde{X} 〉 - μ f^{T} 〈 \tilde{X} 〉 K_{d} \dot{X} - μ f^{T} 〈 \tilde{X} 〉 K_{p} \tilde{X} \\ - μ f^{T} 〈 \tilde{X} 〉 (Y_{g} 〈 X 〉 θ_{g} - Y_{g} 〈 X_{d} 〉 {\hat{θ}}_{g}) + {\tilde{θ}}_{g}^{T} Γ {\dot{\hat{θ}}}_{g}, \end{array}

where we used property 2 to simplify it. Utilizing property 4 and property 5 together with properties of

f 〈 \tilde{X} 〉

, with some manipulations, the upper bound of

\dot{V}

\begin{array}{l} \dot{V} \leq - (λ_{m} {K_{d}} - μ (δ λ_{M} {M} + κ_{c})) ‖ \dot{X} ‖^{2} \\ + μ λ_{M} {K_{d}} ‖ f 〈 \tilde{X} 〉 ‖‖ \dot{X} ‖ - μ f^{T} 〈 \tilde{X} 〉 K_{p} \tilde{X} \\ - μ {(Y_{g} 〈 X 〉 θ_{g} - Y_{g} 〈 X_{d} 〉 {\hat{θ}}_{g})}^{T} f 〈 \tilde{X} 〉 + {\dot{X}}^{T} Y_{g} 〈 X_{d} 〉 {\tilde{θ}}_{g} + {\tilde{θ}}_{g}^{T} Γ {\dot{\hat{θ}}}_{g} . \end{array}

(14)

The last four terms can be rewritten as follows

\begin{array}{l} - μ f^{T} 〈 \tilde{X} 〉 K_{p} \tilde{X} - μ {(g 〈 X 〉 - g 〈 X_{d} 〉)}^{T} f 〈 \tilde{X} 〉 + {\tilde{θ}}_{g}^{T} Y_{g}^{T} 〈 X_{d} 〉 \\ (\dot{X} + μ f 〈 \tilde{X} 〉) + {\tilde{θ}}_{g}^{T} Γ {\dot{\hat{θ}}}_{g} \leq - μ λ_{m} {K_{p}} f^{T} 〈 \tilde{X} 〉 \tilde{X} \\ + μ κ_{g} ‖ \tilde{X} ‖‖ f 〈 \tilde{X} 〉 ‖ \leq - μ (λ_{m} {K_{p}} - κ_{g}) ‖ f 〈 \tilde{X} 〉 ‖^{2}, \end{array}

(15)

in which

\pm μ g^{T} (X_{d}) f 〈 \tilde{X} 〉

was added to derive the first inequity,

{\dot{\hat{θ}}}_{g}

was replaced by the adaptation law (9), property 5 was used and the following inequity obtained from p1 and p2, was utilized

- f^{T} 〈 \tilde{X} 〉 K_{p} \tilde{X} \leq - λ_{m} {K_{p}} f^{T} 〈 \tilde{X} 〉 \tilde{X} \leq - λ_{m} {K_{p}} ‖ f 〈 \tilde{X} 〉 ‖^{2} .

Finally, replacing (15) into (14), yields

\begin{array}{l} \dot{V} \leq - μ [\begin{matrix} ‖ f 〈 \tilde{X} 〉 ‖ & ‖ \dot{X} ‖ \end{matrix}] \\ [\begin{matrix} λ_{m} {K_{p}} - κ_{g} & - \frac{1}{2} λ_{M} {K_{d}} \\ - \frac{1}{2} λ_{M} {K_{d}} & \begin{matrix} \frac{1}{μ} λ_{m} {K_{d}} - δ λ_{M} {M} \\ - κ_{c} \end{matrix} \end{matrix}] [\begin{matrix} ‖ f 〈 \tilde{X} 〉 ‖ \\ ‖ \dot{X} ‖ \end{matrix}] \end{array}

which is a globally negative definite function with respect to (10).■

Remark 1

It is easy to show that using the control law (8) and the following adaptation law ${\dot{\hat{θ}}}_{g} = - Γ^{- 1} Y_{g}^{T} 〈 X_{d} 〉 \dot{X}$ , derivative of the following Lyapunov candidate

V = \frac{1}{2} {\dot{X}}^{T} M \dot{X} + U + \frac{1}{2} {\tilde{X}}^{T} K_{p} \tilde{X} - {\tilde{X}}^{T} g 〈 X_{d} 〉 + \frac{1}{2} {\tilde{θ}}_{g}^{T} Γ {\tilde{θ}}_{g},

becomes

\dot{V} = - {\dot{X}}^{T} K_{d} \dot{X}

. Using Krasovskii–Lasalle theorem,

\dot{X}

and thus

{\dot{\hat{θ}}}_{g}

converge to zero. The residual dynamic is given as

- Y_{g} 〈 X_{d} 〉 {\tilde{θ}}_{g} + K_{p} (X^{*} - X_{d}) = 0,

with X* as the equilibrium point of the closed-loop system. Hence, X* = X_d if the identification error $Y_{g} 〈 X_{d} 〉 {\tilde{θ}}_{g}$ is zero. This unnecessary requirement is removed in Theorem 1 utilizing the strict Lyapunov function (12).□

Remark 2

The closed-loop system (11) in stationary condition $(\ddot{X} = \dot{X} = 0)$ is $Y_{g} {\tilde{θ}}_{g} = K_{p} \tilde{X}$ . In Theorem 1, it was shown that $\tilde{X}$ converges to zero. Therefore, $Y_{g} {\tilde{θ}}_{g}$ converges to zero. This means that the gravity in desired pose (g⟨X_d⟩) is correctly estimated. However, ${\tilde{θ}}_{g}$ does not necessarily converge to zero. In other words, ${\tilde{θ}}_{g} \in ker {Y_{g}}$ ; where ker denotes null space. Thus, ${\tilde{θ}}_{g}$ converges to zero if Y_g has not a kernel that a sufficient condition for this property is $Y_{g}^{T} Y_{g}$ to be full rank.□

3.2 Output feedback time-varying PD-DGC with bounded inputs

One of the most important disadvantages of PD-DGC is its dependency on the exact value of gravitational torque/force in the desired pose. Utilizing an adaptation law was an idea to remedy this problem merely when the gravity term parameters are unknown. In other words, APD-DGC proposed in Theorem 1 needs a precise model of g⟨X⟩. In order to overcome these shortcomings, a simple novel output feedback PD-DGC controller with a time-varying gain is introduced such that asymptotic stability with bounded input is still ensured, and the need for exact gravity compensation in X = X_d is reduced with the expense of an unbounded gain. Note that the purpose of introducing time-varying gain in the controller is forcing the system to stabilize in the desired pose. Before presenting the controller, we need the following assumption.

Assumption 1

The robot is capable to compensate the gravity in the workspace, $τ_{i_{\min}} < \min_{X} {(J^{- T} 〈 X 〉 g 〈 X 〉)}_{i}, τ_{i_{\max}} > \max_{X} {(J^{- T} 〈 X 〉 g 〈 X 〉)}_{i}$ , or conservatively: $τ_{i_{\min}} < \min_{X, Y} {(J^{- T} 〈 X 〉 g 〈 Y 〉)}_{i}, τ_{i_{\max}} > \max_{X, Y} {(J^{- T} 〈 X 〉 g 〈 Y 〉)}_{i}$ , in which X, Y are inside of workspace, $τ_{i_{\min}}$ and $τ_{i_{\max}}$ denote the minimum and maximum required torques, respectively, and (⋅)_i means ith value of (⋅).□

In the following theorem, it is assumed that the exact model of g⟨X⟩ is not known; thus, g*⟨X_d⟩ with g* ≠ g is replaced in the control law. This controller leads to the system stabilize in another pose denoted by X* ≠ X_d. Hence, it may be interpreted that g⟨X*⟩ is replaced in the control law. See Remark 3 for another viewpoint of this representation.

Theorem 2

Consider a parallel robot with dynamic formulation (1) satisfying properties 1–5 with the following control law

\begin{array}{l} τ & = J^{- T} 〈 X 〉 (g 〈 X^{*} 〉 - ζ_{1} tanh 〈 K_{p} 〈 t 〉 \tilde{X} 〉 - ζ_{2} tanh 〈 K_{d} ν 〉) \end{array}

(16a)

\begin{array}{l} \dot{ν} & = - A ν + B \dot{\tilde{X}} \end{array}

(16b)

in which (16b) denotes the dynamic of observer, A and B are positive definite matrices, it is assumed that gravity is not compensated at X = X_d, diagonal time-varying matrix K_p⟨t⟩ should satisfy the following properties:

p1:ζ₁λ_m{K_p⟨t⟩}≥ κ_g,

p2: lim_t→+∞ λ_m{K_p⟨t⟩} = + ∞,

the matrices A, B and the diagonal matrix K_d are positive definite. Additionally, Assumption 1 is satisfied. The scalars $ζ_{1}, ζ_{2} \in ℝ^{+}$ should be chosen such that the following inequality holds

τ_{\min} < \max_{X} {J^{- T} 〈 X 〉} g 〈 X^{*} 〉 + ζ_{1} + ζ_{2} < τ_{\max} .

Then X_d is asymptotically stable with a bounded input position feedback controller.□

Note that for simplicity, we incorporate hyperbolic tangent in the control law. One can define other functions satisfying the properties introduced in Theorem 1.

proof

Boundedness of the inputs is clear with respect to Assumption 1 and suitable choice of ζ₁ and ζ₂. Dynamic equation of the closed-loop system is

\begin{array}{l} M 〈 X 〉 \ddot{X} + C 〈 X, \dot{X} 〉 \dot{X} + g 〈 X 〉 - g 〈 X^{*} 〉 \\ + ζ_{1} tanh 〈 K_{p} 〈 t 〉 \tilde{X} 〉 + ζ_{2} tanh 〈 K_{d} ν 〉 = 0. \end{array}

(17)

In order to prove the stability, consider the following Lyapunov candidate

\begin{array}{l} V = \frac{1}{2} {\dot{X}}^{T} M 〈 X 〉 \dot{X} + U 〈 X 〉 - U 〈 X_{d} 〉 - {\tilde{X}}^{T} g 〈 X^{*} 〉 \\ + ζ_{1} \int_{0}^{\tilde{X}} \tanh 〈 K_{p} 〈 t 〉 r 〉 d r + ζ_{2} \int_{0}^{ν} B^{- 1} \tanh 〈 K_{d} r 〉 d r . \end{array}

The first and last terms are positive definite with respect to $\dot{X}$ and ν, respectively. The other remaining terms, those are as follows

V_{n} : = U 〈 X 〉 - {\tilde{X}}^{T} g 〈 X^{*} 〉 + ζ_{1} \int_{0}^{\tilde{X}} tanh 〈 K_{p} 〈 t 〉 r 〉 d r,

are positive definite respect to $\tilde{X}$ if ${\frac{\partial V_{n}}{\partial \tilde{X}} |}_{\tilde{X} = 0} = 0$ and ${\frac{\partial^{2} V_{n}}{\partial {\tilde{X}}^{2}} |}_{\tilde{X} = 0}$ is positive definite. Note that

\frac{\partial V_{n}}{\partial \tilde{X}} = 0 \to tanh 〈 K_{p} 〈 t 〉 \tilde{X} 〉 = \frac{(g 〈 X^{*} 〉 - g 〈 X 〉)}{ζ_{1}}

(18)

Let us show it by contradiction. Assume that X** ≠ X_d is the solution of the above equality. The right hand side of (18) is constant while K_p⟨t⟩(X** − X_d) is changing based on p2. Hence this contradicts the assumption ${\frac{\partial V_{n}}{\partial \tilde{X}} |}_{X = X^{* *}} = 0$ and thus, it is necessarily zero in X = X_d. In other words

‖ {\frac{\partial^{2} V_{n}}{\partial {\tilde{X}}^{2}} |}_{\tilde{X} = 0} ‖ \geq ζ_{1} λ_{m} {K_{p} 〈 t 〉} - ‖ \frac{\partial g 〈 X 〉}{\partial \tilde{X}} ‖

is clearly positive definite respect to property 5 and p1. Thus, the proposed Lyapunov candidate is suitable. Time derivative of Lyapunov function is

\begin{array}{l} \dot{V} = {\dot{X}}^{T} M 〈 X 〉 \ddot{X} + \frac{1}{2} {\dot{X}}^{T} \dot{M} 〈 X 〉 \dot{X} + {\dot{X}}^{T} g 〈 X 〉 - {\dot{X}}^{T} g 〈 X^{*} 〉 \\ + ζ_{1} {\dot{X}}^{T} \tanh 〈 K_{p} 〈 t 〉 \tilde{X} 〉 + ζ_{2} {\dot{ν}}^{T} B^{- 1} \tanh 〈 K_{d} ν 〉 . \end{array}

Replacing closed-loop dynamic (17) and observer (16b) in it yields $\dot{V} = - ζ_{2} ν^{T} A B^{- 1} tanh 〈 K_{d} ν 〉$ which means ν converges asymptotically to zero since $\dot{V}$ is negative definite with respect to ν. Substitute ν = 0 in the dynamic of observer (16b), results in $\dot{X} \to 0$ since $\dot{ν}$ is also zero and X_d is constant. Hence, the system is settled in a pose. By considering the dynamic of closed-loop system (17) with $\ddot{X} = \dot{X} = ν = 0$ , property p1 and using Krasovskii-Lasalle theorem, it is inferred that X_d is the only equilibrium point of the system. Hence, $\tilde{X}$ converges to zero asymptotically.■

Note that a simple version of control law (16) with full state feedback and unbounded inputs is

τ = J^{- T} 〈 X 〉 (g 〈 X^{*} 〉 - K_{p} 〈 t 〉 \tilde{X} - K_{d} \dot{X}),

(19)

and with output feedback yields

\begin{array}{l} τ & = J^{- T} 〈 X 〉 (g 〈 X^{*} 〉 - K_{p} 〈 t 〉 \tilde{X} - K_{d} ν), \\ \dot{ν} & = - A ν + B \dot{\tilde{X}}, \end{array}

(20)

which are called time-varying PD-DGC (TVPD-DGC) and output feedback time-varying PD-DGC (OFTVPD-DGC), respectively. Notice that ν only depends on

\tilde{X}

; thus, (16) is implementable without velocity measurement.

Remark 3

It is possible to rewrite closed-loop dynamic (17) in the following form

\begin{array}{l} M 〈 X 〉 \ddot{X} + C 〈 X, \dot{X} 〉 \dot{X} + g 〈 X 〉 - g 〈 X_{d} 〉 \\ + ζ_{1} tanh 〈 K_{p} 〈 t 〉 \tilde{X} 〉 + ζ_{2} tanh 〈 K_{d} ν 〉 + d = 0, \end{array}

(21)

where d = g⟨X_d⟩ − g⟨X*⟩ may be interpreted as a bounded disturbance. In other words, we may infer that the desired gravity is exactly compensated, but an external bounded disturbance is applied to the system. Hence, we can argue that the proposed controller is also robust against bounded disturbance.

Remark 4

In Remark 3, it is shown that the proposed controller in Theorem 2 is capable to reject disturbance such as inexact gravity compensation. Another interpretation of d is uncertain Jacobian matrix. In this case, the control law (16) is in the following form

\begin{array}{l} τ & = J^{*^{- T}} 〈 X 〉 (g 〈 X^{*} 〉 - ζ_{1} tanh 〈 K_{p} 〈 t 〉 \tilde{X} 〉 - ζ_{2} tanh 〈 K_{d} ν 〉) \end{array}

(22a)

\begin{array}{l} \dot{ν} & = - A ν + B \dot{\tilde{X}} \end{array}

(22b)

in which J* ≠ J. The closed-loop dynamic is in the following form

\begin{array}{l} M 〈 X 〉 \ddot{X} + C 〈 X, \dot{X} 〉 \dot{X} + g 〈 X 〉 - g 〈 X^{*} 〉 \\ + ζ_{1} tanh 〈 K_{p} 〈 t 〉 \tilde{X} 〉 + ζ_{2} tanh 〈 K_{d} ν 〉 + {\tilde{J}}^{- T} 〈 X 〉 \\ (g 〈 X^{*} 〉 - ζ_{1} tanh 〈 K_{p} 〈 t 〉 \tilde{X} 〉 - ζ_{2} tanh 〈 K_{d} ν 〉) = 0, \end{array}

(23)

in which

J^{*} = J + \tilde{J}

was replaced. Therefore, it is possible to represent the closed-loop dynamic (23) in the form of (21) with d as follows

\begin{matrix} g 〈 X_{d} 〉 - g 〈 X^{*} 〉 + + {\tilde{J}}^{- T} 〈 X 〉 \\ (g 〈 X^{*} 〉 - ζ_{1} \tanh 〈 K_{p} 〈 t 〉 \tilde{X} 〉 - ζ_{2} \tanh 〈 K_{d} ν 〉) . \end{matrix}

Hence, the proposed controller can also reject the uncertainties in kinematic parameters inside the Jacobian matrix.

Remark 5

Consider Lyapunov candidate (7) for the system with control law (19) and assume that K_p is constant. At first look, one may infer that a time-varying increasing K_d will lead to asymptotic stability since

\dot{V} \leq λ_{m} {K_{d}} ‖ \dot{X} ‖^{2} + ‖ \dot{X} ‖‖ d ‖, d = g 〈 X_{d} 〉 - g 〈 X^{*} 〉

and the ultimate bound will decrease by increasing K_d. Although this is a fundamental result in robotics Qu (1994); however, increasing K_d is not practically advisable due to noisy velocity signal. Hence, the proposed controller overcomes this issue while asymptotic stability is ensured. Notice that practically, we may not increase K_p⟨t⟩ to very large values that leads to a pretty negligible error remains. Despite positive features, this is a disadvantage of this controller compared to APD-DGC.□

3.3 Adaptive output feedback time varying PD-DGC with bounded inputs

Up to now, two different developments of PD-DGC for parallel robots were designed that possess special properties. One may conceive that incorporation of both methods will lead to a stronger controller since the adaptation law compensate the unknown parameters in gravity and time-varying gain is merely used to reject the external disturbance d. However, the resulting controller is applicable with bounded inputs if similar to Harandi et al. (2021a); López-Araujo et al. (2013a), a projection algorithm is defined for adaptation law while the unknown parameters should be estimated properly. This analysis is given in the following theorem.

Theorem 3

Given the parallel robotic system (1) satisfying properties 1–5 and Assumption 1 in the presence of bounded disturbance d⟨t⟩. Consider the following controller

\begin{array}{l} τ = J^{- T} 〈 X 〉 (Y_{g} 〈 X_{d} 〉 {\hat{θ}}_{g} - ζ_{1} tanh 〈 K_{p} 〈 t 〉 \tilde{X} 〉 - ζ_{2} tanh 〈 K_{d} ν 〉), \\ \dot{ν} = - A ν + B \dot{\tilde{X}}, - Γ^{- 1} Y_{g}^{T} (X_{d}) \dot{X} ≜ υ, \\ {\dot{\hat{θ}}}_{g} = {\begin{cases} 0 if {\hat{θ}}_{g} = θ_{g_{M}} & υ > 0 \\ 0 if {\hat{θ}}_{g} = θ_{g_{m}} & υ < 0 \\ υ else \end{cases} \end{array}

(24)

with the parameters and conditions on the gains given in Theorem 1 and 2, while

θ_{g_{m}}

and

θ_{g_{M}}

are the lower and upper bounds of θ_g, respectively. Then, the configuration variables asymptotically converge to their constant desired values with bounded input if the estimated parameters converge to real values.□

Notice that based on Remark 2, Theorem 3 can be applied to robots with a full rank matrix $Y_{g}^{T} Y_{g}$ such as cable-driven robots.

proof

The inputs are clearly bounded due to Assumption 1 and simple projection rule defined in (24) for adaptation law that results in $θ_{g_{m}} \leq {\hat{θ}}_{g} \leq θ_{g_{M}}$ . In order to perform the stability analysis, let us consider the following Lyapunov candidate

\begin{array}{l} V = \frac{1}{2} {\dot{X}}^{T} M 〈 X 〉 \dot{X} + U 〈 X 〉 - {\tilde{X}}^{T} g 〈 X_{d} 〉 + \frac{1}{2} {\tilde{θ}}_{g}^{T} Γ {\tilde{θ}}_{g} \\ + ζ_{1} \int_{0}^{\tilde{X}} \tanh 〈 K_{p} 〈 t 〉 r 〉 d r + ζ_{2} \int_{0}^{ν} B^{- 1} \tanh 〈 K_{d} r 〉 d r . \end{array}

Its derivative along the closed-loop trajectory of the system is derived as follows

\dot{V} = - ζ_{2} ν^{T} A B^{- 1} tanh 〈 K_{d} ν 〉 + {\dot{X}}^{T} Y 〈 X_{d} 〉 {\tilde{θ}}_{g} + {\dot{\hat{θ}}}_{g}^{T} Γ {\tilde{θ}}_{g} .

Substitute adaptation law (24), results into $\dot{V} = - ζ_{2} ν^{T} A B^{- 1} tanh 〈 K_{d} ν 〉$ . Hence, ν, and furthermore, $\dot{X}$ converge to zero. The residual dynamic is $- Y_{g} 〈 X_{d} 〉 {\tilde{θ}}_{g} + K_{p} (X - X_{d}) + d = 0$ which shows that X converges to X_d based on Remark 3 and the assumption that ${\tilde{θ}}_{g} \to 0$ . A sufficient condition for correct estimation is persistently excitation of inputs Harandi et al. (2021a). Note that if θ_g is equal to its lower or upper bound, then

\dot{V} = - ζ_{2} ν^{T} A B^{- 1} tanh 〈 K_{d} ν 〉 + {\dot{X}}^{T} Y 〈 X_{d} 〉 {\tilde{θ}}_{g},

which is negative definite due to definition of ${\dot{\hat{θ}}}_{g}$ in (24).■

Notice that one of the advantages of the proposed methods is that they are independent of the matrices M⟨X⟩ and $C 〈 X, \dot{X} 〉$ . Hence, it is not important whether they are precisely known or uncertain. Furthermore, the proposed time-varying controllers essentially improve the performance in flexible link manipulators such as cable-driven robots since in regulation problem, a sudden movement may occur that leads to vibrations of the link, but increasing the gains smoothly can reduce this unpleasant response essentially.

In this section, some novel controllers with exact proof for parallel robots were designed. Since the proposed methods directly control the end-effector, more practical challenging problems are arisen compared to the controllers in joint space such as uncertainties in kinematical parameters and Jacobian matrix, difficulty of accurate measuring of the end-effector’s position, etc. By this means, in the next section, their performance is analyzed through some experiments on two parallel mechanisms with different properties.

4. Experimental Results

4.1 Diamond eye surgery robot

In order to analyze the performance of the proposed controllers, the 2-DOF Diamond eye surgery robot is considered. The schematic of the robot is depicted in Figure 1. It is a complex parallel spherical robot that consists of 4 links and has two degrees of freedom. This robot has an RCM (remote center of motion) point, which is a significant property needed for the implementation of minimally invasive surgeries. According to the spherical structure of the robot, all the links just have a pure rotational motion around the RCM point. The vitrector is mounted on the fourth link and has translational motion along the radial direction to the RCM point. X = [ϕ,γ]^T denotes the configuration variables, and α and β equal to π/4 are the geometric parameters of the robot. Figure 2 shows different parts of the Diamond robot.

A full analysis of the robot includes forward and inverse kinematics, Jacobian matrix, and dynamic equations are proposed in Abedloo et al. (2014); Bataleblu et al. (2020). Due to the enormous number of equations, a brief review of these relations is presented here for ease of reference. The curious reader is referred to the main reference for more details. For kinematic analysis, $q = {[θ_{1}, θ_{2}]}^{T}$ and X are considered as joint and task space variables, respectively. The inverse kinematics relations are as follows

\begin{array}{l} θ_{1} = φ + \cos^{- 1} (\frac{\cos β - \cos γ \cos α}{\sin γ \sin α}), \\ θ_{2} = φ - \cos^{- 1} (\frac{\cos β - \cos γ \cos α}{\sin γ \sin α}) . \end{array}

The Jacobian matrix of the robot is

J = [\begin{matrix} 1 & ξ \\ 1 & - ξ \end{matrix}], ξ = \frac{\cot α - \cos \hat{A} \cot γ}{\sin \hat{A}}, \hat{A} = \frac{θ_{1} - θ_{2}}{2} .

Gravity torque g〈X〉 is

g 〈 X 〉 = - \sum_{i = 1}^{4} m_{i} (J_{ω_{i}}^{T} {}^{0}R_{i} S 〈 {}^{i}ρ_{i} 〉 {}^{0}R_{i}^{T}) g,

in which m_i denotes the mass of each link, S(⋅) shows skew-symmetric matrix representation of vector (⋅), ⁱρ_i is the vector from RCM to center of mass of each link,

J_{ω_{i}}

is Jacobian matrix of each link, and rotation matrices are as follows

\begin{array}{l} {}^{0}R_{1} = [\begin{matrix} \frac{(a \times b) \times a}{‖ (a \times b) \times a ‖} & \frac{a \times b}{‖ a \times b ‖} & a \end{matrix}], {}^{0}R_{2} = [\begin{matrix} \frac{(a \times c) \times a}{‖ (a \times c) \times a ‖} & \frac{a \times c}{‖ a \times c ‖} & a \end{matrix}], \\ {}^{0}R_{3} = [\begin{matrix} \frac{(b \times d) \times b}{‖ (b \times d) \times b ‖} & \frac{b \times d}{‖ b \times d ‖} & b \end{matrix}], {}^{0}R_{4} = [\begin{matrix} \frac{(c \times d) \times c}{‖ (c \times d) \times c ‖} & \frac{c \times d}{‖ c \times d ‖} & c \end{matrix}] . \end{array}

The mass of the links is

m_{1} = 0.2706, m_{2} = 0.2366, m_{3} = 0.1587, m_{4} = 0.3808 .

X₀ = [π/2,π/4]^T is initial value and desired value is X_d = [2π/3,5π/12]^T. In order to measure the position of the end-effector in the spherical coordinate system, joint variables are measured by encoders and transformed by forward kinematics to task space variables. Note that this transformation is one to one, since the analytical solution to forward kinematics has a unique solution.

Experimental results are reported in Figure 3. In order to analyze the condition (6), the PD-DGC with different gains is implemented. K_p = 0.7I₂ and K_p = 5I₂ are, respectively, the incorrect and correct PD-DGC gains, depicted by yellow and green color, while K_d = 0.5I₂ is the gain of damping term in all of the controllers. It is clear that if λ_m{K_p} < κ_g, the robot converges to another position. Otherwise, the stabilization error is about two-degree that is acceptable with simple PD-DGC. To improve the results, we have implemented the developed versions of this controller. These controllers are tested with 10% perturbation of masses.

The results of APD-DGC proposed in Theorem 1 with K_p = 5I₂, Γ = 0.5I₂, μ = 0.2 such that condition (10) is satisfied, are shown by dotted red. The stabilization error of this controller is about one degree which is less than simple PD-DGC with exact values of the system. Note that the unknown parameters are not estimated correctly. Hence, Theorem 3 is not applicable in this case. TVPD-DGC and OFTVPD-DGC proposed in (19) and (20), respectively, are the other controllers depicted by dotted-dash purple and solid black, respectively. K_p = (5 + 5t)I₂ with saturation at K_p = 10I₂ is considered in both controllers while the parameters of the observer are A = 4I₂ and B = I₂. Since K_p is larger than previous controllers, a small overshoot is seen in the response in Figure 3(a) while the stabilization errors are sufficiently small.

OFTVPD-DGC-BI proposed in Theorem 2 is another controller with ζ₁ = 0.9 and ζ₂ = 1, which is shown by the dash line in cyan color. These gains are considered such that τ_min = − 2 and τ_max = 2 are accessible refer to Assumption 1. As indicated in Figure 3(a), the stabilization errors are significantly minor, the response is without overshoot, and as shown in Figure 3(b), the maximum value of control efforts is about 1.5, which is almost half of the maximum values of other controllers with negligible error with the expense of modicum slower convergence rate. This is an outstanding achievement such that with a lower control effort, a suitable response is achieved. Notice that the gains of all controllers are well-tuned such that settling time is about one second.

4.2 ARAS cable-driven robot

Here we implement the proposed controllers to test their performance on a flexible link robot that, in contrast to its nature, is represented by a simple model. ARAS-CAM is a deployable suspended parallel cable-driven manipulator with three translational degrees of freedom. The robot has three tower units mounted on three corners of the room. Each tower is connected to the end-effector via an actuated cable. Figure 4(a) illustrates the overall structure of the robot while the elements of each tower are depicted in Figure 4(b). Each tower contains a 750W AC servo actuator capable of delivering 20kgcm of torque. The actuator couples to a drum collecting the cables. The drum acts as a nut for a screw at its central axis, which causes it to move side to side as it turns to prevent cable cluttering. Each actuator is equipped with an incremental encoder with 5000 pulses per revolution. This encoder measures the drum’s angular displacement, which is linearly related to the cable’s length variation. Thus, given an initial bias, the cable length may be measured accurately. As each cable leaves a drum, it passes through a pulley structure that guides it to the anchor point and redirects its force on the sensitive axis of a load-cell. Each load-cell measures its corresponding cable tension. Each load-cell in the robot measures cable forces up to 50 Kg. Additionally, a 100 frame/second stereo vision camera is utilized to measure the position of the end-effector.

The Jacobian and dynamical matrices of the robot with mas-less infinity stiffness cables are given Harandi et al. (2019)

\begin{matrix} M = m I, C = 0, g = {[\begin{matrix} 0 & 0 & m g \end{matrix}]}^{T}, \\ J = [\begin{matrix} \frac{x_{a 1} - x}{l_{1}} & \frac{y_{a 1} - y}{l_{1}} & \frac{z_{a 1} - z}{l_{1}} \\ \frac{x_{a 2} - x}{l_{2}} & \frac{y_{a 2} - y}{l_{2}} & \frac{z_{a 2} - z}{l_{2}} \\ \frac{x_{a 3} - x}{l_{3}} & \frac{y_{a 3} - y}{l_{3}} & \frac{z_{a 3} - z}{l_{3}} \end{matrix}] \end{matrix}

(25)

where X = [x,y,z]^T denotes configuration variables of the robot, m = 4.5 Kg and the coordinate of anchor points are

\begin{matrix} x_{a 1} = - x_{a 2} = x_{a 3} = \frac{b}{2} = \frac{3.56}{2}, y_{a 1} = - y_{a 2} = \\ - y_{a 3} = \frac{a}{2} = \frac{7.05}{2}, z_{a 1} = z_{a 2} = z_{a 3} = h = 4.26. \end{matrix}

Note that as indicated in Harandi et al. (2021a), these values are derived from calibration and thus, they are not exact.

The initial and desired position of the robot are [1.15,−2.15,2.5]^T and [0.9,−1.9,2]^T, respectively. Note that typically in cable-driven robots a gravity compensator is used as the controller in order to set the robot at the initial position, and the initial condition of the robot is set manually. Apparently, this leads to a small variation in the initial conditions in different implementations. These small variations are inevitable and not significant in cable-driven robots due to the flexibility of cables and the large workspace of the robot. Six different controllers are implemented on the robot. Note that since in this robot the gravity force is constant, the condition (6) is satisfied with all positive K_p. The mass of the end-effector is perturbed about 5% in all of the tests. The results are illustrated in Figure 5. Two simple PD-DGC are applied with low and high gains K_p = 50I₃ and K_p = 300I₃, and the results are depicted by solid blue and dash black, respectively. APD-DGC proposed in Theorem 1 with the gains K_p = 50I₃, Γ = I₃ and μ = 0.1 is shown by dotted red. OFTVPD-DGC-BI and AOFTVPD-DGC-BI proposed in Theorem 2 and Theorem 3 are shown by solid green and dotted-dash cyan, respectively. The gains are K_p = (50 + 50t)I₃ with a saturation at 300I₃, A = 10I₃, B = 2I₃, ζ₁ = 15 and ζ₂ = 5 in both controllers while $θ_{g_{m}} = 4$ and $θ_{g_{M}} = 5$ with θ_g = m are in complement to controller proposed in Theorem 3. The gain of damping term in all the tests is K_d = 25I₃. Furthermore, adaptive sliding mode controller Harandi et al. (2021a) is implemented and depicted by solid yellow to compare the results. Note that all the gains are determined by simulation such that a suitable performance with positive tension is achieved. As indicated in Figure 5(a), the stabilization errors with all of the controllers are acceptable except PD-DGC with low gain. Note that in contrast to the proposed theorems, a limited steady-state error is observed in the responses. This is due to the fact that the sensors have a measurement accuracy and also K_p⟨t⟩ is not unbounded that as denoted in Remark 5, an ultimate bound for the error is obtained such that this bound is proportional to the inverse of K_p. Considering Figure 5(a), it is clear that the transient response of time-varying controllers is better and smoother than that compared to the constant-gains ones, while the settling time is a bit larger than that of the other PD-based controllers. Additionally, the minimum value of stabilization errors corresponds to the time-varying controllers together with the proposed one in Harandi et al. (2021a). Note that the proposed controller in Harandi et al. (2021a) requires a large number of adaptation laws for kinematics and dynamics parameters and is computationally intense. In this case, the number of adaptation laws is more than 100 which is clearly very difficult to be implemented. However, the proposed controllers are much simpler than that of Harandi et al. (2021a) while the closed-loop responses are as good as the adaptive sliding mode controller. Hence, we may conclude that simple practical modifications in PD-DGC leads to a strong controller with a promising performance. Note that this outstanding achievement is attained while the effects of flexible links such as their fluctuations which are intensified in the regulation problem were not modeled in (25).

Considering control efforts depicted in Figure 5(b), it is seen that the variations of control efforts of time-varying controllers are less. Notice that one of the features of a cable-driven robot is that cables can merely push. Hence, in the process of controller design, it should be ensured that control efforts will be non-negative. It is clear that the minimum value of τ_is with constant-gains controllers is more near to zero. This shows the superiority of the controllers obtained from Theorem 2 and Theorem 3 in which boundedness of the inputs was ensured. The minimum value of control efforts of constant-gains controllers will be decreased if higher gains are chosen, while in bounded input time-varying controllers’ larger values of K_p and K_v do not result in large variations in τ. Note that in this case $θ_{g} \in ℝ^{1}$ , and based on the discussion in Remark 2, we could apply the controller of Theorem 3. Notice that to the best of the authors’ knowledge, so far an output feedback regulator had not been designed and implemented on cable-driven robots. Hence, the proposed scheme provides a new practical method to stabilize such systems with merely position feedback.

5. Discussion

According to the experimental results of Diamond and ARAS-CAM robots, it is understood that the main aim of this paper has been achieved in practice. We have introduced newly developed versions of PD controller such that not only a rigorous mathematical stability proof was presented for each controller, but also the experimental results show that the performance of the practical controllers is very appropriate on both rigid and flexible links manipulators in the presence of uncertainties/external disturbance since the dynamic of the actuators is not modeled, the masses were perturbed in both cases, the dynamic of cables, their fluctuations, rotational motions of the end-effector and inexact values of kinematic parameters in ARAS robot were neglected. Additionally, the comparison of the results of the proposed methods with the state-of-the-art controller shows that it is possible to obtain a suitable response while the simplicity of the controllers, the selection of gains, boundedness of inputs, and implementation with merely position feedback are other superiority of the proposed controllers. Finally, it should be noted that if a proposed controller in joint space, for example, Zavala-Rio and Santibanez (2007), is applied to a parallel robot in joint space, the results are worsen compared to the proposed controllers since as indicated in Paccot et al. (2009), designing a controller directly in task space leads to a better response. Dependency to the precise model of the system and the exact value of the velocity, absence of unmodeled dynamics/external disturbance, and symmetry of the upper and lower bounds of actuators are other disadvantages of Zavala-Rio and Santibanez (2007) compared to this work.

6. Conclusion and future works

In this paper, PD with desired gravity compensation for parallel robots with revolute joints and also cable-driven robots was designed. In order to improve the performance of this controller, two advanced versions of it, including adaptive PD plus desired gravity compensator (APD-DGC) and output feedback time-varying PD-DGC with bounded input (OFTVPD-DGC-BI), were proposed, and their stability was analyzed by Lyapunov direct method. Incorporation of both controllers, which results in position feedback robust adaptive controller with bounded input under an assumption was also designed. Experimental results on 2-DOF Diamond eye surgery and 3-DOF ARAS cable-driven robot verify the effectiveness of the proposed methods and their applicability in experimental implementation on both rigid and flexible link robots. Future works include expansion of this controller such that the Jacobian matrix is merely required in the desired configuration.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

M Reza J Harandi

A Hassani

Note

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