Visual PID Control of a redundant Parallel Robot - Google Sites

Visual PID Control of a redundant Parallel Robot Rubén Garrido, Alberto Soria, Miguel Trujano Department of Automatic Control, CINVESTAV-IPN, Mexico D.F., Mexico Phone (5255) 5747-3744

Fax (5255) 5747-3982

Abstract ––In this paper, we study an image-based PID control of a redundant planar parallel robot using a fixed camera configuration. The control objective is to move the robot end effector to the desired image reference position. The control law has a PD term plus an integral term with a nonlinear function of the position error. The proportional and integral actions use image loop information whereas the derivative action adds damping using joint level measurements. The Lyapunov method and the LaSalle invariance principle allow assessing asymptotic closed loop stability. Experiments show the performance of the proposed approach. Keywords –– Parallel Robots, Visual Servoing, Robot Control, PID Control.

I. INTRODUCTION Parallel robots are generally composed of a fixed base connected to a common end effector by means of several mechanical chains. This type of structure results on a closed kinematics possessing high structural stiffness, high bandwidth motion capability, high load capacity, and high precision compared to open chain kinematics robots. Furthermore, position errors due to mechanical friction are averaged and are not accumulated as in open chain kinematics robots. It is interesting to point out that many control schemes employ the robot forward kinematics to determine position and orientation of the robot end effector; thus the control loop is closed via the direct kinematics [1], [2]. Therefore, errors in the forward kinematics solution will produce errors in determining the position and orientation of the end effector. In this work, we will consider the redundant parallel robot studied in [2] and [3]. In order to avoid the use of direct kinematics for closing the loop, a Visual Servoing approach allows measuring directly the effector position. Hence, it would not be necessary to know the robot kinematics parameters exactly when computing the control law. Visual Servoing systems use visual information to close the control loop. One of the interesting features when using Visual Servoing is the fact that it allows in some cases to cope with mechanical uncertainties [4]. There exist two Visual Servoing schemes, namely, fixed camera and camera in hand. [5]. Robot systems using a fixed camera, the vision system takes images related to the robot workspace coordinate system. The goal is to move the robot towards a desired position or object. In the camera in hand configuration, the camera mounts on the robot end effector and the vision system takes images of the robot workspace. The goal of this configuration is to move the robot so the projection of a static or moving object remains at a desired position in the image plane. Furthermore, another way to classify Visual Servoing is as position-based and image-based schemes. Position-based Visual Servoing uses

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the vision system to determine the object and/or robot position relative to the robot workspace coordinate frame consequently requiring camera calibration and a geometric model of the robot workspace. On the other hand, in the Image-based Visual Servoing the vision system determines the object and/or robot position relative to the camera coordinate frame thus not requiring camera calibration or a geometric model to move the robot to the desired position. The approach proposed in this work employs a fixed camera image-based Visual Servoing. A well known in Robot Manipulator control is the fact that a linear Proportional Integral Derivative (PID) controller is able to compensate for unknown constant disturbances and to provide local asymptotic stability [6] in the case of Robot Manipulator joint control. Reference [7] proposes a class of global regulators having a linear PD feedback term plus a non-linear integral term depending on the position error. Reference [8] studies a class of non-linear global PID regulator for robot manipulator employing a nonlinear proportional term produced by a non-quadratic artificial potential. The control law proposed here has a similar structure as in [7] and [8]. However, in the present case the aim is to solve the regulation problem for redundant parallel planar robots with revolute joint using a direct measurement of the robot end effector. A vision system measures the end effector position of the robot. The integral action goal is to compensate constant disturbance terms as stiction. The Lyapunov method together with the LaSalle’s invariance theorem allows concluding asymptotic stability. The rest of the paper describes further details of the proposed control scheme. Section 2 presents the dynamic model of the redundant parallel robot. Section 3 shows the vision system model. Section 4 deals with the proposed control law and the closed loop stability study. Section 5 exhibits some experimental results to show the feasibility of the approach. The paper ends with some comments and future work.

II. Parallel Robot Dynamic Model with Redundant Actuators In this work, the symbols λm { A} y λM { A} stand for

the minimum and maximum eigenvalues of a matrix respectively of a positive definite matrix A . The norm of vector x is

A =

defined

as x =

T

x x and

the

term

λM { AT A} defines the induced norm of matrix A .

In accordance with [2], modelling of a planar paralleloveractuated robot with rotational joints is attained using an equivalent open chain mechanism. Thus, the robot is decomposed into three planar two-link arms having each a well-known dynamic model. Supposing that the robot moves in the horizontal plane, the equivalent mechanism has the following set of equations  º ªτ a º ªΘ ªΘ i º i Mi (1) + Ci «¬ ϕi »¼ «¬ ϕi »¼ + N = «¬τ p »¼ ; i = 1, 2, 3 Combining these equations gives the dynamic model of the mechanism (2) Mq + Cq + N = τ where M is the inertia matrix, C correspond to the Coriolis and centripetal forces matrix, N is a vector that comprises constant disturbances. The robot joint and torque variables are q=ª ¬ qa

q p º¼ = [ Θ

τ = ª¬τ aT

τ Tp º¼ = [τ a1

T

T

T

1

T

Θ2

Θ3

ϕ1

τ a2

τ a3

τ p1

ϕ2

τ p2

τ p3 ]

T

τ p correspond to the active and passive joint torques respectively. It should be noted that if friction at the passive joints is ignored, then, τ and τ a are related by W τ = S τa

(3)

where the Jacobian matrices W and S relate end effector velocities with velocities of the active joints ª ∂qa º q =

∂q  « ∂X »  X =« »X ∂ q ∂X p « »

«¬ ∂X »¼ ª ∂qa º « ∂X » ∂q W =« »= ∂ q p « » ∂X «¬ ∂X »¼ S=

∂qa ∂X

M = W MW T T C = W MW + W CW

N =W N T

III. Model of the Vision system The camera optics maps the robot workspace to the camera screen coordinate system [9]. Hence, the position X of the end effector in the robot coordinated frame

X i = [ xi

(4)

yi ] is given by T

­ ª x º ªOx º½ ª C x º ª xi º X i = « » = η hR ( β ) ® « » − « » ¾ + « » (10) ¬ yi ¼ ¯ ¬ y ¼ ¬Oy ¼¿ ¬C y ¼

T

q p denotes the passive joint position vector. Vectors τ a and

T

a

T

ϕ3 ]

Variable qa is the active joint position vector and

T

Consequently, using (4) and (8) gives the dynamic model of the redundant parallel robot in terms of the end effector position X T MX + CX + N = S τ (9)

Vector [ Cx

C y ] is the centre of the image in the screen of T

the camera, η is the camera scaling factor given in pixels/m and is assumed negative and h is the magnification factor defined by f (11) h=