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Sliding computed torque control based on

passivity for a haptic device: PHANToM premium

1.0Fabian Alfonso Daz Lopez*, Omar Domnguez Ramirez**, L. E. Ramos Velasco***

*Masters Program in Mechatronics

Polytechnic University of Pachuca

Email: [email protected]

**Research Center on Information and Systems Technology

Hidalgo State University


***Laboratory Research in Robotics and Advanced Electronic

Polytechnic University of Pachuca


Resumen In this paper we design a controller, of systemsEuler-Lagrange, based on dynamic properties, and basicallyin stability of Lyapunov method and energy analysis.The main contribution of these controllers is that the systemnonlinearities are taken into consideration, unlike traditionalmethods of control, who regard all systems as linear, whichfor a robot manipulator is not the most suitable.The systems Euler-Lagrange counts in its dynamics withcharacteristic very special, which are taken advantage ofby a great variety of authors to carry out control, thistopic intends taking advantage of some properties, as the

skew-symmetric property, passivity and other very commondefinitions, like Lyapunov functions and of kinetic energy.This paper presents experimental results using as anexperimental platform an haptic interface: PHANToMPremium 1.0

Palabras clave: Redes neuronales, control PID, Interfazhaptica, PHANToM.

I. INTRODUCTION

In recent years the development of robotic technology

as a support on many issues, has been a great impact on

different sectors. The scope is extensive and robot control

techniques have been the subject of multiple investigations.

Among these techniques regularly seeks the optimal controlas possible. Conducting research in this field requires to

begin using classical control techniques that exist in the

literature such as proportional-derivative (PD), proportional-

integral (PI), and proportional-integral-derivative (PID) (?).

Some methodologies have been proposed for the control

of nonlinear systems, on the literature review, is well

known that the PD plus gravity compensation controller can

globally stabilize a manipulators (?), and for the parametric

uncertainly an adaptive version of PD controller has intro-

duced, the main drawback of this approach is the gravity

regressor matrix has to be known.

In this paper a sliding mode control is applied to a haptic

interface (PHANToM premium 1.0) for movement and

tracking of trajectories.

I-A. Justification

As in the dynamic analysis of Euler-Lagrange (E-L)

systems tribological effects are not considered (which is an

inherent effect of electromechanical systems such as robot

manipulators), there are currently more complex control

techniques which reduce significantly the influence of these

phenomena, e.g. computed torque control and first-ordersliding modes are very efficient strategy in the trajectory

follow-up compensated or injecting dynamic friction.

One advantage in the E-L systems is the property of

passivity (?), which is not to generate more energy that

is supplied to the system, this property is very useful for

the control analysis, since that can be included within the

design to induce stability.

I-B. The problem

Since the classical control techniques (such as P, PD,

PI and PID) are designed for linear systems but do not

compensate efficiently nonlinear dynamics that occur indynamic systems such as electromechanical mechanisms,

which present influences of inertial forces, potential energy,

centripetal and Coriolis forces, and friction between others.

The proportional controller P with state feedback is the

easiest closed-loop controller that may be used in the control

of robot manipulators. The application of this strategy is

used for position control, however, has certain constrains

because its design fails the interaction of conservative

forces (such as potential energy) resulting in a performance

limitation of a regulation task.

PD type controllers with gravity compensation do not

compensate the dynamics of the system effectively causing


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a steady-state error. These controllers are very applied with

low performance.

I-C. Our proposal

In this paper we design a control strategy which is able

to satisfy the interaction of conservative forces (such as

potential energy) and using approximations to Lyapunovfunctions to compensate another nonlinearities, for this

requires prior knowledge of the dynamic model system.

We propose one of the most popular controllers in the field

of robotics as it is the computed torque and at this paper

we add a first order sliding mode for this controller, we

perform the experimental implementation in our case in a

haptic device PHANToM premium 1.0 of 3 DOF (degree

of freedom) (?), which free motion control techniques for

robot manipulators can be applied, which aim is to show

how the problem can be solved.

II. THE PHANTOM PREMIUM 1.0

The PHANToM premium 1.0 is a haptic interface manu-

factured by Sensable Technologies, we can use this robot to

evaluate many control laws, and, for this paper we evaluated

the experiments with it. PHANToM premium provides 3

DOF, positional sensing, serial type and open kinematic

chain consisting of three rigid links and revolute-type joints.

As the vector of viscous friction forces inertial effects

inherent to the device are relatively slight, making this

system a high performance haptic device. Optical encoders

that provide feedback position and velocity joint guarantee

changes in the operational space (workspace) of 0.03mm

in any of the Cartesian axes X, Y and Z (?). PHANToM

premium can be modeled as a robot manipulator (?), thedynamic properties are presented in the next section.

III. DYNAMIC MODEL AND ITS PROPERTIES

We consider the class of rigid, fully actuated, uncons-

trained mechanical systems which can be modeled by

the Euler- Lagrange principle. This results in a class of

nonlinear systems modeled by a set of highly coupled

nonlinear differential equations over the entire domain of

the Euclidian space in n defined below:

1. Kinetic energy computation: K(q, q)2. Potential energy computation: U(q)

3. Lagrangian computation: L(q, q)4. Development of the Euler-Lagrange equations:

E(q, q)=K(q, q) + U(q) (1)

where E(q, q) is the systems total energy (Hamiltonian),and q = [q1...qn]

T , describes a generalized coordinates

vector as a function of time.

The difference between kinetic energy K(q, q) and poten-tial energy U(q) is so-called Lagrangian L(q, q) of a robotmanipulator, so that:

L(q, q) = K(q, q) U(q) (2)

Kinetic energy is obtained as follows:

K =n

i=1

1

2migv

2

i=

1

2qTD(q)q (3)

Potential energy U(q) is obtained as follows:

u =n

i=1

1

2migh

2

i(4)

where mi is the i mass of the i link, hi is the i height

of the i link respect to mass center and g is a gravitational

constant.

Considering U(q) as conservative forces, the Euler- La-grange equations of motion are:

d

dt

L(q, q)

qi

L(q, q)

qi= i (5)

where i corresponds to the torque of the i-th actuator.The Lagrangian L(q, q), given by 1 is rewritten as:

L(q, q) =1

2qTH(q)q U(q) (6)

Thus the resulting equation can be written in scalar form:

nj=1

Dij qj +n

j=1

nk=1

Cijk qj qk + Gi =i (7)

which shows the structure of the velocity product terms.

Cijk are known as Christoffel symbols of the first type, and

are given by:

Cijk =1

2

Hij

qk+

Hik

qj+

Hik

qi

(8)

They are functions of only the position variables, qi. The

elements of L(q, q) can be defined as:

Cik =n

k=1

Cijk qk (9)

However C(q, q) is not unique, and other definitions arepossible. Thus synthetic, the dynamic model of a rigid

n- link serial non-redundant robot manipulator, with all

actuated revolute joints described in joint coordinates, is

given as follows:

D(q)q+ C(q, q)q+ G(q) = (10)

Where D(q) denotes a symmetric positive definiteinertial matrix, C(q, q) is a Coriolis and centripetal forcesmatrix, G(q) models the gravity forces vector and standsfor the torque input.

PHANToM Premium 1.0, is a joint low friction haptic

device equivalent to 0.04 N, so, not considered to be the

vector of viscous friction force because of its magnitude

specified. The computed force in cartesian space (thimble


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gimbal), must be transformed into torques in the actuator

space. Typically the compute is:

= JT(q)f (11)

and stands a force of 8.5N.

III-A. Properties of Euler Lagrange systems

The dynamic equation 10 have the following properties

(?):

There exists some positive constant such that

D(q) I (12)

where I denotes the nn identity matrix. The D(q)1

exist and this is positive definite.

The matrix C(q, q) have a relationship with the inertialmatrix as:

qT

D(q) 2C(q, q)

q = 0 (13)

this property is known as the anty-symmetry property.

From the passivity property we have that:

V(x) V(x0)

t0

yT(s)u(s)ds (14)

where V(x) is a storage function, y(s) is the output,and u(s) is the input of the system, and s is a va-riable change. For the Euler-Lagrange system, energy

function E as the storage function, and we have the

passivity property as:

E(t) E(0)

t0

qT dt (15)

where q is the output and is the input

IV. SLIDING MODE CONTROL

The term sliding mode control first appeared in the

context of variable-structure systems. In the course of the

history of automatic control theory, the investigation ofsystems with discontinuous control action has been main-

tained at a high level. In particular, at the first stage relay

or on-off regulators ranked highly for design of feedback

systems. The reason was twofold: easy of implementation

and high efficiency of hardware. Due to its order reduction

property and its low sensitivity to disturbances and plant

parameter variations, sliding mode control is an efficient

tool to control complex high-order dynamic plants operating

under uncertainty conditions.

Sliding modes as a phenomenon may appear in a dynamic

system governed by ordinary differential equations with

discontinuous right-hand sides (?).

IV-A. Outline of sliding mode control methodology

The sliding mode dynamics depends on the switching

surface equations and do not depends on control. Hence

the design procedure should consist of two stages. First,

the equation of the manifold with sliding mode is selected

to design the desire dynamics of this motion in accordancewith some performance criterion.

Then, the discontinuous control should be found such that

the state would reach the manifold and such the sliding

mode exists in this manifold. As a result, the design is deco-

upled into two subproblems of lower dimension, and after a

finite time interval preceding the sliding motion, the system

will possess the desired dynamic. The deviation from the

ideal model may be caused by imperfections of switching

devices such as small delays, dead zones and hysteresis,

which may lead to highfrequency oscillations. The same

phenomenon may appear due to small time constants of

sensors and actuators having been neglected in the ideal

model. This is the so-called chattering phenomenon, andfor its suppression, is implemented a lowpass digital filter

(wavelet filter).

IV-B. Hyperbolic functions and their properties

Analogous to the simplest first-order tracking relay sys-

tem with state variable x(t):

x(t) = f(x) + (16)

with the bounded function f(x), |f(x)| f0 = 0 andthe control as a relay function of the tracking error e =

r(t) x; r(t) is the reference input and is given by:

= 0sign(e)

0, e > 0discontinuity, e = 0

0, e < 0(17)

where 0 is constant, we have the hyperbolic tangent

function, described by the limit:

lm

tanh(e) sign(e) (18)

The problem with the use of sign function (stability in

the sense of Lyapunov) is the discontinuity presented at

the origin; however, with the use of hyperbolic functionsis possible to override this problem. These functions have

certain properties, which are used for stability analysis;

some of them are described below:

The hyperbolic sine is growing exponentially, also

meets the following criteria senh(x) = 0 x =0, |senh(x)| > 0, x = 0The hyperbolic cosine is not radially unbounded and

positive definite, which implies cosh(x) 1, x The hyperbolic tangent is radially unbounded, in ad-

dition: tanh(x) = 0, x = 0

The combination of hyperbolic functions with other fun-

ctions is very interesting, for example:


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In(cosh(x)), x is positive definiteIn(cosh(x)) = 0 for x = 0.x tanh(x), x , tanh(x) = 0 , is positive definite,tanh(x) = 0 for x = 0

V. DESIGN OF THE SLIDING COMPUTED

TORQUE CONTROL BASED ON PASSIVITY

Since robot manipulators keep the passivity dynamic

property (?), we can design a nonlinear control law through

an analysis involving this property. To make this analysis it

is necessary to consider also the Lyapunov stability theory.

Thus, errors should be considered directly on the system

dynamics, making a variable change on the dynamics

involving q

For controllers design, first we propose a candidate fun-

ction to be Lyapunov, i.e. a function that is positive defined.

From the properties of E-L systems (?) it is known that

kinetic energy has certain characteristics, and by definition

(?) K(q, q) is positive definite, so therefore this functioncan be part of the Lyapunov function as follows:

V(x) =1

2qTD(q)q+

1

2qTKpq (19)

where x represents the states (q, q), Kp is a positivedefined and symmetric matrix. From the Lyapunov stability

theory it is known that for a system to be stable in a

breakeven it requires to satisfy with certain properties, such

as to find a Lyapunov function that satisfies with V(x) > 0,

that to be continuously differentiable and its first temporaryderivative meets V(x) < 0.

To achieve these properties we suggest a Lyapunov

function involving joint acceleration, i.e. that we consider

the second order of the system at the error equations of the

form q = q qd, and that it complies with the followingconditions:

lmt

q(t) = lmt

q qd 0,

lmt

q(t) = lmt

q qd 0,

lmt

q(t) = lmt

q qd 0

(20)

Given that ?? is a positive semidefinite function, obtainthe derivative of the Lyapunov candidate function, so that

is follows:

V(x) = 12

qTD(q)q+ 12qT

D(q)q+ 12

qTD(q)q+ 12qT

Kpq+1

2qTKpq

V(x) = qT

D(q)q+ 12qT

D(q)q+ qTKpq(21)

Replacing the reference of the acceleration error in the

above equation it:

V(x) = qT

D(q)(q qd) +1

2

qT

D(q)q+ qTKpq (22)

wich can be developed in the following form:

V(x) = qT

D(q)q D(q)q+ 12qT

D(q)q+ qTKpq

V(x) = qT

[ C(q, q)q G(q)] D(q)qd +1

2qT

D(q)q+ qTKpq(23)

VI. PLATAFORMA E XPERIMENTAL

VII. RESULTADOS EXPERIMENTALES

VII-A. Resultados de la identificacion del sistema

VII-B. Resultados del controlador PID wavenet

VII-C. Resultados de la autosintonizacion

VI II . CONCLUSIONES Y TRABAJO FUTURO

IX. AGRADECIMIENTOS

El primer autor agradece a CONACYT por la beca de

maestra otorgada durante el periodo septiembre 2010 -

septiembre 2012, con numero de registro 372724.

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