Trajectory Tracking Control of Parallel Manipulator Based on Udwadia-Kalaba … · 2019. 8. 28. · ResearchArticle Trajectory Tracking Control of Parallel Manipulator Based on Udwadia-Kalaba

Research ArticleTrajectory Tracking Control of Parallel Manipulator Based onUdwadia-Kalaba Approach

Chenming Li,1 Han Zhao,1 Shengchao Zhen,1,2 Kang Huang,1 Hao Sun,1,2

Ke Shao,1 and Bin Deng1

1School of Mechanical Engineering, HeFei University of Technology, Hefei 230009, China2School of Mechanical Engineering, Georgia Institute of Technology, Atlanta 30332, USA

Correspondence should be addressed to Shengchao Zhen; [email protected]

Received 1 August 2017; Accepted 27 November 2017; Published 14 December 2017

Academic Editor: Emiliano Mucchi

Copyright © 2017 Chenming Li et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

There have been many approaches for achieving the trajectory tracking control of parallel manipulator. However, these approachesare complex for calculating Lagrangian multipliers. In this paper, unlike the former approaches, a new approach of trajectory track-ing control which is based on Udwadia-Kalaba approach is presented. Using this methodology, we can obtain a concise and explicitequation of motion and consider holonomic and nonholonomic constraint whether it is ideal or nonideal simultaneously.Themostimportant difference is that we divide constraints into structural constraints and performance constraints in this paper. Structuralconstraints are used to establish dynamic model without regard for trajectory control. And performance constraints are used torepresent the desired trajectory. For the parallel manipulator, a nonlinear dynamics system, its constraint forces are obtained bysecond-order constraints.And the numerical simulation inMATLAB shows the parallelmanipulator’smovementmeets the require-ment; tracking trajectory is exact and perfect. Through this paper, we can see that the method can simplify calculation availably.

1. Introduction

Parallel manipulator is a kind of closed-loop kinematicchain mechanism in which the end-effector is linked tothe base by several independent kinematic chains [1]. Incomparison with the serial manipulator which has open-loop structure, the parallelmanipulator hasmany advantages,such as higher stiffness, higher accuracy, higher payloadcapacity, lower inertia, and smaller workspace. Therefore, ithas attracted a great amount of attention and interest amongthe researchers and engineers and has been applied to manyindustrial applications in the past decade, like milling [2],rapid machining [3], medical resuscitation [4], and so on.

In order to achieve the great potential performances exist-ing in the parallel manipulator system, the controller designis essential and indispensable. To deal with the trajectorytracking control problem of the parallel manipulator, it is easyto find that many advanced controllers have been proposedand successfully implemented, like proportional-integral-derivative control [5], sliding mode control [6], adaptive con-trol [7, 8], robust control [9, 10], robust PID control [11], adap-tive sliding mode control [12], adaptive robust control [13],

fuzzy control [14] neural network control [15], fuzzy neuralnetwork control [16], and so on.

Parallel manipulators can be designed with variousmechanical structures because the designers want them torealize different control requirements according to the actualdemands. In this paper, a 3-degree-of-freedom (3-DOF)planar parallel manipulator (3-HSS, H-helix pair, S-sphericalpair) has been designed and constructed. To deal with thetrajectory tracking control problem of this manipulator,our methodology is based on a novel, concise approach toexplicitly formulate the equations of motion for constrainedsystems proposed by Udwadia and Kalaba [17–20].

In this paper, we will use the basic kinematics equationsproposed byUdwadia and Kalaba based onGauss’s theory formodeling and trajectory tracking of the parallel manipula-tors. There are several key points by using the method in thispaper compared with the traditional control method. First,there is no need to obtain the complex Lagrangian multipliersduring the process of dynamic modeling of the parallelmanipulators. Second, we define the constraint equations bya new perspective which means the constraints are dividedinto structural constraints and performance constraints.

HindawiMathematical Problems in EngineeringVolume 2017, Article ID 8975743, 12 pageshttps://doi.org/10.1155/2017/8975743

https://creativecommons.org/licenses/by/4.0/https://doi.org/10.1155/2017/8975743

2 Mathematical Problems in Engineering

Structural constraints are used to set up the dynamic modelin the absence of trajectory control. And performanceconstraints are used to reflect the desired trajectory of theparallel manipulator. Third, we can obtain the needed motortorque for realizing the trajectory of manipulator directly bycalculating the Udwadia and Kalaba equation. At last, thesimulation results in this paper show that this method cancontrol the parallel manipulator accurately.

2. Udwadia and Kalaba Equation forConstrained Mechanical Systems

In this section, we talk about the fundamental equation,called Udwadia-Kalaba equation, for constrained mechanicalsystem. We can obtain the explicit closed-form equation ofmotion of the constrained mechanical system in three steps[18].

First, we consider the unconstrained mechanical systemin which the coordinates are all assumed to be independentof each other. The equation of motion of this system can beexpressed, by using Newtons or Lagranges equation, as

𝑀(𝑞, 𝑡) ̈𝑞 = 𝑄 (𝑞, ̇𝑞, 𝑡) (1)with the initial conditions

𝑞 (𝑡 = 0) = 𝑞0,̇𝑞 (𝑡 = 0) = ̇𝑞0 (2)

where 𝑞 is the generalized coordinate 𝑛-vector and 𝑡 is thetime;𝑀(𝑞, 𝑡) is a positive definite inertia 𝑛 × 𝑛matrix; ̇𝑞 is an𝑛-vector of velocity; ̈𝑞 is an 𝑛-vector of acceleration; 𝑄(𝑞, ̇𝑞, 𝑡)is an 𝑛-vector called the given force which could includecentrifugal force, gravitational force, and control input. From(2), the generalized acceleration of the unconstrained systemis given by

̈𝑞 = 𝑎 (𝑞, ̇𝑞, 𝑡) = 𝑀−1 (𝑞, 𝑡) 𝑄 (𝑞, ̇𝑞, 𝑡) . (3)Second, a set of control requirement or trajectory require-

ments is imposed as constraints on this uncontrolled system.We shall assume that the system is subjected to ℎ holonomicconstraints of the form

𝜑𝑖 (𝑞, 𝑡) = 0, 𝑖 = 1, 2, . . . , ℎ; (4)meanwhile there are 𝑚 − ℎ nonholonomic constraints in theform of

�̇�𝑖 (𝑞, ̇𝑞, 𝑡) = 0, 𝑖 = ℎ + 1, ℎ + 2, . . . , 𝑚. (5)Then, we can differentiate these constraint equations in

Lagrangian mechanics which are usually in Pfaffian form.Under the assumption of sufficient smoothness, 2-orderconstraint equations can be acquired by taking the timederivative once on nonholonomic constraints and twice onholonomic constraints, which can be written in the form ofmatrix as

𝐴 (𝑞, ̇𝑞, 𝑡) ̈𝑞 = 𝑏 (𝑞, ̇𝑞, 𝑡) (6)where 𝐴(𝑞, ̇𝑞, 𝑡) is an 𝑚 × 𝑛 matrix denoting the constraintmatrix and 𝑏(𝑞, ̇𝑞, 𝑡) is an𝑚 × 1 vector.

Remark 1. Theconstraint is in either the zeroth-order or first-order form which is expressed in the form of Lagrangianequation, Maggi equation, Boltzmann and Hamel equa-tion, and Gibbs and Appell equation. However, for furtherdynamic analysis and control design, second-order con-straints are considered to be the most suitable form. The rea-son is that the acceleration will be linear by using the second-order constraint. And there is no need to worry about the lossof information during the differentiation (e.g., a constant). Infact, making the zeroth-order or holonomic constraint meetthe initial condition means the initial condition retained themissing information. Thus, when we use Udwadia-Kalabaequation, the first step is to find all constraints includingholonomic constraints and nonholonomic constraints in theunconstrained system. Then, we could obtain the second-order form of constraints by differentiating the constraintswith respect to time once or twice. Finally, we could trans-form these constraints into second-order matrix equationslike (6).

Eventually, the explicit equations of motion with con-straints will be formed. Notice that the additional generalizedforces of constraints need to be imposed on the system as aresult of the presence of constraints. We therefore assume theactual explicit equation of motion of the constrained systemin the form of

𝑀(𝑞, 𝑡) ̈𝑞 = 𝑄 (𝑞, ̇𝑞, 𝑡) + 𝑄𝐶 (𝑞, ̇𝑞, 𝑡) (7)where 𝑄𝐶(𝑞, ̇𝑞, 𝑡) ∈ 𝑅𝑛 is the additional generalized forcewhich is imposed on the unconstrained system, arising bythe holonomic and nonholonomic constraints denoting a 𝑛by 1 constraint matrix. Determining the general explicit formof 𝑄𝐶(𝑞, ̇𝑞, 𝑡) ∈ 𝑅𝑛 is our goal with the 𝑄(𝑞, ̇𝑞, 𝑡) known atany time 𝑡. In Lagrangian mechanics, 𝑄𝐶(𝑞, ̇𝑞, 𝑡) ∈ 𝑅𝑛 isconsidered to be ideal constraints because this kind of forcesof constraint does zero work under virtual displacementaccording to the D’ Alembert’s Principle. However non-ideal constraints also exist in an actual mechanical system,and the nonideal constraint forces including friction force,electromagnetic force, will be generated. Udwadia expressed𝑄𝐶(𝑞, ̇𝑞, 𝑡) ∈ 𝑅𝑛 in the form of

𝑄𝐶 (𝑞, ̇𝑞, 𝑡) = 𝑄𝐶id (𝑞, ̇𝑞, 𝑡) + 𝑄𝐶nid (𝑞, ̇𝑞, 𝑡) (8)where 𝑄𝐶id(𝑞, ̇𝑞, 𝑡) represents the ideal constraint force and𝑄𝐶nid(𝑞, ̇𝑞, 𝑡) represents the nonideal constraint.

Furthermore, Udwadia and Kalaba have proved the gen-eral “explicit” equations of ideal and nonideal constraintforces at any time 𝑡 can be expressed in the form of

𝑄𝐶id (𝑞, ̇𝑞, 𝑡) = 𝑀1/2𝐵+ (𝑏 − 𝐴𝑀−1𝑄)𝑄𝐶nid (𝑞, ̇𝑞, 𝑡) = 𝑀1/2 (𝐼 − 𝐵+𝐵)𝑀−1/2𝑐

(9)

where 𝐵 = 𝐴𝑀−1/2 and the superscript “+” denotes theMoore-Penrose generalized inverse.

Mathematical Problems in Engineering 3

From (7), (8), and (9), the general “explicit” equation ofmotion can be expressed as

𝑀 ̈𝑞 = 𝑄 +𝑀1/2𝐵+ (𝑏 − 𝐴𝑀−1𝑄)+𝑀1/2 (𝐼 − 𝐵+𝐵)𝑀−1/2𝑐. (10)

Remark 2. In particular, 𝑐(𝑞, ̇𝑞, 𝑡) is a suitable 𝑛-vector in(10) which is determined by the mechanical system. Now wedefine a given nonzero 𝑛-vector virtual displacement V in aconstrained system at the instant time 𝑡 and the work done𝑊 generated by 𝑐(𝑞, ̇𝑞, 𝑡) under virtual displacement V. Andthe work done 𝑊 = V𝑇𝑐(𝑞, ̇𝑞, 𝑡) which in any displacementV which is subject to 𝐴(𝑞, ̇𝑞, 𝑡)V = 0 is the same as the workdone by𝑄𝐶(𝑞, ̇𝑞, 𝑡) ∈ 𝑅𝑛. Hence, we put the equation in formof

𝑊 = V𝑇𝑄 (𝑞, ̇𝑞, 𝑡) = V𝑇𝑐 (𝑞, ̇𝑞, 𝑡) (11)which means we extend Lagrange’s form of D’ Alembert’sPrinciple. The work done by the ideal constraint force𝑄𝐶id(𝑞, ̇𝑞, 𝑡) and the nonideal constraint force 𝑄𝐶nid(𝑞, ̇𝑞, 𝑡)under virtual displacements is

V𝑇𝑄𝐶id = 0V𝑇𝑄𝐶nid ̸= 0

(12)

when the constraints are ideal and the total work done undervirtual displacement is zero which means 𝑐(𝑞, ̇𝑞, 𝑡) equalszero; according to D’ Alembert’s Principle, (10) becomes

𝑀 ̈𝑞 = 𝑄 +𝑀1/2𝐵+ (𝑏 − 𝐴𝑀−1𝑄) . (13)Therefore, at any instant of time, the constrained system

is subjected to an additional constraint force 𝐹𝐶(𝑡), given by𝐹𝐶 (𝑡) = 𝑀1/2𝐵+ (𝑏 − 𝐴𝑀−1𝑄) . (14)

If the unconstrained acceleration 𝑀−1𝑄 is zero with theconstant diagonal matrix𝑀 = 𝑚𝐼 (14) becomes

𝐹𝐶 (𝑡) = 𝑚𝐴+𝑏. (15)Remark 3. There are two main advantages of this equationwhich Udwadia and Kalaba have provided. First, by usingthis equation, there is no need to determine the Lagrangianmultipliers which are always complex. On the contrary, theparameters in the equation are all simple to obtain. Second,this equation applies to all holonomic and/or nonholonomicconstrained systems including ideal or nonideal constraints.

3. Trajectory Tracking Control ofthe Parallel Manipulator

3.1. Dynamic Model of the Parallel Manipulator. The 3-HSSparallel manipulator is composed of a moving platform andthree sets of sliding saddle chain structure. Each branchedchain includes two equilong and parallel bars. One end of the

1 2

3

P

Figure 1: The 3-HSS mechanism structure.

bar is connectedwith the saddle by rolling spherical joint, andthe other end is connected to the moving platform. As for thesliding saddle which is driven by the servomotor-screw-nutpair, it is moving along the ball screw which is mounted onthe guide block. Hence the moving platform could achievethe 3D-motion. In addition, the platform is capable of highspeed machining with the high speed motorized spindle onit.

The law of movement of each bar in branched chain is thesame based on the parallelogram strut structure. Therefore,in this paper, we abstract the original mechanism into single-bar equivalent model as shown in Figure 1 during thekinematics modeling process. In this equivalent mechanism,each branched chain contains only one bar. And two endsof the bar are connected with the moving platform and thesaddle by Hooke joint. Beside the saddle is driven by activepair (helix pair). Based on the above structure, the equivalentmechanism has the same degree of freedom as the originalmechanism.

Next, we employ the classic universal joint, prismatic jointand spherical joint to illustrate the dynamic modeling of thedelta parallel structure (as show in Figure 1). Following theprocedure, we segment the delta parallel structure into foursubsystems: three arm subsystems (1, 2, 3) and one platformsubsystem (𝑃). We shall then derive the equations of motionof the “unconstrained” subsystems.

Select reference frame 𝑜𝑥𝑦𝑧. And assume the mechanismfixed on the plane 𝑜𝑥𝑦. Then consider the arm-subsystem inFigure 1 separately. Denote the following.

As shown in Figure 2, the end of the branched chainwhich is connected with the moving platform can bedescribed with coordinates 𝑞𝑖 = [ℎ𝑖, 𝜃𝑖, 𝜑𝑖]𝑇, 𝑖 = 1, 2, 3.And the moving platform can be described with coordinates𝑞𝑃 = [𝑥𝑃, 𝑦𝑃, 𝑧𝑃]𝑇.ℎ𝑖 is height of the other end of the 𝑖th branched chain. 𝜃𝑖is the angle between 𝑖th branched chain and 𝑧-axis. 𝜑𝑖 is theangle between the 𝑖th branched chain’s projection in the 𝑜𝑥𝑦plane and 𝑥-axis. 𝑙𝑖 is the length of the 𝑖th branched chain.

Thus, when we consider the arm-subsystem separately,the motor’s coordinate of location can be expressed as(0, 0, ℎ) and the end of the branched chain’s coordinate oflocation can be expressed as ((𝑙/2) sin 𝜃 cos𝜑, (𝑙/2) sin 𝜃 sin 𝜑,


l

P

ℎ

X

Y

Z

O

∅

Figure 2: The arm-subsystem.

ℎ + (𝑙/2) cos 𝜃). Then the kinetic and potential energy of thesystem can be written as

𝑇𝑠 = 𝑇1 + 𝑇2 = 12𝑚1V21 + 12𝑚2V22 + 12𝐽𝜔2 = 12𝑚1ℎ̇2

+ 12𝑚2(( 𝑙2 ( ̇𝜃 cos 𝜃 cos𝜑 − �̇� sin 𝜃 sin 𝜑))2

+ ( 𝑙2 ( ̇𝜃 cos 𝜃 sin 𝜑 + �̇� sin 𝜃 cos𝜑))2

+ (ℎ̇ − 𝑙2 ̇𝜃 sin 𝜃)2) + 12𝐽 ( ̇𝜃2 + �̇�2)

𝑉𝑠 = 𝑉1 + 𝑉2 = 𝑚1𝑔ℎ + 𝑚2𝑔(ℎ + 𝑙2 cos 𝜃)

(16)

where 𝑇𝑠, 𝑉𝑠 is the system’s kinetic/potential energy, 𝑇1, 𝑉1 isthe motor’s kinetic/potential energy, 𝑇2, 𝑉2 is the branchedchain’s kinetic/potential energy, and 𝑚1, 𝑚2 is the mass ofmotor/branched chain.

Next, adopting the standard Lagrangian approach, wehave the following equation of motion for each “uncon-strained” arm subsystem (𝑖 = 1, 2, 3):

𝑀𝑖 (𝑞𝑖, 𝑡) ⋅ ̈𝑞 + 𝐶𝑖 (𝑞𝑖, ̇𝑞𝑖, 𝑡) = 𝜏𝑖 (𝑡) (17)where

𝑀𝑖 =[[[[[[[

𝑚1 + 𝑚2 −12𝑚2𝑙 sin 𝜃1 0−12𝑚2𝑙 sin 𝜃1 𝐽1 + 14𝑚2𝑙2 0

0 0 𝐽1 + 14𝑚2𝑙2

]]]]]]]

𝐶𝑖 = [[[[[

−12𝑚2𝑙 ̇𝜃21 cos 𝜃1 − (𝑚1 + 𝑚2) 𝑔−𝑚2ℎ̇1 ̇𝜃1𝑙 ⋅ cos 𝜃1 + 12𝑚2𝑔𝑙 sin 𝜃10

]]]]],

𝜏𝑖 = [[[

𝑓𝑎𝑐𝑖00]]].

(18)

For the “unstrained” platform, that is, the subsystem 4simplified as mass, by choosing the system coordinates as inFigure 2, the equation of motion is

𝑀𝑖 (𝑞𝑖, 𝑡) ⋅ ̈𝑞 + 𝐶𝑖 (𝑞𝑖, ̇𝑞𝑖, 𝑡) = 𝜏𝑖 (𝑡) , 𝑖 = 𝑝. (19)We have

[[[[

𝑚𝑝 0 00 𝑚𝑝 00 0 𝑚𝑝

]]]][[[[

�̈�𝑝�̈�𝑝�̈�𝑝]]]]= [[[[

00

𝑚𝑝𝑔]]]]. (20)

So, the dynamic equations of the system is

𝑀(𝑞) ̈𝑞 + 𝐶 (𝑞, ̇𝑞) + 𝐺 (𝑞) = 𝜏. (21)Now we cluster all the 4 subsystems together by letting

𝑀 =[[[[[[

𝑀1𝑀2

𝑀3𝑀𝑃

]]]]]]12×12

,

𝐶 =[[[[[[

𝐶1𝐶2𝐶3𝐶𝑃

]]]]]]12×1

,

𝜏 =[[[[[[

𝜏1𝜏2𝜏3𝜏𝑃

]]]]]]12×1

,

(22)

where

𝑀1 =[[[[[[[


0 0 𝐽1 + 14𝑚2𝑙2

]]]]]]],


𝑀2 =[[[[[[[


0 0 𝐽1 + 14𝑚2𝑙2

]]]]]]],

𝑀3 =[[[[[[[


0 0 𝐽1 + 14𝑚2𝑙2

]]]]]]],

𝑀𝑃 = [[[[

𝑚𝑝𝑚𝑝

𝑚𝑝]]]],

𝐶1 = [[[[[


]]]]],

𝐶2 = [[[[[


]]]]],

𝐶3 = [[[[[


]]]]],

𝐶𝑃 = [[[[

00

−𝑚𝑝𝑔]]]].

(23)

3.2. Structural Constraint of the ParallelManipulator. Wenowintroduce all the kinematic constraints among the subsystemsand convert them into the second-order form.

Since the spherical joint only fixes the three translationalDOF’s between the connected links, we have the kinematicconstraints, for 𝑖 = 1, 2, 3, 𝑃 as follows:

𝑥ℎ1 = 𝑥𝑝,𝑥ℎ2 = 𝑥𝑝,𝑥ℎ3 = 𝑥𝑝𝑦ℎ1 = 𝑦𝑝,𝑦ℎ2 = 𝑦𝑝,𝑦ℎ3 = 𝑦𝑝𝑧ℎ1 = 𝑧𝑝,

𝑧ℎ2 = 𝑧𝑝,𝑧ℎ3 = 𝑧𝑝.

(24)

Choosing generalized coordinate system

𝑞 = [ℎ1, 𝜃1, 𝜑1, ℎ2, 𝜃2, 𝜑2, ℎ3, 𝜃3, 𝜑3, 𝑥𝑝, 𝑦𝑝, 𝑧𝑝]𝑇 . (25)So, we can get

𝑥𝑝 = 𝑙 sin 𝜃1 cos𝜑1 = 𝑙 sin 𝜃2 cos𝜑2 = 𝑙 sin 𝜃3 cos𝜑3𝑦𝑝 = 𝑙 sin 𝜃1 sin 𝜑1 = 𝑙 sin 𝜃2 cos𝜑2 = 𝑙 sin 𝜃3 cos𝜑3𝑧𝑝 = ℎ + 𝑙 cos 𝜃1 = ℎ + 𝑙 cos 𝜃2 = ℎ + 𝑙 cos 𝜃3.

(26)

Differentiating (26) with respect to 𝑡 twice, we can get1𝑙 �̈�𝑝 = ̈𝜃𝑖 cos 𝜃𝑖 cos𝜑𝑖 − �̈�𝑖 sin 𝜃𝑖 sin 𝜑𝑖

− ̇𝜃2𝑖 sin 𝜃𝑖 cos𝜑𝑖 − �̇�2𝑖 sin 𝜃𝑖 cos𝜑𝑖− 2 ̇𝜃𝑖�̇�𝑖 cos 𝜃𝑖 sin 𝜑𝑖

1𝑙 𝑦𝑝 = ̈𝜃𝑖 cos 𝜃𝑖 sin 𝜑𝑖 + �̈�𝑖 sin 𝜃𝑖 cos𝜑𝑖− ̇𝜃2𝑖 sin 𝜃𝑖 sin 𝜑𝑖 − �̇�2𝑖 sin 𝜃𝑖 sin 𝜑𝑖+ 2 ̇𝜃𝑖�̇�𝑖 cos 𝜃𝑖 cos𝜑𝑖

�̈�𝑝 = ℎ̈ − ̈𝜃𝑖𝑙 sin 𝜃𝑖 − ̇𝜃2𝑖 𝑙 cos 𝜃𝑖𝑖 = 1, 2, 3.

(27)

Let 𝑖 = 1, (27) be written as matrix in the form of 𝐴 ̈𝑞 = 𝑏:

𝐴1 = [[[[[

1𝑙 0 0 0 − cos 𝜃1 cos 𝜑1 sin 𝜃1 sin 𝜑10 1𝑙 0 0 − cos 𝜃1 sin 𝜑1 − sin 𝜃1 cos 𝜑10 0 1 −1 𝑙 sin 𝜃1 0

]]]]],

̈𝑞1 = [�̈�𝑝, ̈𝑦𝑝, �̈�𝑝, ℎ̈1, ̈𝜃1, �̈�1]𝑇 ,𝑏1

= [[[[

− ̇𝜃21 sin 𝜃1 cos𝜑1 − �̇�21 sin 𝜃1 cos 𝜑1 − 2 ̇𝜃1�̇�1 cos 𝜃1 sin 𝜑1− ̇𝜃21 sin 𝜃1 sin 𝜑1 − �̇�21 sin 𝜃1 sin 𝜑1 + 2 ̇𝜃1�̇�1 cos 𝜃1 cos 𝜑1

− ̇𝜃21𝑙 cos 𝜃1]]]].

(28)

Assume matrix, 𝑖 = 1, 2, 3:

𝐴 𝑖 = [[[

0 − cos 𝜃𝑖 cos𝜑𝑖 sin 𝜃𝑖 sin 𝜑𝑖0 − cos 𝜃𝑖 sin 𝜑𝑖 − sin 𝜃𝑖 cos𝜑𝑖−1 𝑙 sin 𝜃𝑖 0

]]]


𝑙 = [[[[[

1𝑙 0 00 1𝑙 00 0 1

]]]]]

𝑏𝑖

= [[[[

− ̇𝜃2𝑖 sin 𝜃𝑖 cos𝜑𝑖 − �̇�2𝑖 sin 𝜃𝑖 cos𝜑𝑖 − 2 ̇𝜃𝑖�̇�𝑖 cos 𝜃𝑖 sin 𝜑𝑖− ̇𝜃2𝑖 sin 𝜃𝑖 sin 𝜑𝑖 − �̇�2𝑖 sin 𝜃𝑖 sin 𝜑𝑖 + 2 ̇𝜃𝑖�̇�𝑖 cos 𝜃𝑖 cos𝜑𝑖

− ̇𝜃2𝑖 𝑙 cos 𝜃𝑖]]]].

(29)

The total parallel manipulator system can be written in amatrix form:

𝐴𝑆 = [[[

𝑙 𝐴1𝑙 𝐴2𝑙 𝐴3

]]]9×12

̈𝑞 = [�̈�𝑝, �̈�𝑝, �̈�𝑝, ℎ̈1, ̈𝜃1, �̈�1, ℎ̈2, ̈𝜃2, �̈�2, ℎ̈3, ̈𝜃3, �̈�3]𝑇 ,

𝑏𝑆 = [[[

𝑏1𝑏2𝑏3]]]9×1

.

(30)

4. Trajectory Tracking Simulation ofthe Parallel Manipulator

In this section, we aim to use the Udwadia-Kalaba approachto obtain an explicit, closed-form expression for the controlforce required to precisely meet the trajectory requirementsof the 3-HSS parallel manipulator.

4.1. Performance Constraint of the Parallel Manipulator

Case 1. Assume that the prespecified trajectory of the mobileplatform is represented by the following equation:

𝑥𝑝 = −2√33 − 𝑅 cos (𝑤𝑡)𝑦𝑝 = 2 + 𝑅 sin (𝑤𝑡)𝑧𝑝 = 0.05𝑡.

(31)

Differentiating the constraint equation (31) with respectto 𝑡 twice, we can get

̈𝑥𝑝 = −𝑅𝑤2 cos (𝑤𝑡)̈𝑦𝑝 = −𝑅𝑤2 sin (𝑤𝑡)

�̈�𝑝 = 0.(32)

Hence the system constraints can be written in the formof

𝐴𝑝 (𝑞, 𝑡) ̈𝑞 = 𝑏𝑝 (𝑞, ̇𝑞, 𝑡) , (33)

where

𝐴𝑃 = [[[

1 0 00 1 00 0 1

]]],

𝑏𝑃 = [[[[

−𝑅𝑤2 cos (𝑤𝑡)−𝑅𝑤2 sin (𝑤𝑡)

0]]]].

(34)

Equation (31) is the performance constraints in thispaper. Thus, the three servomotors which are on the saddlemust make the platform satisfy (31) by controlling the threegroups of branches. Finally, we consider the structural andperformance constraints at the same time, which means weput 𝐴𝑆, 𝐴𝑃, and 𝑏𝑆, 𝑏𝑃 in one matrix. So, we obtain

𝐴 = 𝐴𝑆 + 𝐴𝑃 =[[[[[[

𝑙 𝐴1𝑙 𝐴2𝑙 𝐴3

𝐴𝑃

]]]]]]12×12

,

𝑏 = 𝑏𝑆 + 𝑏𝑃 =[[[[[[

𝑏1𝑏2𝑏3𝑏𝑃

]]]]]]12×1

.

(35)

According toUdwadia-Kalaba approach, the closed-formcontrol force can be expressed as

𝑄𝑐 = 𝑀1/2 (𝐴𝑀−1/2)+ (𝑏 − 𝐴𝑀−1𝑄)𝜏 = 𝑀1/2 (𝑞, 𝑡) (𝐴 (𝑞, 𝑡)𝑀−1/2 (𝑞, 𝑡))+

⋅ (𝑏 (𝑞, ̇𝑞, 𝑡) − 𝐴 (𝑞, 𝑡)𝑀−1 (𝑞, 𝑡) 𝐶 (𝑞, ̇𝑞, 𝑡) ̇𝑞) .(36)

Case 2. In Case 2, we let the movement of platform bemore complex in order to verify the control approach. Themovement in 𝑥 direction is uniform motion, in 𝑦 direction istrigonometric function, and in 𝑧 direction is parabola. Thus,the desired trajectory is given as follows.

𝑥𝑝 = −2√33 − 0.001𝑡𝑦𝑝 = 2 + 𝑅 sin (𝑤𝑡)𝑧𝑝 = − 1250000𝑡2 + 1250𝑡.

(37)

Similarly, differentiating the constraint equation (37) withrespect to 𝑡 twice, we can get

�̈�𝑝 = 0̈𝑦𝑝 = −𝑅𝑤2 sin (𝑤𝑡)

�̈�𝑝 = − 1125000 .(38)


Table 1: Parameters for simulation.

Parameter Name Value𝑚𝑒1, 𝑚𝑒2, 𝑚𝑒3 Mass of saddle 10 kg𝑚𝑐1, 𝑚𝑐2, 𝑚𝑐3 Mass of branched chain. 2 kg𝑚𝑃 Mass of moving platform 15 kg𝐼1𝑥, 𝐼1𝑦, 𝐼2𝑥, 𝐼2𝑦, 𝐼3𝑥, 𝐼3𝑦 Moment of inertia around 𝑥- and 𝑦-axis 𝑚𝑐1𝑙2/3𝑙 Length of branched chain (4√6/3)m𝑅 Radius of constrained motion (√3/5)m𝑆 Distance between two frame 4m𝜔 Angular velocity of moving platform 0.02𝜋

Table 2: Initial conditions of Case 1 for simulation.

ℎ1 4√3/3 ℎ3 4√3/3 ℎ̇1 −√3𝑅𝜔/2 ℎ̇3 √3𝑅𝜔/2𝜃1 3𝜋/4 𝜃3 3𝜋/4 ̇𝜃1 −√6𝑅𝜔/2𝑙 ̇𝜃3 √6𝑅𝜔/2𝑙𝜑1 2𝜋/3 𝜑3 4𝜋/3 �̇�1 −√2𝑅𝜔/2𝑙 �̇�3 −√2𝑅𝜔/2𝑙ℎ2 4√3/3 𝑥𝑝 −2√3/3 ℎ̇2 0 ̇𝑥𝑝 0𝜃2 3𝜋/4 𝑦𝑝 2 ̇𝜃2 0 ̇𝑦𝑝 𝑅𝜔𝜑2 2𝜋 𝑧𝑝 0 �̇�2 √2𝑅𝜔/𝑙 �̇�𝑝 0.05

Therefore, the matrices 𝐴𝑃, 𝑏𝑃 can be obtained. Thecontrol force can be obtained as well.

4.2. Simulation Results. In this section, the explicit analyticresults given in Section 4 are verified by numerical simula-tions. The numerical integration through-out this paper isdone in the MATLAB environment using a variable time stepode 15i integrator and the time of simulation is 1000 seconds.The values of the parameters are defined in Table 1.

We give the initial conditions of Case 1 including initialcoordinate and velocity of the generalized coordinates inTable 2.

Figures 3 and 4 show the simulated trajectory of theparallel manipulator in different views. From the two figures,we can see the moving platform can move on the giventrajectory by solving the Udwadia-Kalaba equation (36). InFigures 5, 6, and 7, we show the variation of the threeparameters ℎ, 𝜃, 𝜑, respectively. Figure 8 proved the distancebetween the platform and three saddles is constant; in otherwords, the length of branched chains is unchangeable whichmeans the method in this paper is right. Figures 9 and10 show the error of the platform in 𝑥 and 𝑦 directions,respectively. The errors are really small which means theplatform can move along the desired trajectory. Figure 11shows the forces generated by motors which verified the newapproach validity.

In Case 2, we give the initial conditions in Table 3.Figures 12 and 13 show the simulated trajectory in Case

2. From the Figures 13(a), 13(b), and 13(c), we can see thetrajectory in 𝑥, 𝑦, and 𝑧 directions clearly. Figures 14, 15,and 16 show the variation of the three parameters ℎ, 𝜃, 𝜑,respectively. Figures 17, 18, and 19 show the error of theplatform in 𝑥, 𝑦, and 𝑧 directions, respectively. The errors are

0

1

0

2

3

1

4

2 03 4

−1

−2−3

Figure 3: The simulated trajectory of the parallel manipulator inCase 1.

all very small which are, respectively, in the order of 10−11,10−5, and 10−10. Figure 20 shows the forces generated bymotors in Case 2.

5. Conclusion

In this paper, Udwadia-Kalaba approach is applied for tra-jectory tracking control of the 3-HSS parallel manipulatorfor the first time. First, we establish the nonlinear dynamic


Table 3: Initial conditions of Case 2 for simulation.

ℎ1 4√3/3 ℎ3 4√3/3 ℎ̇1 −√3𝑅𝜔/2 ℎ̇3 √3𝑅𝜔/2𝜃1 3𝜋/4 𝜃3 3𝜋/4 ̇𝜃1 −√6𝑅𝜔/2𝑙 ̇𝜃3 √6𝑅𝜔/2𝑙𝜑1 2𝜋/3 𝜑3 4𝜋/3 �̇�1 −√2𝑅𝜔/2𝑙 �̇�3 −√2𝑅𝜔/2𝑙ℎ2 4√3/3 𝑥𝑝 −2√3/3 ℎ̇2 0 ̇𝑥𝑝 −0.001𝜃2 3𝜋/4 𝑦𝑝 2 ̇𝜃2 0 ̇𝑦𝑝 𝑅𝜔𝜑2 2𝜋 𝑧𝑝 0 �̇�2 √2𝑅𝜔/𝑙 �̇�𝑝 1/250

0

0 4321

−1

−2.5

−3.5

−0.5

−1.5

−2

−3

Figure 4: The simulated trajectory of the parallel manipulator indifferent view in Case 1.

0 100 200 300 400 500 600 700 800 900 1000

2

3

4

The v

ertic

al d

ispla

cem

ent (

m)

Time (s)

3.5

2.5

1.5

Figure 5: The variation of parameter ℎ in simulation in Case 1.

model by dividing the system into four “unconstrained”subsystems which simplifies the modeling process. Next, thesystem structure and desired trajectory are regarded as the

0 100 200 300 400 500 600 700 800 900 1000115

120

125

130

135

140

145

150

155

The a

ngle

ch

ange

(deg

)

Time (s)

Figure 6: The variation of parameter 𝜃 in simulation in Case 1.

0 100 200 300 400 500 600 700 800 900 1000100

150

200

250

300

350

400

Time (s)

The a

ngle

ch

ange

(deg

)

Figure 7: The variation of parameter 𝜑 in simulation in Case 1.

structure constraints and performance constraints, respec-tively. Finally, this approach provides an explicit, closed-form analytical expression for the control force by solvingUdwadia-Kalaba equation. The simulation results of twocases show that the servoconstraint forces based onUdwadia-Kalaba equation make the platform’s movement meets thedesigned trajectory precisely.


0 100 200 300 400 500 600 700 800 900 10000

1

2

3

4

5

6

The l

engt

h of

bra

nche

d ch

ains

(m)

Time (s)

Figure 8: The variation of branched chains’ length in simulation inCase 1.

0 100 200 300 400 500 600 700 800 900 10000

1

2

The e

rror

ofx

×10−6

Time (s)

2.5

1.5

0.5

Figure 9: The error of platform in the 𝑥 direction in Case 1.

0 100 200 300 400 500 600 700 800 900 1000

0

1

2

3

4

The e

rror

ofy

×10−7

−0.5

Time (s)

2.5

3.5

1.5

0.5

Figure 10: The error of platform in the 𝑦 direction in Case 1.

0 100 200 300 400 500 600 700 800 900 1000120

125

130

135

140

145

150

Sadd

le fo

rces

Time (s)

Figure 11:The simulated lift forces generated bymotors as functionsof time in Case 1.

0

1

2

0

3

1

4

2

5

3 0

6

4 15

−1

−1−2

−3−4

Figure 12: The simulated trajectory of the parallel manipulator inCase 2.

In summary, there are two main advantages of thisapproach in this paper. First, this paper provides an easydynamic modeling process for 3-HSS parallel manipulator.The constraints in the system which can be holonomic aswell as nonholonomic are divided into structure constraintsand performance constraints. What we need do is to getthe second-order-form constraints. It is more convenientcompared with the common approach. Second, the controlperformance is excellent due to the exact equation of motionby this approach. However, in another approach, there areapproximations of the equation of motion or modificationof the controller such as Lagrange multiplier which makesthe equation complicated and the tracking performance notoptimistic.


0

1

0 54321−1

−1

−2

−3

−4

(a) The projection on 𝑥-𝑦 plane

0

1

2

3

4

5

6

0 54321−1(b) The projection on 𝑦-𝑧 plane

00

1

2

3

4

5

6

−1−2−3−4 1(c) The projection on 𝑥-𝑧 plane

Figure 13: The simulated trajectory of the parallel manipulator in different views in Case 2.

0 100 200 300 400 500 600 700 800 900 1000

1

2

3

4

The v

ertic

al d

ispla

cem

ent (

m)

Time (s)

3.5

2.5

1.5

0.5

Figure 14: The variation of parameter ℎ in simulation in Case 2.

0 100 200 300 400 500 600 700 800 900 1000100

110

120

130

140

150

160

Time (s)

The a

ngle

ch

ange

(deg

)

Figure 15: The variation of parameter 𝜃 in simulation in Case 2.

0 100 200 300 400 500 600 700 800 900 1000100

150

200

250

300

350

400

Time (s)

The a

ngle

ch

ange

(deg

)

Figure 16: The variation of parameter 𝜑 in simulation in Case 2.

0 100 200 300 400 500 600 700 800 900 1000

0

1

2

3

4

5

6

7

8×10−11

−1

−2

Time (s)

The e

rror

ofx

Figure 17: The error of platform in the 𝑥 direction in Case 2.


0 100 200 300 400 500 600 700 800 900 1000

0×10−5

−0.2

−0.4

−0.6

−0.8

−1.2

−1

Time (s)

The e

rror

ofy

Figure 18: The error of platform in the 𝑦 direction in Case 2.

0 100 200 300 400 500 600 700 800 900 1000

0

2

−2

−4

−6

−8

−10

−12

×10−10

Time (s)

The e

rror

ofz

Figure 19: The error of platform in the 𝑧 direction in Case 2.

0 100 200 300 400 500 600 700 800 900 1000115

120

125

130

135

140

145

150

155

160

165

Sadd

le fo

rces

Time (s)

Figure 20: The simulated lift forces generated by motors as func-tions of time in Case 2.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The theoretical foundation of this paper is consistent with (1)National Natural Science Foundation of China (51505116); (2)the Fundamental Research Funds for the Central Universities(JZ2016HGTB0716); (3) Natural and Science Foundation ofAnhui Province (1508085SME221); (4) China PostdoctoralScience Foundation (2016M590563). In addition, the projectis supported by these foundations.

References

[1] J. P. Merlet, Parallel Robots, Kluwer Academic Publishers,Dordrecht, The Netherlands, 2006.

[2] J. Wu, J. Wang, T. Li, and L. Wang, “Performance analysis andapplication of a redundantly actuated parallel manipulator formilling,” Journal of Intelligent & Robotic Systems, vol. 50, no. 2,pp. 163–180, 2007.

[3] J. Kim, F. C. Park, S. J. Ryu et al., “Design and analysis of aredundantly actuated parallel mechanism for rapidmachining,”IEEETransactions on Robotics and Automation, vol. 17, no. 4, pp.423–434, 2001.

[4] Y. Li and Q. Xu, “Design and development of a medicalparallel robot for cardiopulmonary resuscitation,” IEEE/ASMETransactions on Mechatronics, vol. 12, no. 3, pp. 265–273, 2007.

[5] Y. Su, D. Sun, L. Ren, and J. K. Mills, “Integration of saturatedPI synchronous control and PD feedback for control of parallelmanipulators,” IEEE Transactions on Robotics, vol. 22, no. 1, pp.202–207, 2006.

[6] N.-I. Kim, C.-W. Lee, and P.-H. Chang, “Sliding mode controlwith perturbation estimation: Application to motion control ofparallel manipulator,”Control Engineering Practice, vol. 6, no. 11,pp. 1321–1330, 1998.

[7] W. Shang and S. Cong, “Nonlinear adaptive task space controlfor a 2-DOF redundantly actuated parallel manipulator,” Non-linear Dynamics, vol. 59, no. 1-2, pp. 61–72, 2010.

[8] V. Vinoth, Y. Singh, andM. Santhakumar, “Indirect disturbancecompensation control of a planar parallel (2-PRP and 1-PPR)robotic manipulator,” Robotics and Computer-Integrated Manu-facturing, vol. 30, no. 5, pp. 556–564, 2014.

[9] K. Fu and J. K.Mills, “Robust control design for a planar parallelrobot,” International Journal of Robotics & Automation, vol. 22,no. 2, pp. 139–147, 2007.

[10] W. W. Shang and S. Cong, “Robust nonlinear control of aplanar 2-DOF parallel manipulator with redundant actuation,”Robotics and Computer-Integrated Manufacturing, vol. 30, no. 6,pp. 597–604, 2014.

[11] M. A. Khosravi and H. D. Taghirad, “Robust PID control offully-constrained cable driven parallel robots,” Mechatronics,vol. 24, no. 2, pp. 87–97, 2014.

[12] M. Zeinali and L. Notash, “Adaptive sliding mode control withuncertainty estimator for robot manipulators,”Mechanism andMachine Theory, vol. 45, no. 1, pp. 80–90, 2010.

[13] X. Zhu, G. Tao, B. Yao, and J. Cao, “Adaptive robust posturecontrol of parallel manipulator driven by pneumatic muscleswith redundancy,” IEEE/ASME Transactions on Mechatronics,vol. 13, no. 4, pp. 441–450, 2008.


[14] A. Noshadi, M. Mailah, and A. Zolfagharian, “Intelligent activeforce control of a 3-RRR parallel manipulator incorporatingfuzzy resolved acceleration control,” Applied MathematicalModelling, vol. 36, no. 6, pp. 2370–2383, 2012.

[15] H. Sadjadian, H. D. Taghirad, and A. Fatehi, “Neural net-works approaches for computing the forward kinematics ofa redundant parallel manipulator,” International Journal ofComputational Intelligence, vol. 2, no. 1, pp. 40–47, 2005.

[16] D. Zhu and Y. Fang, “Adaptive control of parallel manipulatorsvia fuzzy-neural network algorithm,” Control Theory and Tech-nology, vol. 5, no. 3, pp. 295–300, 2007.

[17] F. E. Udwadia and R. E. Kalaba, “A new perspective onconstrained motion,” Proceedings: Mathematical and PhysicalSciences, vol. 439, no. 1906, pp. 407–410, 1992.

[18] F. E. Udwadia and R. E. Kalaba, Analytical dynamics: A newapproach, Cambridge University Press, Cambridge, UK, 1996.

[19] F. E. Udwadia and R. E. Kalaba, “Explicit equations of motionfor mechanical systems with nonideal constraints,” Journal ofApplied Mechanics, vol. 68, no. 3, pp. 462–467, 2001.

[20] F. E. Udwadia and R. E. Kalaba, “What is the general form ofthe explicit equations of motion for constrained mechanicalsystems?” Journal of Applied Mechanics, vol. 69, no. 3, pp. 335–339, 2002.

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Trajectory Tracking Control of Parallel Manipulator Based on Udwadia-Kalaba … · 2019. 8. 28. · ResearchArticle Trajectory Tracking Control of Parallel Manipulator Based on Udwadia-Kalaba

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