A Sliding Mode Control For A Planar 4-cable Cddr.pdf

A Sliding Mode Control for a Planar 4-Cable Direct Driven Robot Xavier Aguas, Marco Herrera, Oscar Camacho and Paulo Leica Departamento de Automatizaci´on y Control Industrial Escuela Polit´ecnica Nacional, Ecuador, Quito Email: {xavier.aguas, marco.herrera, oscar.camacho, paulo.leica}@epn.edu.ec

Abstract—Cable Direct Driven Robots (CDDRs) are structurally similar to parallel robots but these are formed by replacing all the supporting rigid legs with cables, where the motion of the end-effector is controlled through cables which are pulled from actuators placed off-board the robot. CDDRs control is a challenge due to the physical characteristics of cables because these can only apply tensile forces and no compression. The aim of this paper is to present a Sliding Mode Control (SMC) for a Planar 4-Cable Direct Driven Robot, SMC is designed in order to obtain fast system response and the robustness against the model uncertainty. For the purpose of validating the proposed controller the trajectory tracking test and model uncertainty test were performed. Simulation was carried out for the proposed controller and the results were compared with a PD Controller in terms of Integral Square Error (ISE) index. Index Terms—Cable Direct Driven Robots, Sliding Mode Control, parallel manipulator, robustness, Tracking Trajectory

I. I NTRODUCTION In recent years, Cable Direct Driven Robots (CDDRs) have made huge advances into many industrial sectors [1] because these robots are easy to install and well adapted to their operating environment [2]. CDDRs are good candidates for performing a broad range of applications such as pick-andplace, cleanup of disaster areas, large radiotelescope, high speed manipulation, sandblasting, suspended actuated cameras for sport events and coconut farms [3]. CDDRs are a type of parallel robot, wherein cables are used as its links [4], while the cables length is changing, the end-effctor is forced toward the desired position and orientation. CDDRs possess some useful characteristics such as large workspace capability, transportability, economical structure and maintenance than conventional parallel robots, these advantages are presented in [5]. However, substituting rigid links by cables presents new defiances in the study of cable driven robots which are distinct to conventional robots because the cables can only apply tensile forces and no compression [6]. The majority CDDRs are designed with actuation redundancy, i.e. more cables than degrees of freedom in try to avert configurations where certain wrenches require an impossible pushing force in one or more cables [7]. Control for CDDRs have catched the attention of many investigators since its high effect on the performance of the robotic systems and different control methods have been proposed for these robots as in [8] shows the efficiency of the adaptive passivity-based controller when there is no enough knowing about system parameters and

disturbances or as in [9] presents experimental performance of robust PID using cables as an alternative to be used for very large workspace applications to different desired trajectories. Trajectory planning can be executed by considering only kinematic and geometric constraints. However, for CDDRs, dynamic constraints must also be considered in most cases, due to the unilateral tension property of cables which can exert only unidirectional forces on the payload [10]. The principal difficulty in control is keeping positive tensions in cables while the loads move, for this reason the Sliding Mode Controllers (SMC) have been designed to obtain fast system response for cable suspended loads [11]. In the process design of CDDRs the most important stage is the simulation phase. Simulators permit us to quickly test a new conception and observe how this implementation fits our design wherein the highly flexible nature of the cables is extremely challenging to simulate [12]. In this paper the Planar 4-Cable CDDR model [13] is considered since no rotational move and no moment resistance are required at the end-effector. All 4 cables convene in a single point and the end-effector is modelled as a point mass. In order to obtain fast system response and the robustness against the model uncertainty and external disturbances a SMC is designed, wherein the control actions are torques to the motors and not forces on end-effector as in other researches. This approach provides faster tracking capacity than the classical PD Controller as is showed in [14]. For the design of the proposed controller, end-effector and motor dynamics of the system was modelled and integrated. With the full system model, effects of motor parameters can be considered to improve control. The performance of this controller is compared with a PD Controller in several parameter variations test in terms of integral square error (ISE) index and a simulator was developed for this research to observe the behavior of the cables during its trajectory. The simulation results show the ability the proposed control law in comparison with the classic control. The organization of this paper is as follows: Section (II) describes the dynamics model of Planar 4-Cable CDDR. Section (III), control strategies are presented. Section (IV) provides the simulation results. Finally, the conclusions are presented. II. DYNAMICS M ODEL OF A P LANAR 4-C ABLE CDDR This section shows a dynamics model of a Planar 4-Cable CDDR for enhanced control [9]. The dynamics model refers

to relationship between the translational motion of the endeffector and the required active torques [13]. The Planar 4Cable CDDR considered is shown in Fig. 1. The mass of cables is very small compared to the end-effector and it can therefore be neglected.

J4

TABLE I D ESCRIPTION OF P LANAR 4-C ABLE CDDR PARAMETERS

3 cJ3 3

A4 c4

A3

4 L3

L4

LB x, y

J1

c1

Ai

Motor position

θi

Cable angle

Ri

Pulley radio

ti

Cable tension

βi β˙ i

Pulley angle Angular velocity of pulley

β¨i

Angular acceleration of pulley

ci

Viscous damping coefficients of motor

Ji

Rotational inertial of motor Torque

2

L2

1

A1

Description

τi

m

L1

Parameter

i = 1, ..., 4

c2

A2

LB

J2

Fig. 1. Planar 4-Cable CDDR Diagram [13].

The dynamics model for the end-effector with 2-dof is: ¨ = FR MX (1)   m 0 T where M = is the mass matrix, X = [x y] 0 m is the end-effector position and FR is the resultant force of cables acting on the end-effector. In this paper is considered the dynamics behavior of the lumped motor shaft/cable pulley. Fig. 2 presents the motor shaft/cable pulley system where all βi are zero when the end-effector is situated at the origin of frame (centroid of the square base). From this position, a right-handed positive angle βi causes a change in cable length. The Planar 4-Cable CDDR parameters are shown in Table I.

    c1 0 J1 0     .. .. where, J =   are  and C =  . . 0 c4 0 J4 diagonal matrices. J is the rotational inertia matrix and C is the rotational viscous damping coefficients matrix. β is the pulley angles vector; τ is the torques vector produced by motors; T is the cable tensions vector and R is the pulley radio. In order to present the cable tensions as a function of angular position and the motor torques, (2) is expressed as: T =

1 ˙ (τ − J β¨ − C β) R

(3)

Considering the dynamics model of the end-effector and the motor shaft/cable pulley system, we can find the full dynamics model of Planar 4-Cable CDDR. A right-handed positive angle βi on a pulley will incite a negative change ∆Li = Li − L0i in cable length i. The equation (4) shows this behavior. βi R = −∆Li

(4)

ti

i

Ai

q

2

2

(x − Aix ) + (y − Aiy ) is the length for cable q i and L0i = Aix 2 + Aiy 2 is the initial length for cable i. The βi can be written as: where Li =

Ri 

ci  i

i

q  2 2 L01 − (x − A1x ) + (y − A1y )  1  .. βi (X) =  . R q 2 2 L04 − (x − A4x ) + (y − A4y )

i Ji

   

(5)

The derivative with respect to time of (5) is calculated as:

Fig. 2. Scheme of motor shaft/cable pulley system [13].

The dynamics equation for motor shaft/cable pulley is shown as follow: J β¨ + C β˙ = τ − RT



(2)

∂β ˙ β˙ = X ∂X and d β¨ = dt



∂β ∂X



X˙ +

(6) 

∂β ∂X



¨ X

(7)

T can be rewritten as in (8):       1 d ∂β ∂β ¨ ∂β ˙ ˙ T = τ −J X+ X −C X (8) R dt ∂X ∂X ∂X The dynamics model of Planar 4-Cable CDDR can be expressed in equations of motion for robotic system using translational statics jacobian matrix S in function of the cable angles.   −cos(θ1 ) −cos(θ2 ) −sin(θ3 ) −cos(θ4 ) S= (9) −sin(θ1 ) −sin(θ2 ) −sin(θ3 ) −sin(θ4 ) The procedure to find the equations of motion for robotic system is shown as follows: Equation (8) can be rewritten as:       ∂β ¨ ∂β ˙ d ∂β ˙ X −J X −C X (10) RT = τ − J dt ∂X ∂X ∂X ¨ By grouping (10) in terms of X˙ and X:        ∂β ˙ ∂β d ∂β ¨ RT = τ − J +C X −J X dt ∂X ∂X ∂X (11) With the statics relationship ST = FR given in [13], T can be represented in the form: +

T = S FR

TABLE II P LANAR 4-C ABLE CDDR PARAMETERS Parameter m=0.5 Ri =0.03 ci =0.03 Ji =0.008 A1 =(-0.20 -0.20) A2 =(0.20 -0.20) A3 =(0.20 0.20) A4 =(-0.20 0.20)

¨ T = S+M X

In order to design a PD controller, it is necessary to know the error. The control scheme of this controller is presented in Fig. 3. X d (t )

e(t ) +

Kp

-

d

+ +

Plant

X (t )

Kd

dt

Fig. 3. PD controller scheme.

The tracking error vector is defined as follows:

(12)

(13)

Description End-Effector mass Pulley radio Viscous damping coefficient Rotational inertia of motors Motor position 1 Motor position 2 Motor position 3 Motor position 4

A. PD Controller

e(t) = Xd (t) − X(t)

T −1

where S + = S T (SS ) is the right pseudoinverse of statics jacobian matrix. Replacing (1) in (12), T is written as:

Unit [kg] [m] [N.s/m] [kg.m2 ] [m] [m] [m] [m]

(16)

where Xd (t) is the desired position vector and X(t) is the end-effector position vector. PD controller has the following form:

By combining (13) and (11), the equation of motion can be expressed as:

de(t) (17) dt Applying this controller to dynamics model of Planar 4-Cable         CDDR and in order to have a input as torques vector τ ∂β ˙ ∂β ¨ J d ∂β X+ X = τ is necessary multiply by S + because the controller directly S + RM + J +C ∂X dt ∂X ∂X (14) applies a virtual force as control actions on end-effector. The equation of motion can be written alternatively in the de(t) uP D (t) = τ = S + (K p e(t) + Kd ) (18) following general form: dt where Kp and Kd are tuning parameters. ¨ + N (X, X) ˙ X˙ = τ Meq (X)X (15) B. Sliding Mode Controller ∂β In this equation, Meq (X) = S + RM + J ∂X In this subsection, a sliding mode controller for Planar 4  ∂β ∂β Cable CDDR based on the analysis dynamic of the robot in d ˙ = J and N (X, X) + C ∂X where Meq (X) is dt ∂X section II. The scheme of this controller is illustrated in Fig. 4. ˙ is the matrix of Coriolis. the inertial matrix and N (X, X) The parameters considered of Planar 4-Cables CDDR in this X (t ) X d (t ) e(t ) SMC 0 research are presented in Table II. Plant u (t)  u (t) P D(t) = Kp e(t) + Kd

+

c

III. D ESIGN OF C ONTROLLERS FOR A P LANAR 4-C ABLE CDDR This section designs different controllers for Planar 4Cable (CDDR) based on the dynamics model. The designed controllers are a PD controller and a SMC.

d

dt

d

1

Fig. 4. Sliding mode controller scheme.

The tracking error vector is defined as follows: e(t) = Xd (t) − X(t)

Therefore, by analysing: (19)

if

σ(t) > 0

→

sign(σ(t)) > 0

In order to apply a SMC, the sliding surface as in [15] is considered as:

if

σ(t) < 0

→

sign(σ(t)) < 0

σ(t) = λ0 e(t) + λ1 e(t) ˙

(20)

where λ0 and λ1 are positive parameters. This controller has two components: a continuous uc and a discontinuous ud . u(t) = uc (t) + ud (t)

(21)

Now, the surface must be derived for development of the controller: σ(t) ˙ = λ0 e(t) ˙ + λ1 e¨(t)

(22)

By substituting (19) in (22): ¨ d (t) − X(t)) ¨ σ(t) ˙ = λ0 e(t) ˙ + λ1 (X

(23)

The continuous part of the controller is provided with the condition to keep the output on the sliding surface σ(t) ˙ = 0: ¨ ¨ d (t) + λ0 e(t) ˙ X(t) =X λ1

(31)

Finally, to reduce the chattering effect, (30) can be rewritten as a sigmoid function:   σ(t) (32) ud (t) = Meq (X) k1 λ1 |σ(t)| + δ where δ is chattering reduction tuning parameter. With this controller based on the model we directly obtain torques as control actions and do not consider a virtual force like a PD controller. IV. SIMULATION RESULTS A PD controller and a SMC were implemented in Matlab/Simulink 2018a using the full dynamics model of Planar 4-Cable CDDR. To compare the control performance, a PD controller was tuned as similar as the proposed controller. The simulation has a duration of 80 seconds with a sampling time of 0.1 which uses the ODE45 solver method(Solve non-stiff differential equations). Fig. 5 shows the simulator developed to observe the behavior of the cables during the trajectory. Movement CDDR

(24)

0.5 0.4

By replacing (24) in (15) , uc (t) can be written as:

0.3 0.2

To design ud (t), a positive-definite Lyapunov function V is defined:

0.1

axis Y

¨ d + λ0 Meq (X)e(t) ˙ X˙ (25) uc (t) = Meq (X)X ˙ + N (X, X) λ1

0 -0.1 -0.2 -0.3

1 T σ(t) σ(t) > 0 (26) 2 The derivative of the function V must be negative-definite: V =

T V˙ = σ(t) σ(t) ˙ <0

(27)

By substituting (23) in (27) and considering that ¨ = Meq (X)−1 (u(t) − N (X, X) ˙ X): ˙ X T ¨d V˙ = σ(t) [λ0 e(t) ˙ + λ1 X

− λ1 Meq (X)

−1

˙ X)] ˙ (u(t) − N (X, X) (28)

By replacing (21) and (25) in (28), V˙ can be rewritten as: T −1 V˙ = σ(t) (−λ1 Meq (X) ud (t))

(29)

To satisfy (27), ud (t) should be: ud (t) = k1 Meq (X)sign(σ(t))

(30)

where k1 > 0 is a tuning parameter and Meq (X) is symmetric and positive-definite matrix, these properties of Meq (X) is shown [16].

-0.4 -0.5 -0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

axis X

Fig. 5. Planar 4-Cable CDDR Simulator designed in Matlab/Simulink.

In order to prove the superiority of the proposed control law, ISE index is presented. The variation ∆% corresponds to the comparison between PD Controller and the proposed control law. ∆% =

e1 − e2 e1 +e2 2

· 100

(33)

where e1 , e2 represent ISE index values for PD controller and SMC respectively. Three tests were carried out: • Tracking Trajectory • Mass change • Rotational inertia change Finally, the values of Kp = 10 and Kd = 0.8 for PD controller and λ0 = 90, λ1 = 5, k1 = 6 and δ = 0.1 for SMC have been selected by trial and error until achieving the lowest ISE index.

A. Square Trajectory Test In this test, the end-effector follows a square whose side length is 0,4 [m]. This trajectory shows the response of the controller to the sudden changes which exist in each corner of the trajectory. Fig. 6 shows the trajectory reference and real movement of the end-effector.

Fig. 8 and Fig. 9 demonstrate performance of the end-effector trajectory tracking in workspace.

Reference PD SMC

0.1

0.05

axis y [m]

Square Trajectory

0.2

Circular Trajectory

0.15

Reference PD SMC

0.15 0.1

0

-0.05

axis y [m]

0.05 -0.1 0 -0.15 -0.15

-0.05

-0.1

-0.05

0

0.05

0.1

0.15

axis x [m] -0.1

Fig. 8. XY graph, circular trajectory when mp = 0.15m.

-0.15 -0.2 -0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

Circular Trajectory

0.2

axis x [m]

Reference PD SMC

0

Tracking performance is shown in Fig. 7. From this graph, one can notice that PD controller has degraded tracking performance in the X direction.

axis y [m]

Fig. 6. XY graph, square trajectory. -0.05

-0.1

Square Trajectory Reference PD SMC

-0.1

-0.15 -0.02 -0.01

0

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

axis y [m]

-0.11

axis x [m]

Fig. 9. XY graph, beginning of the circular trajectory when mp = 0.15m

-0.12

-0.13

Table IV shows ISE index in mass variation test when mp = 0.15m. PD controller presents oscillations on reaching the trajectory while the proposed SMC reaches faster the reference and without oscillation problems.

-0.14

-0.15

0.146

0.147

0.148

0.149

0.15

0.151

0.152

0.153

0.154

axis x [m]

TABLE IV ISE M ASS VARIATION T EST

Fig. 7. XY graph, corner of square trajectory.

Table III shows ISE index in traking trajectory test. The result shows which PD controller has a constant error in steady state. TABLE III ISE S QUARE T RAJECTORY T EST. Position X Y

PD 0.004063 0.004158

SMC 0.003291 0.003291

∆% 20.995 23.278

B. Mass Variation Test In this test, the end-effector mass is changed mp = 0.15m, where mp is the test mass and m is the end-effector mass.

Position X Y

PD 0.00136 0.00925

SMC 5.491e-7 0.003289

∆% 199.8386 95.079

C. Rotational Inertial Variation Test The Planar 4-Cable CDDR model is simulated using the following parameters: The rotational inertia is changed in motor 1 and 4 (J1 = 25J and J4 = 25J). Fig. 10 and Fig. 11 llustrate dynamic behavior with the proposed control algorithm. The result shows which the proposed SMC again has an advantage over the conventional PD Controller. The position outputs track the desired values pretty good and the steady state errors are very small. In the Table V presents the comparison between both controllers.

ment of the project PIJ-15-17 ”Development and construction of a PID-robust prototype based on advanced control techniques to improve the robustness of industrial processes”

Circular Trajectory

0.2

Reference PD SMC

0.15 0.1

R EFERENCES

axis y [m]

0.05 0 -0.05 -0.1 -0.15 -0.2 -0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

axis x [m]

Fig. 10. XY graph, tracking performance when J1 = 25J and J4 = 25J.

Circular Trajectory 0

Reference PD SMC

-0.02 -0.04

axis y [m]

-0.06 -0.08 -0.1 -0.12 -0.14 -0.16 -0.18 -0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

axis x [m]

Fig. 11. X Y graph, beginning of the circular trajectory J1 = 25J and J4 = 25J.

TABLE V ISE ROTATIONAL I NERTIAL VARIATION T EST Position X Y

PD 0.00631 0.00845

SMC 5.491e-07 0.00328

∆% 199.965 88.15

C ONCLUSION In this paper a sliding mode control using a PD controller as sliding surface for the end-effector position control of Planar 4-Cable CDDR is developed. In order to show the effectiveness of this controller several tests are performed for circular and square trajectories. The simulation results indicate that controller proposed presents robustness against the model uncertainty when some parameters of robot model were changed. The results of SMC controller for square trajectory test, rotational inertial test and mass variation test present lower ISE index than PD controller. In brief, SMC shows the robustness to the parameter uncertainties. ACKNOWLEDGMENT The authors gratefully acknowledge the financial support provided by the Escuela Polit´ecnica Nacional, for the develop-

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