ADE690470 1 10

ADE690470 1 10 Research Article Advances in Mechanical Engineering 2017, Vol 9(2) 1–10 � The Author(s) 2017 DOI 10 1177/1687814017690470 journals sagepub com/home/ade An approach to the dynamic modeli[.]

Advances in Mechanical Engineering

2017, Vol 9(2) 1–10

Ó The Author(s) 2017 DOI: 10.1177/1687814017690470 journals.sagepub.com/home/ade

An approach to the dynamic modeling

and sliding mode control of the

constrained robot

Heng Shi, Yanbing Liang and Zhaohui Liu

Abstract

An approach to the dynamic modeling and sliding mode control of the constrained robot is proposed in this article On the basis of the Udwadia–Kalaba approach, the explicit equation of the constrained robot system is obtained first This equation is applicable to systems with either holonomic or non-holonomic constraints, as well as with either ideal or non-ideal constraint forces Second, fully considering the uncertainty of the non-ideal force, that is, the dynamic friction

in the constrained robot system, the sliding mode control algorithm is put forward to trajectory tracking of the end-effector on a vertical constrained surface to obtain actual values of the unknown constraint force Moreover, model order reduction method is innovatively used in the Udwadia–Kalaba approach and sliding mode controller to reduce variables and simplify the complexity of the calculation Based on the demonstration of this novel method, a detailed robot system example is finally presented

Keywords

Constrained robot, Udwadia–Kalaba equation, sliding mode control, dynamic modeling, simulation

Date received: 29 September 2016; accepted: 3 January 2017

Academic Editor: Elsa de Sa Caetano

Introduction

A constrained robot system is a typical mechanical

sys-tem The control of this kind of system usually needs

some dynamic equations It is known that the robot

system has characteristics of high coupling,

nonlinear-ity, and uncertainty in trajectory tracking As a result,

it is almost impossible to build a model of the robot

dynamics perfectly Fortunately, this solution of this

problem is vigorously worked since the constrained

movement technique was proposed by Lagrange.1 He

developed a Lagrange multiplier method for solving

constrained movements However, in practical

engi-neering applications, it is difficult to get the Lagrange

multiplier, which leads to the difficulty of obtaining this

equation Then, Gauss2provided a new common

prin-ciple for motions of constrained mechanical systems,

which can be applied in constrained robot systems

Gibbs3 and Appell4 also present the comprehensive

equation of movement which is apprised highly by Pars;5 however, the equation is difficult to deal with large degree of freedom (DOF)

Professors Udwadia and Kalaba6–9 proposed the equation of the multi-body system motion under the constraint condition, which is one of the important achievements in Lagrange mechanics field This equa-tion is applicable to a variety of constraints, such as holonomic and non-holonomic constraints Later on, they extended their work to the non-ideal constraint system and general mechanical system The merit of

Xi’an Institute of Optics and Precision Mechanics, Chinese Academy of Sciences, Xi’an, China

Corresponding author:

Heng Shi, Xi’an Institute of Optics and Precision Mechanics, Chinese Academy of Sciences, Xi’an 710119, China.

Email: shiheng@opt.ac.cn

Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 3.0 License

(http://www.creativecommons.org/licenses/by/3.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/ open-access-at-sage).

their method is that the system they focus on may not

meet D’Alembert’s principle,10 while other works are

almost on the basis of D’Alembert’s principle

Attributing to the simple and general expression, this

equation has attracted more and more attentions and

has been applied in many different fields

In recent years, researchers have done much work to

obtain the dynamic model and control of the

con-strained robot Liu and Liu11got dynamic modeling of

industrial robot subject to constraint using the

Udwadia–Kalaba equation, and the ideal constraint

force is taken into consideration Su et al.12used a

slid-ing mode control algorithm in the constrained robot

system, but the approach ignored the constraint force

which is indispensable in practical application Wang

et al.13 primarily studied the variable control of

non-holonomic constraints Wanichanon et al.14proposed a

general sliding control scheme on the holonomic and

non-holonomic constraints On the foundation of their

work, the ideal force and non-ideal force are both

taken into consideration and a novel dynamic model is

established

Actually, a constrained robot system usually suffers

from the constraint force, which is commonly caused

by the end-effector of the robot being constrained on a

surface In practice, the constraint force which is

pro-duced on the constraint surface cannot be ignored

This constraint force can be divided into two parts

One is the normal force which is regarded as the ideal

constraint force Thanks to the Udwadia–Kalaba

approach, it can be used to obtain the dynamic model

combined with normal force in constrained robot

sys-tem, and the explicit expression of the normal force can

be obtained The other is the tangential force which

can be seen as the non-ideal constraint force Non-ideal

constraint forces often include friction force and

elec-tromagnetic force Note that the friction force cannot

be ignored in the constrained robot system, but

unfor-tunately it cannot be calculated In the simulation, the

friction force can be obtained by first giving an initial

condition and then introducing the sliding mode

con-trol method to track the trajectory and force

The constrained robot system is a very complicated

multi-input multi-output (MIMO) nonlinear system,

which has several dynamic characteristics of

time-vary-ing, coupltime-vary-ing, and nonlinear For these characteristics,

neural adaptive control and sliding mode control are

regularly used to control the constrained robot, and

lots of researchers have achieved good results S Frikha

et al.15 proposed an adaptive neural sliding mode

con-trol scheme with Lyapunov criterion for typical

uncer-tain nonlinear systems, and neural network was used to

estimate the structural model of the system H Wei

et al.16used adaptive neural network control with

full-state feedback for an uncertain constrained robot,

which can effectively guarantee the performance

and improve the robustness of closed-loop system

R Garcı´a-Rodrı´guez and V Parra-Vega17 designed a neural sliding mode control scheme for constrained robots based on Lyapunov function, which can prove the robustness of closed-loop system and finally con-clude convergence of position and force tracking errors Liu et al.18 proposed neural network control which is based on adaptive learning design for nonlinear sys-tems with state constraints, in which signals of the closed-loop system are bounded and the tracking error converges to a bounded compact set

In this article, the sliding mode control19–21 is used

to trajectory tracking of the end-effector on a vertical constrained surface to obtain actual values of the unknown constraint force The sliding mode control is also called the variable structure control, which is pro-posed by Soviet scholars Utkin and Emeleyanov The structure of the sliding mode control system is con-stantly changing as the current state, so that the system

is moving according to a predetermined trajectory The sliding mode control method is suitable for the robot control because of its two benefits On one hand, the sliding mode control does not need the accurate mathe-matical model of the controlled object As mentioned above, in the constrained robot system, the non-ideal force can only be obtained by experimentation or observation Therefore, utilizing this benefit, the sliding mode control algorithm is appropriate to achieve tra-jectory and force tracking On the other hand, the slid-ing mode control is invariant to uncertainty factors such as parameters perturbation and the external dis-turbance This benefit can ensure the control perfor-mance of the system due to the random interference The main contributions of this article are as follows:

1 By the Udwadia–Kalaba equation, the ideal and non-ideal forces are both taken into consid-eration Because the non-ideal constraint force

is hardly calculated but only can be obtained by the experiment or the experience, the sliding mode control algorithm is developed for track-ing the non-ideal force (the dynamic friction) in the constrained robot system In this way, the dynamic equation of the constrained robot is more complete and the non-ideal force can be obtained theoretically This contribution gives a theoretical basis for the future experiment investigation

2 The major innovation of this article is the estab-lishment of a new order reductive dynamic equation and sliding mode controller to describe the constrained robot motion The model order reduction method not only can simplify the complexity of the calculation but also can be extended to multi-degree of the constrained robot system

The outline of this article is organized as follows.

First, Udwadia–Kalaba approach is described in detail

Second, the dynamic model of the constrained robot

system is obtained by the order reduction and

Udwadia–Kalaba approach and the sliding mode

con-trol algorithm is derived Third, the 2-DOF robot with

vertical constraint is used as the example to specify and

verify the correctness of the Udwadia–Kalaba

approach and sliding mode control approach Fourth,

some conclusions are presented

Detail the Udwadia–Kalaba approach

To the robot system without constraint, the dynamic

equation of n-DOF robot is established with Lagrange

method22

M (q, t)€q + C(q, _q) _q + G(q) = t ð1Þ

where q =½q1, q2, , qnT describes joint

displace-ments t denotes applied joint torques M(q, t) is the

symmetric and positive inertia or mass matrix C(q, _q)

represents coriolis and centrifugal torques G(q) denotes

gravitational torques Here, assume joint displacements

are independent of each other Equation (1) is written

in the following form

M (q, t)€q = Q(q, _q, t) ð2Þ Q(q, _q, t) can be thought of the external resultant force

of the constraint system From equation (2), when

q, _q, and t are known, the acceleration can be

obtained as follows

a(q, _q, t) = M1(q, t)Q(q, _q, t) ð3Þ

It is assumed that the constraint form of this robot

system can be described by m = m1+ m2equations6

ui(q, t) = 0 i = 1, 2, , m1 ð4Þ

and

cj(q, _q, t) = 0 j = 1, 2, , m2 ð5Þ

where u is a m1 vector and c is a m2 vector Equations

(4) and (5) include all the usual varieties of holonomic

and/or non-holonomic constraints Differentiating

equation (4) twice with respect to time and equation (5)

once with respect to time, the following matrix form23

can be obtained

A(q, _q, t)€q = b(q, _q, t) ð6Þ where A referred to as m 3 n constraint matrix and b is

a m-vector

When the system is constrained and additional set of

forces act on the robot system, which can be called the

constrained system, the motion equation of the con-strained robot system is given by

M (q, t)€q = Q(q, _q, t) + Qc(q, _q, t) ð7Þ where Qc(q, _q, t) is n-vector, which is caused by the additional constraint force and satisfies constraint conditions

According to D’Alembert’s principle, constraint forces can do positive, negative, or zero work under virtual displacement in the constrained system When constraint forces do no work, they are called ideal con-straint forces While concon-straint forces do work, they can

be named as the non-ideal constraint force which is the dynamic friction in this article Therefore, when the constrained robot system exists ideal and non-ideal con-straints in the same time, the Qc(q, _q, t) can be given by

Qc(q, _q, t) = Qc

id(q, _q, t) + Qc

nid(q, _q, t) ð8Þ where Qc

id(q, _q, t) denotes the ideal constraint force and

Qcnid(q, _q, t) represents the non-ideal constraint force Assuming that the virtual displacement24 is y, the work done by the ideal constraint force Qc

idis shown as

While the work done by the non-ideal constraint force Qc

nid is

yTQcnid 6¼ 0 ð10Þ The form of the ideal and non-ideal constraint force has been given by Udwadia and Kalaba6

Qcid= M12B+(b AM1Q) ð11Þ and

Qcnid= M12(I B+B)M12c ð12Þ where B = AM1=2and the superscript ‘‘ + ’’ represents the Moore–Penrose inverse matrix The vector c is a known vector, which can be obtained by experimenta-tion or observaexperimenta-tion in a certain mechanical system.8 From above all, the equation of the constrained robot system is given by

M €q = Q + M12B+b AM1Q

+ M12I B+B

M12c ð13Þ

Remark Non-holonomic constraint is the constraint that contains time derivatives of the generalized coordinates of the system, which is not integrable While, non-ideal constraint is the one that does virtual work which is not equal to zero in any particle system

So, non-holonomic and non-ideal constraints are

naturally different in the aspect of definition In this

article, holonomic and non-holonomic constraints, as

well as ideal and non-ideal constraints are used to

indicate different kinds of constraints in the constrained

robot systems And according to the Udwadia–Kalaba

approach, the explicit equation of the constrained robot

system is applicable to all holonomic and

non-holonomic (ideal and non-ideal) constrained systems no

matter whether they satisfy D’Alembert’s principle

Model reduction

The constrained robot is a typical mechanical system

The 2-DOF robot with vertical constraints is shown in

Figure 1 below

As shown in Figure 1, it is the schematic diagram of

2-DOF robot with the vertical constraint Let

x =½x1, x2T denotes the end-effector coordinate of the

constrained robot in Cartesian coordinate system

q =½q1, q2Tis the generalized coordinate of the system

There are two perpendicular forces acting on the

end-effector by the vertical plane The first force is the ideal

constraint force Qc

id, which is the positive pressure on the contact surface and is perpendicular to the

strained surface The second force is the non-ideal

con-straint force Qc

nid, which remains tangent to the constrained surface Therefore, the ideal constraint

force Qc

idprovides holding power to guarantee the

end-effector for moving on the contact surface The

non-ideal constraint force Qc

nid provides the tangential accel-eration along the constrained surface

The equation of the constraint is written as20

where f is two times continuously differentiable Assuming that the vector x and q have such a relation

where h is two times continuously differentiable, then the equation of constraints in joint space is obtained

F(q) = f h(q)ð Þ = 0 ð16Þ

In the constrained robot system, Qc

id can be obtained

by equation (11) Qc

nid includes the dynamic friction only in the end-effector When the end-effector is mov-ing on the vertical plane, it is the dynamic friction act-ing on the robot Due to the uncertainty of the Qc

nid, the non-ideal constraint force is expressed as

Qcnid= JT(q, t)F(t) = JT(q, t)l ð17Þ where F(t) is the dynamic friction in Cartesian space l

is the associated Lagrangian multiplier JT(q, t) is the Jacobian matrix of equation (17), which can be given by

J (q, t) = ∂F(q)

∂q =

∂f(x)

∂x

∂x

∂q=

∂f(x)

∂x

∂h(q)

From equations (1), (7), (8), and (17), the dynamic equation of the 2-DOF constrained robot is given by

M (q, t)€q + C(q, _q) _q + G(q) + Qcid(q, _q, t) + JT(q, t)l = t

ð19Þ Since the end-effector of the robot is constrained in the vertical surface, DOFs of the robot system changed from two to one Here, choose q1 as the variable to describe the motion of the constrained robot So, q2 is the remaining joint redundant variable And q2 can be donated by q1

And then from equation (20), one can obtain

_q = _q1 _q2

= ∂c(q_q11)

∂q 1 _q1

" #

= L(q1) _q1 ð21Þ

and

€

q = _L(q1) _q1+ L(q1)€q1 ð22Þ where L(q1) = 1

∂c(q1)=∂q1

Therefore, equation (19) is expressed in the reduced form as

Figure 1 2-DOF robot with vertical constraint.

M1(q1, t)€q1+ C1(q1, _q1) _q1+ G1(q1) + Qcid1(q1, _q1, t)

+ JT(q1, t)l = t ð23Þ where M1(q1, t) = M(q, t)L(q1), C1(q1, _q1) = M (q, t)

_L(q1) + C(q, _q)L(q1), G1(q1) = G(q), Qc

id1(q1, _q1, t) =

Qc

id(q, _q, t), and JT(q1, t) = JT(q, t)

Remark Equation (23) is the basis for the control

purpose of the constrained robot system

Now multiply both sides of equation (23) with

LT(q1), the following equation is obtained

LT(q1)M1(q1, t)€q1+ LT(q1)C1(q1, _q1) _q1+ LT(q1)G1(q1)

+ LT(q1)Qc

id 1(q1, _q1, t)

+ LT(q1)JT(q1, t)l = LT(q1)t

ð24Þ Equation (24) can be simplified as

ML(q1, t)€q1+ CL(q1, _q1) _q1+ GL(q1) + QcidL(q1, _q1, t) = LTt

ð25Þ

By exploiting the structure of equations (23) and

(25), three properties can be obtained:25

Property 1: Define the matrix ML(q1, t) = LT(q1)

M1(q1, t), ML(q1, t).0

Property 2: Define the matrix CL(q1, _q1) = LT(q1)

C1(q1, _q1) and then M_L(q1, t) 2CL(q1, _q1) is the

skew symmetric matrix

Property 3: J (q1, t)L(q1) = LT(q1)JT(q1, t) = 0

Three properties are the basis of the design for the

sliding mode control laws

To obtain the practical dynamic friction in

simula-tion, the l value can be obtained using the following

three situations

Situation 1: When J (q) = 0, the value of l can be

obtained by the expression of JT(q1, t) or the

defini-tion of the dynamic fricdefini-tion

Situation 2: When J (1)6¼ 0, the value of l is given

by

l= 1

J (1) t(1) M1(1)€q1 C1(1) _q1 G1(1) Qc

id1(1)

ð26Þ Situation 3: When J (2)6¼ 0, the value of l is given

by

l= 1

J (2) t(2) M1(2)€q1 C1(2) _q1 G1(2) Qc

id1(2)

ð27Þ

To be sure, the value of l can be measured by the force sensor in the practical engineering

Sliding mode control for the constrained robot system

In this section, a general tracking problem for the con-strained robot system is considered As the desired joint position qd(t) and the desired constraint force ld are known, the desired constraint force which is known for the desired dynamic friction JT(q, t)ld is known Note that the objective of the control is to track the desired joint position and the constraint force within an accep-table error range and to satisfy the imposed constraints,

Qcnidd= JT(q, t)ld and F(qd) = 0 A sliding mode con-trol law is designed to make q(t) track qd(t), and Qc

nid track Qc

nidd as t! ‘ Since q2= c(q1), a sliding mode control law is only need to be found to satisfy q1(t) track qd1(t) as t! ‘

Defining

_qr1= _qd1+ Le1 ð29Þ

where e1 denotes the tracking error, el represents the force tracking error, qr1is the reference trajectory, and L.0 is the tunable matrix

The sliding surface is defined as

s1= _qr1 _q1= _e1+ Le1 ð31Þ

sL1= L(q1)s1 ð32Þ The sliding controller is defined as

t= M1(q1, t)€qr1+ C1(q1, _q1) _qr1+ G1(q1) + Qcid1(q1, _q1, t)

+ KpsL1+ JT(q1, t)lr ð33Þ where Kp.0

The item for controlling the dynamic friction is given by

lr= ld+ Klel ð34Þ where Kl.0

From equation (34), the following equation is obtained

lr l = el+ Klel= 1 + Kð lÞel ð35Þ Using equations (23), (31), and (33), the following equation can be obtained

M1(q1, t)_s1+ C1(q1, _q1)s1+ KpsL1= JT(q1, t) lð  lrÞ

ð36Þ

Multiply both sides of equation (36) with LT(q1) and

using the Property 3, equation (36) can be written as

ML(q1, t)_s1+ CL(q1, _q1)s1+ LT(q1)KpsL1= 0 ð37Þ

The Lyapunov function is taken as

V = 1

2ML(q1, t)s

2

Differentiating equation (38) with respect to time

gives

_

V = s1 ML(q1, t)_s1+1

2M_L(q1, t)s1

ð39Þ

Considering the skew symmetric property of

_

ML(q1, t) 2CL(q1, _q1), equation (39) is expressed as

_

V = s1ðML(q1, t)_s1+ CL(q1, _q1)s1Þ ð40Þ

Substituting equation (37) into equation (40), one

can obtain

_

V = s1ðML(q1, t)_s1+ CL(q1, _q1)s1Þ = s1LT(q1)KpsL1

= sT

L1KpsL1= Kps2L1 0

ð41Þ Since _V is negative semi-definite and Kp is positive

definite, when _V [0, the following are obtained

sL1[0, _sL1[0

s1[0, _s1[0 _e1[e1[0

ð42Þ

According to equations (35), (36), and (42), it is

obvi-ous that

l lr[0

The following can be known by LaSalle theorem

e1! 0, _e1! 0, l! ld as t! ‘

Simulated example

As shown in Figure 1, the 2-DOF robot with vertical

constraint is used to verify the correctness and

reliabil-ity of the proposed dynamic model and the sliding

mode control method Detailed matrices in equation

(1) are shown in the following

M (q, t) = p1+ p2+ p3+ 2p4cos q2 p3+ p4cos q2

p3+ p4cos q2 p3

ð44Þ

C(q, _q) = p4_q2sin q2 p4( _q1+ _q2) sin q2

p4_q1sin q2 0

ð45Þ

G(q) =

3

2p1e1cos q1+ p2e1cos q1+ p4e1cos (q1+ q2)

p4e1cos (q1+ q2)

ð46Þ where e1= g=l1, g is the acceleration of gravity And the four unknown parameters p1, p2, p3, and p4 are functions of physical parameters

p1=1

3m1l

2 1

p2= m2l21

p3=1

3m2l

2 2

p4=1

2m2l1l2

ð47Þ

According to equation (2), the external resultant force without constraint can be obtained as26

Q(q, _q, t) = C(q, _q) _q  G(q) ð48Þ

In equation (48), because the robot has not been con-strained, the vector of applied joint torque is t =½0, 0T

As shown in Figure 1, the position of the end-effector is obtained as

X1= 0

X2=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

l2+ l2 2l1l2cos (2q1)

where ½X1 X2 is the coordinate of the end-effector in the global coordinate system

Considering the end-effector of the robot is con-strained with the vertical surface, one can get

l1cos q1+ l2cos (q1+ q2) = X1

l1sin q1+ l2sin (q1+ q2) = X2 ð50Þ Taking time derivation twice on equation (50), one can obtain

 l1sin q1€1 l2sin (q1+ q2)€q1 l2sin (q1+ q2)€q2

= l1cos q1_q21+ l2cos (q1+ q2)( _q1+ _q2)2+ €X1

l1cos q1€1+ l2cos (q1+ q2)€q1+ l2cos (q1+ q2)€q2

= l1sin q1_q21+ l2sin (q1+ q2)( _q1+ _q2)2+ €X2

ð51Þ Differentiating equation (49) with respect to time twice, the following equation is given by

€

X1= 0

€

X2=2n

m€1+

4m2p 4n2

m3 _q21 ð52Þ

where the three parameters m, n, and p are functions of

physical parameters

m =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

l12+ l2

2 2l1l2cos (2q1) q

n = l1l2sin (2q1)

p = l1l2cos (2q1)

ð53Þ

Substituting equation (52) into equation (51), the

second-order constraint equation23is shown as

A(q, _q, t)€q = b(q, _q, t) ð54Þ

in which

A = l1sin q1 l2sin (q1+ q2) l2sin (q1+ q2)

l1cos q1+ l2cos (q1+ q2)2n

m l2cos (q1+ q2)

b = l1cos q1_q

2+ l2cos (q1+ q2)( _q1+ _q2)2

l1sin q1_q2+ l2sin (q1+ q2)( _q1+ _q2)2+4m2p4nm3 2_q2

ð55Þ Now, M, A, b, and Q have been obtained, so the

ideal constraint force Qc

id can be obtained according to equation (11)

As shown in Figure 1, the constraint function is

X1= f(x) = 0

f(q1) = l1cos q1+ l2cos (q1+ q2) = 0 ð56Þ

Since the constraint equation is

F(q) = f h(q)ð Þ = f(q1) = l1cos q1+ l2cos (q1+ q2)

ð57Þ According to equation (18), the following is obtained

J (q, t) =∂F(q)

∂q =½l1sin q1 l2sin (q1+ q2)

l2sin (q1+ q2) ð58Þ

Two situations to get the value of l are shown as follows:

Situation 1: When q1+ q2= p and q1= 0, q = p,

we get J (q, t) = 0 From Figure 1, one can easily find that two links of the robot are placed with over-lapping each other In this situation, l = 0 is obtained, which means the constrained dynamic friction is zero

Situation 2: When q1+ q26¼ p and J (1) 6¼ 0,

J (2)6¼ 0, l can be obtained by equations (26) and (27)

In simulation, it is assumed that l1= l2= 1,

m1= m2= 1 According to equation (56), one can obtain

cos q1+ cos (q1+ q2) = 0 ð59Þ The robot is constrained by the vertical surface, so 0\q1 p=2, 0 q2\p Then

cos (q1+ q2) = cos (p q1) ð60Þ The relationship between q1 and q2 can be obtained as

According to equation (61), one can obtain

L(q1) = 1

2

ð62Þ

For the simulation, the reduced order model is used

as the controlled object The initial position is

qd1= 0:7 + 0:6 cos t and the desired dynamic friction is

ld= 8 sin t; therefore, the initial value is chosen as

q = 1:3½ p 2:6 The controller parameters are

Kp= 5 0

0 5

, Kl= 8, and L = 10:0 The simulation time is set to 20 s The control scheme of the proposed control is shown in Figure 2

1

d

1

d

q

1

e

The calculation of the sliding mode surface

The sliding mode controller

1

robot system

d

λ

λ

eλ

−

1

q

1

q

1

q

Λ

p

K

Kλ

L(q1)

−

T(q1,t)

Figure 2 The control scheme of the controlled system.

Simulation results are shown in Figures 3–8.

Figures 3 and 4 show angles and angle speed

track-ing of the constrained robot, in which Figure 3

represents the first link and Figure 4 represents the sec-ond link In Figure 3, solid red lines represent ideal val-ues of q1 and dq1 and dashed blue lines represent practical values of q1 and dq1 It can be found that solid lines are almost coincident with dashed lines Similar to Figure 4, solid red lines represent ideal val-ues of q2 and dq2 and dashed blue lines represent prac-tical values of q2 and dq2 Due to equation 2q1+ q2= p, as shown in Figures 3 and 4, the rela-tionship between q1 and q2 has been validated obviously

Figures 5 and 6 show tracking errors of angles and angle speed of the constrained robot system It is obvi-ous that tracking errors of angles and angle speed are almost equal to zero In the stable region of Figures 5 and 6, the maximum error of q1 and q2 is less than 0.0005 rad and the maximum error of dq1 and dq2 is less than 0.001 rad/s The main cause of this error and stability is the switch of discontinuous features in slid-ing mode control, which causes the chatterslid-ing of the system When the trajectory of the system gets to the switching surface, the moving point can pass through the switching surface forming the chattering Major factors in the production of chattering mainly include four aspects: time delay switch, spatial delay switch, the inertia, and discreteness of the system

Figure 7 represents the torque of the constrained robot, in which the solid red line represents torque val-ues of the first link and the dashed blue line represents torque values of the second link Because the second link is little affected by the static moment and the first link is affected by the static and dynamic moments, t1

is greater than the t2 Figure 8 shows the tracking and tracking error of the constrained dynamic friction, which can be seen as the non-ideal force in Udwadia and Kalaba theory It

is observed that the dashed red line represents desired values of l and the solid blue line represents practical

Figure 3 The angle and angle speed tracking of the first link.

Figure 4 The angle and angle speed tracking of the second

link.

Figure 5 Tracking errors of angles of the robot.

Figure 6 Tracking errors of angle speed of the robot.

values of l It can be found that the tracking error

between l and ldis almost equal to zero

From the above figures, these simulation results

show that the dynamic model and the sliding mode

control algorithm are achieved successfully

Conclusion

In this article, a novel approach to the dynamic

model-ing and slidmodel-ing mode control of the constrained robot

system is proposed By the Udwadia–Kalaba equation,

expressions of the ideal and non-ideal forces are

obtained and then the dynamic equation of the

con-strained robot system is established Because the

non-ideal constraint force is hardly calculated, the sliding

mode control algorithm is presented for tracking the

non-ideal force (the dynamic friction) in the constrained

robot system The major innovation in this article is the establishment of a new order reductive dynamic model and sliding mode controller to describe the constrained robot motion Due to the lack of freedom, model order reduction method is creatively used in the Udwadia– Kalaba approach to complete the simulation of the con-strained robot system The model order reduction method can simplify the complexity of the calculation and can be extended to multi-degree of constrained robot system A simple 2-DOF robot system with the vertical constraint is used to illustrate the methodology proposed in the article From several simulation results,

it can be found that the tracking errors are almost equal

to zero So, the proposed model and control method are feasible, correct, and valid

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.

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