Department of Aerospace Engineering,Ryerson University,350 Victoria Street,
Toronto, ON, M5B 2K3, Canada
Dynamic Modeling and Analysisof a Circular Track-Guided TripodTo enlarge the workspace and improve the motion capability of a parallel robot, the baseof the robot can be guided to move along a linear or curved track. This paper aims atanalyzing how the motion of the base affects the dynamics of a parallel robot. For thispurpose, kinematic and dynamic equations are developed for a circular track-guidedtripod parallel robot. For kinematics, the motion of the base is incorporated into theanalytical formulations of the position and velocity of the tripod. For dynamics, equa-tions of motion are derived using the Lagrangian formulation, and influence factors aredefined to provide a quantitative means to measure the effects of the velocity and accel-eration of the base on the actuator forces of the tripod. As an application of the abovemethod, a circular track-guided tripod is proposed for the automatic riveting in theassembly of an aircraft fuselage. Simulation studies are carried out to investigate thetripod dynamics. It is found that the motion of the base has a strong impact on theactuator forces. The dynamic model provides a useful tool for the design and control ofthe circular track-guided tripod. �DOI: 10.1115/1.4000313�
IntroductionIn the past two decades, researchers have become increasingly
nterested in parallel robots due to their advantages of high stiff-ess, high force transmission, and compact structure. A parallelobot is made up of a moving platform and a base, which areinked together by at least two independent kinematic chainslegs� �1�. The development of these robots has created a need forheir dynamic models, especially in the fields where high speednd large load capabilities are desired. Dynamic models play anmportant role in analyzing the robots’ dynamic performance andelecting their actuators.
Although the kinematics has been extensively studied for par-llel robots, the works on their dynamics are relatively fewer. Theomplexity of the dynamic modeling of parallel robots resultsrom the coupling between their closed-loop kinematic chains andhe existence of many passive joints. Different methods have beensed to derive their dynamic models such as the Newton–Eulerpproach �2,3�, the Lagrangian formulation �4,5�, the principle ofirtual work �6–8�, Kane’s equation �9�, and the natural orthogo-al complement method �10�. Generally speaking, researchersave focused on two types of dynamic problems: inverse dynam-cs and direct dynamics. Inverse dynamics is to determine thectuated joint forces, given the motion of the platform and thexternal forces/moments. On the other hand, direct dynamics is toetermine the motion of the platform if the actuated joint forcesnd the external forces/moments are given. A more detailed litera-ure review on the dynamic analysis of parallel robots can beound in Ref. �1�.
In the above works, the base of the parallel robot was assumedo be fixed on the ground. However, in modern industrial applica-ions, the base can be guided by a linear or curved track to enlargehe workspace and to improve the motion capability of a parallelobot. For example, tripod parallel robots were guided by a linearrack and served as a material transport system for the packagingndustry �11� and as an add-on device of a robotic polishing sys-em �12�. For more complicated tasks, the base of a parallel robotan also be guided to move along a curved track. For example, ave degree of freedom �5DOF� parallel robot was guided by a
1Corresponding author.Contributed by the Design Engineering Division of ASME for publication in the
OURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received August8, 2008; final manuscript received February 10, 2009; published online November
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curve for the welding, machining, and inspection inside a vacuumvessel �13�. In this paper, a circular track-guided tripod is pro-posed for the automatic riveting in the assembly of an aircraftfuselage. As illustrated in Fig. 1, the tripod is mounted on a cir-cular track and performs the riveting operations along the fuselagecross section by holding a rivet gun on its platform. To facilitatethe design and control of these linear or curved track-guided par-allel robots, dynamic models must be developed to investigatehow the motion of the base affects the robot dynamics.
For linear track-guided parallel robots, the base has one or moreextra translational DOFs. The motion along the track can be in-corporated into the velocities of the platform and the legs by su-perimposing the linear velocity of the base. Therefore, the corre-sponding dynamic modeling approach would be straightforwardand will not be discussed in this paper. However, for curved track-guided parallel robots, the base has both linear and angular veloci-ties. This causes much more complicated dynamic models com-pared to the linear track-guided cases. First, the velocitycomputation becomes more complicated due to the coupling be-tween the rotation of the base and the motion of the platform andthe legs. Second, nonlinear inertial force �centrifugal and Coriolis�terms related to the rotation of the base exist in the dynamicmodel. Third, if the curved track is in a vertical plane, the jointforces due to the gravity vary along the track. All these indicatethat the motion of the base can have a significant impact on therobot dynamics. Since existing dynamic models of parallel robotswere derived from the assumption that the base was fixed on theground, only the inertial terms due to the platform and the legswere incorporated into the models, while the motion of the basewas not considered. As a result, limitations exist when using thesemodels for curved track-guided parallel robots. This paper aims atovercoming these limitations and analyzing how the motion of thebase affects the robot dynamics. For this purpose, kinematic anddynamic equations are developed for a circular track-guided tri-pod parallel robot. For kinematics, the motion of the base is in-corporated into the analytical formulations of the position andvelocity of the tripod. For dynamics, equations of motion are de-rived using the Lagrangian formulation, and influence factors aredefined to provide a quantitative means to measure the effects ofthe velocity and acceleration of the base on the actuator forces ofthe tripod.
In the rest of the paper, the architecture of the circular track-guided tripod is first described in Sec. 2, followed by the position
and velocity analysis. The dynamic model is then derived in Sec.
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, based on which influence factors are defined in Sec. 5. Finally,he simulation results are presented in Sec. 6, with an applicationf the tripod for the automatic riveting of an aircraft fuselage.
Architecture Description and Reference FramesFigure 2 shows the tripod under study in this work, which isounted on a circular track with a radius Rc fixed on the ground.he tripod is composed of a base b1b2b3 and a moving platform1p2p3. The base and the platform are equilateral triangles, withenters at B and P, respectively, and with dimensions Bb1=Bb2Bb3=R and Pp1= Pp2= Pp3=r. The base and the platform areonnected by four extensible legs: three active and one passive.s plotted in Fig. 3, each active leg includes two parts: one con-ected to the platform with a spherical joint �S-joint� at pi and thether to the base with a universal joint �U-joint� at bi. These twoarts are linked together by a prismatic joint �P-joint� so that theeg length si can be changed. In particular, the rotation axes of the-joint at pi are represented by two unit vectors a1i and a2i. Vector
1i is in the plane of b1b2b3 and perpendicular to line Bbi, while2i is the normal vector of the plane containing vector a1i and lineipi. The passive leg BP is perpendicular to plane b1b2b3, and it isonnected to the base with a P-joint and to the platform with a-joint at point P. The rotation axes of the U-joint at P are par-
llel to p3p1 and Pp2, respectively. This tripod can be considereds a modified version of the Tricept robot �14�.
Fig. 1 Tripod for automatic riveting of aircraft fuselage
Fig. 2 Circular track-guided tripod
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To describe the motion ability of the robot, we first establishthree reference frames �see Fig. 2�. An inertial frame �OXYZ� islocated at the center of the circular track, with the Z axis posi-tively downward, the X axis perpendicular to the track, and the Yaxis determined from the right hand rule. A local reference frame�Bxyz� is attached on the base at point B, with the x axis parallelto b3b1, the y axis pointing toward b2, and the z axis perpendicularto the plane of b1b2b3. Another local frame �Px�y�z�� is fixed onthe platform at point P, with the x� axis parallel to p3p1, the y�axis pointing toward p2, and the z� axis perpendicular to the planeof p1p2p3. The tripod has four DOFs: three generated by the legsand one by the circular track. With the legs, the platform has onetranslational and two rational DOFs, i.e., the translation with re-spect to the base along BP, represented by the z coordinate ofpoint P, and the rotations about the x� and y� axes. With thecircular track, the robot has an extra DOF, i.e., point B on the basecan rotate about the X axis in plane OYZ, represented by an angle�. Thus, the following generalized coordinates can be chosen todescribe the configuration of the tripod
q = �z,�1,�2,��T �1�
where �1 and �2 are Euler angles, and they are defined by thefollowing sequence of rotations to obtain axes �x�y�z�� from axes�xyz�: first rotate �xyz� an angle �1 about the x axis to axes�x�y�z�� and then rotate �x�y�z�� an angle �2 about the y� axis to�x�y�z��.
The 3�3 rotation matrices from the base-fixed frame �Bxyz� tothe inertial frame �OXYZ� and from the platform-fixed frame�Px�y�z�� to the base-fixed frame �Bxyz� are respectively writtenas follows:
TOB = �1 0 0
0 − sin � − cos �
0 cos � − sin �� ,
TBP = � cos �2 0 sin �2
sin �2 sin �1 cos �1 − cos �2 sin �1
− sin �2 cos �1 sin �1 cos �2 cos �1� �2�
Then the rotation matrix from �Px�y�z�� to �OXYZ� is
TOP = TOBTBP �3�For the calculation of the legs’ kinetic energies in Sec. 4, local
reference frames fixed on the legs are defined �see Fig. 3�. Thelocal frame fixed on the passive leg �Px4y4z4� is defined at pointP, with the x4 axis parallel to the z axis �along the leg directionBP�, the y4 axis parallel to the y axis, and the z4 axis determinedfrom the right hand rule. Then the rotation matrix from �Px4y4z4�
Fig. 3 Legs of the tripod
to �OXYZ� is
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T4 = TOB�0 0 − 1
0 1 0
1 0 0� �4�
he local frames �bixiyizi� and �pixiyizi� are fixed on the base-onnected part at bi and on the platform-connected part at pi of theth active leg, respectively. They are parallel to each other and thexes are defined as follows: the xi axis is along the leg directionipi and represented by a unit vector ui, the yi axis is defined asi�a1i, and the zi axis is determined from the right hand rule.hen, the rotation matrices from �bixiyizi� and �pixiyizi� to the
he calculation of the unit vector ui will be discussed in Sec. 3.1.
Kinematic Analysis
3.1 Position Analysis. Given the configuration of the tripod,he leg length si and the leg length direction vector ui can bebtained from the following equation:
siui = h + pi − bi �6�
here h is the vector representing BP and it can be computed as
h = TOB�0,0,z�T �7�
he vectors bi and pi denote Bbi and Ppi, respectively, and theyre obtained as
bi = TOBbbi, pi = TOP
bpi �8�
here bbi and bpi are the vectors written in the correspondingody-fixed frames �Bxyz� and �Px�y�z��, respectively. Then weave
si = h + pi − bi, ui =h + pi − bi
si�9�
quation �9� represents the analytical solution of the inverse ki-ematics.
3.2 Velocity Analysis. We now analyze the linear and angularelocities of each link. Since point B on the base rotates about theaxis in plane OYZ, the linear and angular velocities of the base
an be easily written as
vB = �0,− Rc� sin �,Rc� cos ��T, �B = ��,0,0�T �10�
From Fig. 3, the position vector of P can be obtained as PB+h. Taking the time derivative of P, we have the linear veloc-
ty of the platform as
vP = vB + �B � h + TOB�0,0, z�T �11�
n particular, the first term in vP is due to the linear velocity of thease, while the second one is a coupling term between the angularelocity of the base and the translational displacement of the plat-orm along the z axis. The angular velocity of the platform isbtained from the aforementioned rotations discussed in Sec. 2,.e.,
�P = ��
0
0� + TOB��1
0
0� + TOB�1 0 0
0 cos �1 − sin �1
0 sin �1 cos �1�� 0
�2
0�
�12�The linear velocities of the base- and platform-connected parts
f the ith active leg can be represented by the velocities of bi and
i respectively, i.e.,
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where the third and fourth terms are related to the linear andangular velocities of the base respectively. Note that both thebase- and platform-connected parts of the ith active leg have iden-tical angular velocities, i.e., �bi=�pi=�i. Furthermore, taking thedot product of both sides of Eq. �14� with ui gives
si = uiTvP + �pi � ui�T�P − ui
TvB − �bi � ui�T�B �16�
Equation �16� will be used in the computation of the generalizedforces due to actuators in Sec. 4.
Lastly, the velocity of the passive leg is investigated. This legmoves along BP but does not rotate about the x� or y� axes. Thus,its linear and angular velocities can be obtained as
v4 = vP, �4 = �B �17�Equations �10�–�13�, �15�, and �17� represent the analytical for-
mulations of the linear and angular velocities of each link writtenin the inertial frame. For the convenience in calculating the kineticenergies, these velocity vectors are written in the correspondinglocal body-fixed frames, i.e.,
bvB = TOBT vB, bvP = TOP
T vP, bv4 = T4Tv4, bvbi = Tbi
T vbi,
bvpi = TpiT vpi
b�B = TOBT �B, b�P = TOP
T �P, b�4 = T4T�4, b�bi = Tbi
T �bi,
b�pi = TpiT �pi
where the superscripts b denote that the vectors are written inlocal body-fixed frames. Furthermore, we define the following 6�1 vectors:
bVB = bvBb�B
�, bVP = bvPb�P
�, bV4 = bv4b�4
� ,
bVbi = bvbib�bi
�, bVpi = bvpib�pi
�Then these vectors can be expressed as linear combinations of thegeneralized velocities, i.e.,
bVB = HBq, bVP = HPq, bV4 = H4q, bVbi = Hbiq ,
bVpi = Hpiq �18�
where HB, HP, H4, Hbi, and Hpi are 6�4 mapping matrices fromthe generalized velocities to the velocities of the links, and theyare obtained as
HB =��bVB�
˙, HP =
��bVP�˙
, H4 =��bV4�
˙,
�q �q �q
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Hbi =��bVbi�
�q, Hpi =
��bVpi��q
�19�
hese velocity mapping matrices depend on the configuration ofhe tripod, and they can be derived from the above analyticalelocity formulations of the links.
Dynamic Modeling
4.1 Kinetic Energy. The kinetic energies of the base, thelatform, and the legs are obtained as follows:
TB = 12
bVBT�bMB�bVB, TP = 1
2bVP
T�bMP�bVP,
T4 = 12
bV4T�bM4�bV4,
Tbi = 12
bVbiT �bMbi�
bVbi, Tpi = 12
bVpiT �bMpi�
bVpi �20�
here bMB, bMP, bM4, bMbi, and bMpi are the 6�6 symmetricass matrices of the links written in the corresponding local
ody-fixed frames, and they can be written in the following form:
bM = mI − m · brG
m · brGbJ
� �21�
here I is a 3�3 identity matrix, m is the mass of the link, the�3 matrix bJ denotes the second moment of inertia, brG repre-ents the 3�3 skew-symmetric matrix form of vector brG �corre-ponding to a cross-product operation�, and brG is the 3�1 posi-ion vector of the center of gravity �CG� of the link from therigin of the corresponding body-fixed frame. Note that both bJnd brG are expressed in the body-fixed frames.
Substituting the velocities in Eq. �18� into Eq. �20�, we canrite the total kinetic energy of the tripod as
T = 12 qTMq �22�
here
M = HBT�bMB�HB + HP
T�bMP�HP + H4T�bM4�H4
+ �i=1
3
HbiT �bMbi�Hbi + �
i=1
3
HpiT �bMpi�Hpi
s a 4�4 symmetric generalized inertial matrix referred to thelatform and is dependent on the generalized coordinates.
4.2 Potential Energy. The potential energies of the base, thelatform, and the legs are due to the gravity and they can beritten as
VB = − mBrGBT g, VP = − mPrGP
T g, V4 = − m4rG4T g
Vbi = − mbirGbiT g, Vpi = − mpirGpi
T g �23�
here g denotes the acceleration of gravity, mB, mP, m4, mbi, andpi are the masses of the base, the platform, the passive leg, the
ase-connected, and the platform-connected parts of the ith activeeg, respectively, and rGB, rGP, rG4, rGbi, and rGpi are the corre-ponding position vectors of their CG written in the inertial frameOXYZ�. Then the total potential energy is
V = VB + VP + V4 + �i=1
3
Vbi + �i=1
3
Vpi �24�
gain, the potential energy is dependent on the robot configura-ion.
4.3 Generalized Forces. The generalized forces include twoarts: the external forces and moments acting on the platform andhe joint forces input by the actuators. All the external loads ap-lied on the platform can be equivalent to a resultant force acting
b
t point P �represented by a 3�1 vector Fe� and a resultant
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moment about P �represented by a 3�1 vector bMe�. These ex-ternal force and moment vectors are written in the platform-fixedframe and they satisfy the following relationship Qe
Tq= b�eT�bVP�,
where Qe is the generalized force due to the external forces andmoments and b�e= �bFe
T , bMeT�T. Substituting bVP=HPq into
QeTq= b�e
T�bVP� leads to
Qe = HPT�b�e� �25�
The actuated joint forces on the active legs and the actuatedmoment on the circular track can be written in a vector �a= �F1 ,F2 ,F3 ,Mc�T, in which F1, F2, and F3 are the actuator forceson the P-joints of the legs, and Mc is the actuator moment drivingthe base along the track. The corresponding generalized force Qa
satisfies QaTq=�a
Tsa, in which sa= �s1 , s2 , s3 , ��T. The velocities s1,s2, and s3 can be expressed as linear combinations of the general-ized velocities from Eq. �16�, and sa is then written as
sa = Jq �26�
where J=�sa /�q is a 4�4 mapping matrix from the generalizedvelocities to the actuated joint velocities. Substituting Eq. �26�into Qa
Tq=�aTsa gives
Qa = JT�a �27�
Equations �25� and �27� complete the formulations of the gener-alized forces.
4.4 Equations of Motion. Once the kinetic and potential en-ergies and the generalized forces are obtained, we proceed to de-rive the dynamic model via the Lagrangian formulation, i.e.,
d
dt �L
�q� −
�L
�q= Qe + Qa �28�
where L=T−V. Substituting T, V, Qe, and Qa leads to the follow-ing equations of motion:
Mq + C�q,q� + G�q� = HPT�b�e� + J T�a �29�
where G�q�=�V /�q is the gravitational force term and
C�q,q� = �i=1
4
qi
�M
�qi�q
−1
2qT�M
�q1q,qT�M
�q2q,qT�M
�q3q,qT�M
�q4q�T
�30�
includes all the nonlinear force terms and depends on q and q. Inparticular, the kth component of C can be written in quadraticform as
Ck = 12 qTckq, k = 1,2,3,4 �31�
where the elements of the 4�4 symmetric matrix ck are computedas
ck,ij =�Mkj
�qi+
�Mki
�qj−
�Mij
�qk, i, j,k = 1,2,3,4 �32�
where Mkj, Mki, and Mij are elements of the matrix M.Now, the inverse dynamic problem can be solved by rearrang-
ing Eq. �29� as
�a = J−T�Mq + C�q,q� + G�q� − HPT�b�e�� �33�
Once the robot configuration q, the velocity q, and the externalloads b�e are given, the actuator forces and moment can be com-puted from Eq. �33�. In addition, the direct dynamic problems canbe solved from the second-order ordinary differential Eq. �29�.Once given b�e and �a, the configuration q can be obtained fromEq. �29� using a numerical integration approach.
It is noted that the actuator forces are nonlinear functions of
velocities �through vector C� and configuration �through mass ma-
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rix M and gravitational force G�, while they are linear with re-pect to the accelerations q. Therefore, increasing the velocities,or example, may lead to a nonlinear effect on the actuator forces,ut this is not true for changing the accelerations.
Influence Factors of the Base’s MotionIn this section, influence factors are proposed to quantitativelyeasure the effects of the base’s motion on the inverse dynamics
f the tripod. The main idea is to separate the terms related to �
nd � in the formulations of F1, F2, and F3. For this purpose, thexternal forces are neglected and Eq. �33� is rewritten as
�a = M�q�q + C�q,q� + G�q� �34�
here M=J−TM, C=J−TC, and G=J−TG. The three terms on theight hand side of Eq. �34� relate to the acceleration, the velocitynd the gravity, respectively. It is noted that the rotation angle �
trongly affects the gravitational force term G, since the orienta-ion of the tripod changes with �. The variation in the gravita-ional forces at different � will be further discussed in Sec. 6.3. Inhis section, we investigate the effects of � and � on the actuatororces on the legs.
First, we study the influence of the angular acceleration of thease �. To this end, the actuated joint torque is written as �a�F ,Mc�T, in which F= �F1 ,F2 ,F3�T and the generalized coordi-ate vector is written as q= �x ,��T, in which x= �z ,�1 ,�2�T rep-
esents the motion due to the legs. Correspondingly, the matrix Ms written as
M = M11 M12
M21 M22
� �35�
here M11 is a 3�3 matrix, M12 is a 3�1 column vector, M21 is
1�3 row vector, and M22 is a scalar. The actuator forces on the
egs related to the acceleration is M11x+M12�, in which the sec-nd term is due to the base’s motion. An influence factor for thecceleration of the base is defined as one-third of the one-norm of¯
12, i.e.,
Ka = 13�
k=1
3
M12,k �36�
here the subscript a denotes acceleration, and M12,k is the kth
omponent of vector M12. The physical meaning of Ka can benterpreted as the mean value of the magnitudes of the actuatororces on the legs due to a unit acceleration �.
Then, we turn our attention to the effects of the angular velocity˙ . If we let A=J−T, the kth component of vector C can be written
s Ck=� j=14 AkjCj, where Akj denotes the element of the matrix A
n its kth row and jth column. Substituting Cj =1 /2qTc jq giveshe following quadratic form:
Ck = 12 qTckq �37�
here ck=� j=14 Akjc j is a symmetric matrix and it can be written as
ck = ck,11 ck,12
ck,12T ck,22
� �38�
here ck,11 is a 3�3 matrix, ck,12 is a 3�1 column vector, and
k,22 is a scalar. Then, we have
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Ck =1
2�xT,��ck,11 ck,12
ck,12T ck,22
� x
�� �39�
Thus the actuator force on the kth leg due to � is 1 /2ck,22�2.Then, an influence factor for the angular velocity of the base isdefined as
Kv = 16�
k=1
3
ck,22 �40�
where the subscript v denotes velocity. The physical meaning ofKv can be interpreted as the mean value of the magnitudes of theactuator forces on the legs due to a unit angular velocity �.
The factors Ka and Kv are used to measure the influence of theangular acceleration and velocity of the base on the inverse dy-namics of the tripod. These factors are independent of the rotationangle � due to the symmetry of the circular track, but they dependon the pose of the tripod �translational displacement along the zaxis and the Euler angles �1 and �2�. In addition, the influencefactors can be normalized with the total mass, i.e.,
ka =Ka
mA, kv =
Kv
mA�41�
where mA is the total mass of the robot. The physical meaning ofka can be interpreted as the mean value of the magnitudes of legactuator forces due to a unit angular acceleration of the base forper unit mass of the robot, and kv is the mean value of the mag-nitudes of actuator forces due to a unit angular velocity of the basefor per unit mass. The normalized factors could be applied tocompare the dynamic characteristics of two different circulartrack-guided robots.
The influence factors presented in this section are nonlinearfunctions of the tripod’s pose �z, �1, and �2�, and they representthe effects of the base’s motion on the inverse dynamics. Whenthe robot is at a configuration with higher Ka or Kv, it requireslarger leg actuator forces to maintain its pose �z, �1, and �2� whilethe base moves along the track with a unit angular acceleration or
velocity. It should be noted that other velocities �z, �1, and �2�cannot be linearly superimposed on the actuator forces due to thenonlinearity discussed in Sec. 4. The coupling between � and z,
�1 and �2 can be investigated by solving the inverse dynamicequation �33�.
6 SimulationThe dynamic model is implemented and used to solve the in-
verse dynamic problem for a circular track-guided tripod. One ofthe advantages of using the Lagrangian formulation to derive thedynamic model is that no acceleration analysis is required, but weneed to calculate the partial derivatives about the generalized co-ordinates q or generalized velocities q in the velocity mappingmatrices HB, HP, Hbi, Hpi, H4, and J and in the nonlinear forcevector C. Since the analytical formulations of the links’ velocitieshave been obtained, these partial derivatives can be derived usingthe symbolic computation package MAPLE. The resulting formula-tions can then be converted into MATLAB codes with the CodeGen-eration package in MAPLE, and these MATLAB codes can be used toimplement the inverse dynamic model.
6.1 Geometric and Inertial Parameters. The geometric pa-rameters of the circular track-guided tripod in the simulation ex-amples are given as follows �unit: m�:
Rc = 1.75, R = 0.2, r = 0.11
The inertial properties �the mass, the position of CG relative to thebody-fixed frame origin, and the second moment of inertia� of
b b
the links are given in Table 1, with rG and J expressed in the
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orresponding body-fixed frames. Finally, the acceleration ofravity acts along the Z direction, i.e., g= �0,0 ,9.81�T �unit:/s2�.
6.2 Influence Factors. Before we analyze the inverse dynam-cs of the tripod, we investigate the distribution of the influenceactors Ka and Kv, which are nonlinear functions of z, �1, and �2.ow Ka and Kv affect the actuator forces will be studied in Sec..4 for an automatic riveting application of the tripod.
Since the factors are independent of the rotation angle � due tohe symmetry of the circular track, the results of Ka and Kv arelotted as functions of �1 and �2 at different values of z in Fig. 4.t can be observed that the translational displacement z stronglynfluences Ka and Kv. As the platform moves closer to the base �zecomes smaller�, the distance from the center of the track to thelatform and the legs becomes larger and, thus, the moments ofnertia of the platform and the legs about the X axis increase. As aesult, the robot requires larger actuator forces to maintain theose �z, �1, and �2�, while the base undergoes a unit angular
−10−5
05
10
−10−5
05
100.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
θ1
(deg)θ2
(deg)
Ka
(N−
s2 )
z=0.25 m
z=0.35 m
z=0.45 m
(a)
Fig. 4 Infl
Table 1 Inertial p
Linkm
�kg�
Base 3.450Platform 0.4891Base-connected part of active legs 0.8778Platform-connected part of active legs 0.2885Passive leg 0.3032
−3 −2 −1 0 1 2 3
−6
−4
−2
0
2
4
6
φ (rad)
Mea
nva
lue
ofgr
avita
tiona
lfor
ces
(N)
z = 0.25m, θ1=θ
2=0
z = 0.35m, θ1=θ
2=0
z = 0.45m, θ1=θ
2=0
(a)
Fig. 5 Gravitational force
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acceleration � or a unit velocity � at z=0.25 m than at z=0.45 m. From the plots of Ka and Kv, as z decreases from 0.45m to 0.25 m, the mean value of the actuator forces due to a unit
acceleration � increases approximately from 0.04 N to 0.11 N,
and that due to a unit velocity � increases from 0.78 N to 0.98 N.For a given displacement along the z axis, the variations in Eulerangles have small effects on the factors Ka and Kv, since �1 and �2have slight influence on the moments of inertia of the platformand the legs about the X axis.
6.3 Gravitational Forces. The actuator forces due to thegravity depend on the rotation angle �, the displacement along thez axis and the Euler angles �1 and �2. Figure 5�a� shows the
variation of the mean value of the gravitational forces, �G1+ G2
+ G3� /3, at different � and z, given �1=�2=0. It is found that therotation angle � has a significant impact on the gravitationalforceterm, since the orientation of tripod varies along the track. If
is from 0 to �, the base is on the lower half circle, and the totalctuator force due to the gravity is positive �along the bipi direc-ion�, representing that the gravity tends to prevent the motion ofhe platform. If � is from −� to 0, the base is on the upper halfircle, and the total actuator force due to the gravity is negativealong the pibi direction�, representing that the gravity tends torive the motion of the platform. The gravitational force termeaches its maximum value at the bottom of the track ��=� /2�nd its minimum value at the top ��=−� /2�. As the translationalisplacement z decreases, the actuator forces due to the gravityncreases. However, compared to the rotation angle �, the dis-lacement z has relatively minor influence on the gravitationalorce term. Figure 5�b� shows the variation in the mean value ofhe gravitational forces, at different �1, �2, and z, given �=� /2. Itan be observed that the Euler angles have small influence on theravitational force term.
6.4 Application: Automatic Riveting of an Aircraftuselage. The circular track-guided tripod is proposed for auto-atic riveting in the assembly of an aircraft fuselage. In general,
he operation of fuselage assembly is divided into two stages: therst step is to put together the fuselage skin panels to form severalections along the fuselage, and the second step is to assemble ahole fuselage by connecting the sections together. Riveting is theajor joining method used in these two steps. In this work, we
ocus our attention on the second step. Due to the high complex-ty, the riveting in this step is mainly done manually. However,
anual riveting operations are tedious, repetitious, prone to error,nd costly, also causing health and ergonomic problems �15�. Tovercome these problems, the circular track-guided tripod is pro-osed to finish the riveting operations along the fuselage crossection, by attaching a rivet gun at the center of the platform �seeig. 1�. Compared to manual riveting, robotic riveting can provideigher efficiency and lower healthy risks �16�.
In this example, riveting tasks are required at ten spots
−101
−1
0
1
Y (m)
Z(m
)A
4
Fuselage
A9
A6
A5
A2
Moving platform
O2
Y O
Z
P
Rivet gun
A1
A10
O1
A8
A7
A’
A3
n
Fig. 6 Fuselage cross section
Table 2 Configura
Pointz
�m� �1 �2 �
A1 0.2515 0 0 − �2
A3 0.2515 0 0 − 5�18
A5 0.2515 0 0 − �18
A7 0.2515 0 0 �6
A9 0.3575 0.0679 0 7�18
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A1 ,A2 , . . . ,A10 along the fuselage cross section from the top to thebottom, as plotted in Fig. 6. At each riveting spot, the rivet gun�line PAi� must stop for a few seconds to complete the rivetingprocess and must be parallel to the local normal vector n of thecross section. The profile of the fuselage cross section is com-posed of three segments of arcs: A1A7 �� from −� /2 to � /6� is anarc with a center at point O; A7A� �� from � /6 to 2� /9� is an arcwith a center at O1; and A�A10 �� from 2� /9 to � /2� is an arcwith a center at O2. Since the cross section is not a single circularcurve and the gun must be parallel to its local normal vector, therobot must provide motions not only to drive the gun along thecircular track, but also to adjust the distance between the gun andthe cross section and to change the gun’s orientation.
The distance between the platform and the fuselage cross sec-tion is PAi=0.02 m, representing the length of the rivet gun. Therequired configurations of the platform at spots A1 ,A2 , . . . ,A10 aregiven in Table 2. From A1 to A7, the profile of the cross section isan arc with the same center of the circular track, so the transla-tional displacement along the z direction and the Euler angles �1and �2 do not change and the leg lengths maintain constant. How-ever, from A7 to A10, the leg lengths must vary in order to obtainthe required position and orientation of the platform for rivetingoperations. The tripod is planned to perform a point-to-point mo-tion from A1 to A10. The motion trajectories of the platform areplanned by the interpolation between each two spots as follows:
q = q�Ai� + �q�Ai+1� − q�Ai���6tn5 − 15tn
4 + 10tn3�, i = 1,2, . . . ,9
�42�
where q�Ai� and q�Ai+1� represent the robot configurations at Ai
and Ai+1 respectively, tn is a normalized time factor and it can beobtained from the time t as
tn =t − �i − 1��T
�T, i = 1,2, . . . ,9 �43�
where �T is the time interval between each two spots. If t= �i−1��T, then tn=0 and q=q�Ai�, representing that the tripod is atpoint Ai; if t= i�T, then tn=1 and q=q�Ai+1�, representing that thetripod moves to point Ai+1. The time histories of �, z, �1, and �2obtained from Eq. �42� are drawn in Fig. 7. In addition, it can bechecked from Eq. �42� that if the initial velocity and accelerationat the first spot A1 are zero, then we have q= q=0 at tn=1. In otherwords, with the path planning in Eq. �42�, the velocity and accel-eration of platform are zero at A1 ,A2 , . . . ,A10, so that the tripodcan stop and finish the riveting at each spot.
To facilitate the design of the robotic riveting system, the dy-namic model developed in Sec. 4 is applied to investigate theinverse dynamics of the tripod for its actuator selection. Once thetime histories of q, q, and q are obtained from Eqs. �42� and �43�,the actuator forces on the legs can be computed from the inversedynamic model in Eq. �34�. Due to the symmetry of the tripod andthe fact that the platform does not rotate about the y� axis ��2=0�, the actuator forces on the first and third legs are identical,i.e., F1=F3. The force results are displayed as functions of � inFigs. 8–10 for three cases: �T=1.2 s, �T=0.6 s, and �T=0.3 s, respectively. A smaller time interval �T represents a
s at riveting spots
Pointz
�m� �1 �2 �
A2 0.2515 0 0 − 7�18
A4 0.2515 0 0 − �6
A6 0.2515 0 0 �18
A8 0.3104 0.1266 0 5�18
A10 0.3740 0 0 �2
tion
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ht
a
gascbsrti�t
s o
0
Downloaded Fr
igher speed of the base. It has been demonstrated in Eq. �34� thathe total actuator force includes three parts: a term related to the
cceleration Mq, a nonlinear term related to the velocity C, and a
ravitational force term G. The results of all these force terms arelso shown in Figs. 8–10. We can observe the following conclu-ions. �1� The gravitational force term on the first and third legs islose to that on the second leg. �2� The gravity has a large contri-ution to the total actuator forces on the legs, particularly at lowpeeds, and as a result, the actuator forces strongly depend on theotation angle �. �3� As the time interval �T decreases, the forceerms related to the acceleration and the velocity become moremportant and can have a significant impact on the actuator forces.4� The force related to the acceleration on the second leg is largerhan those on the first and third legs. �5� The nonlinear force term
(b)(a)0 2 4 6 8
−2
−1
0
1
2
t / ∆Tφ
0 20.2
0.25
0.3
0.35
0.4
z(m
)
Fig. 7 Time histories o
−1.5 −1 −0.5 0 0.5 1 1.5−10
−8
−6
−4
−2
0
2
4
6
8
10
φ (rad)
F1
=F
3(N
)
Total forceAcceleration termVelocity termGravity term
(a)
Fig. 8 Actuator force
−1.5 −1 −0.5 0 0.5 1 1.5−10
−8
−6
−4
−2
0
2
4
6
8
10
φ (rad)
F1
=F
3(N
)
Total forceAcceleration termVelocity termGravity term
(a)
Fig. 9 Actuator forces o
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related to the velocity is positive �along BiPi�, representing thecentripetal effects due to the base’s motion along the circulartrack. 6� The profile of the cross section from A1 to A7 is a singlearc with a center at point O, while the one from A7 to A10 consistsof two arcs with two different centers and radii, thus, the leglengths change from A7 to A10. As such, the base’s motion � iscoupled with z and �1, which strongly affects the actuator forces.
For example, due to the influence of z and �1, the variations in theacceleration terms through A7–A10 are very different from thosethrough A1–A7 �as indicated in Fig. 10�.
Finally, we analyze the effects of the base’s motion on the in-verse dynamics using the influence factors and demonstrate thatthe tripod with higher Ka and Kv requires larger actuator forces tomaintain its pose �z, �1, and �2� while the base moves along the
6 8/ ∆T
0 2 4 6 8−0.05
0
0.05
0.1
0.15
t / ∆T
Eul
eran
gles
θ1
θ2
(c)
eneralized coordinates
−1.5 −1 −0.5 0 0.5 1 1.5−10
−8
−6
−4
−2
0
2
4
6
8
10
φ (rad)
F2
( N)
Total forceAcceleration termVelocity termGravity term
b)
n the legs „�T=1.2 s…
−1.5 −1 −0.5 0 0.5 1 1.5−10
−8
−6
−4
−2
0
2
4
6
8
10
φ (rad)
F2
(N)
Total forceAcceleration termVelocity termGravity term
b)
4t
f g
(
(
n the legs „�T=0.6 s…
Transactions of the ASME
04/2013 Terms of Use: http://asme.org/terms
tba2stwthvcactt
J
Downloaded Fr
rack. To this end, we investigate the actuator forces from A1 to A2ut with two values of z. The generalized coordinates at A1 and A2re given in Table 3 for the two cases. The larger z value in Caserepresents that the radius of the fuselage cross section is 0.1 m
maller than that in Case 1 �see Fig. 6�. The motion trajectories ofhe platform between A1 and A2 are also planned using Eq. �42�ith �T=0.3 s. Using this motion planning, z, �1, and �2 main-
ain constant from A1 to A2. The two cases have the same timeistories of � and identical values of �1 and �2 but have differentalues of z. The influence factors Ka and Kv for both cases areompared in Table 3, and the resulting actuator forces related tocceleration � and velocity � are compared in Figs. 11 and 12. Itan be observed that the magnitudes of the actuator forces relatedo � and � in Case 1 are approximately 74% and 13% higher thanhose in Case 2, respectively. This indicates that when the tripod is
−1.5 −1 −0.5 0 0.5 1 1.5−10
−8
−6
−4
−2
0
2
4
6
8
10
φ (rad)
F1
=F
3(N
)
Total forceAcceleration termVelocity termGravity term
(a)
Fig. 10 Actuator force
Table 3 Configurations a
Case Pointz
�m� �1 �2 � Poi
1 A1 0.2515 0 0 − �2 A2
2 A1 0.3515 0 0 − �2 A2
−1.55 −1.5 −1.45 −1.4 −1.35 −1.3 −1.25−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
φ (rad)
Forc
eson
legs
1an
d3
(N)
Case 1: Ka= 0.0945 N−s2
Case 2: Ka= 0.0542 N−s2
0.908 N
1.584 N
(a)
Fig. 11 Actuator forces related
ournal of Computational and Nonlinear Dynamics
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used for the riveting of a smaller fuselage cross section, the base’sacceleration and velocity have less influence on the inverse dy-namics.
7 ConclusionsIn this paper, a dynamic model is developed for a circular track-
guided tripod robot by the Lagrangian formulation, with a particu-lar focus on the effects of the base’s motion on the tripod dynam-ics. The motion of the base is incorporated into the analyticalformulations of the position and velocity, and the resulting dy-namic equations are separated into three terms related to: the ac-celeration, the velocity, and the gravity, respectively. Influencefactors are proposed to provide a quantitative means to measurehow the base’s velocity and acceleration affect the actuator forces.Simulation studies are carried out to examine the inverse dynam-
−1.5 −1 −0.5 0 0.5 1 1.5−10
−8
−6
−4
−2
0
2
4
6
8
10
φ (rad)
F2
(N)
Total forceAcceleration termVelocity termGravity term
b)
n the legs „�T=0.3 s…
1 and A2 with different z
z�m� �1 �2 �
Ka�N s2�
Kv�N s2�
0.2515 0 0 − 7�18 0.0945 0.9712
0.3515 0 0 − 7�18 0.0542 0.8627
−1.55 −1.5 −1.45 −1.4 −1.35 −1.3 −1.25−4
−3
−2
−1
0
1
2
3
4
φ (rad)
Forc
eson
leg
2(N
)
Case 1: Ka= 0.0945 N−s2
Case 2: Ka= 0.0542 N−s2
3.168 N
1.815 N
b)
(
s o
t A
nt
(
to acceleration „�T=0.3 s…
JANUARY 2010, Vol. 5 / 011005-9
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istaosittitdt
R
0
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cs of a tripod proposed for the automatic riveting along the crossection of an aircraft fuselage. It is found that the rotation angle ofhe base strongly influences the gravity term and the resultingctuator forces on the legs, and that the acceleration and velocityf the base must be considered in the dynamic analysis at highpeeds. The actuator forces at two configurations with differentnfluence factors are compared. It is demonstrated that when theripod is used for the riveting of a smaller fuselage cross section,he base’s acceleration and velocity have less influence on thenverse dynamics. The dynamic model provides a useful tool forhe design and control of the circular track-guided tripod. Theynamic analysis approach presented in this work can be extendedo other curved track-guided parallel robots.
erlands, pp. 1–18 and 277–288.�2� Dasgupta, B., and Choudhury, P., 1999, “A General Strategy Based on the
Newton–Euler Approach for the Dynamic Formulation of Parallel Manipula-tors,” Mech. Mach. Theory, 34�6�, pp. 801–824.
�3� Wang, L.-P., Wang, J.-S., and Li, Y.-W., 2003, “Kinematic and Dynamic Equa-tions of a Planar Parallel Manipulator,” Proc. Inst. Mech. Eng., Part C: J.Mech. Eng. Sci., 217�5�, pp. 525–531.
�4� Beji, L., and Pascal, M., 1999, “The Kinematics and the Full Minimal Dy-namic Model of a 6-DOF Parallel Robot Manipulator,” Nonlinear Dyn., 18�4�,pp. 339–356.
−1.55 −1.5 −1.45 −1.4 −1.35 −1.3 −1.25−2
−1
0
1
2
3
4
5
φ (rad)
Forc
eson
legs
1an
d3
(N)
Case 1: Kv= 0.9712 N−s2
Case 2: Kv= 0.8627 N−s2
4.115 N 4.636 N
(a)
Fig. 12 Actuator forces re
�5� Di Gregorio, R., and Parenti-Castelli, V., 2004, “Dynamics of a Class of Par-
11005-10 / Vol. 5, JANUARY 2010
om: http://computationalnonlinear.asmedigitalcollection.asme.org/ on 05/
allel Wrists,” ASME J. Mech. Des., 126�3�, pp. 436–441.�6� Wang, J., and Gosselin, C., 1998, “A New Approach for the Dynamic Analysis
of Parallel Manipulators,” Multibody Syst. Dyn., 2�3�, pp. 317–334.�7� Geike, T., and McPhee, J., 2003, “Inverse Dynamic Analysis of Parallel Ma-
nipulators With Full Mobility,” Mech. Mach. Theory, 38�6�, pp. 549–562.�8� Li, M., Huang, T., Mei, J., Zhao, X., Chetwynd, D. G., and Hu, S. J., 2005,
“Dynamic Formulation and Performance Comparison of the 3-DOF Modulesof Two Reconfigurable PKM—The Tricept and the TriVariant,” ASME J.Mech. Des., 127�6�, pp. 1129–1136.
�9� Liu, M.-J., Li, C.-X., and Li, C.-N., 2000, “Dynamics Analysis of the Gough-Stewart Platform Manipulator,” IEEE Trans. Rob. Autom., 16�1�, pp. 94–98.
�10� Xi, F., Angelico, O., and Sinatra, R., 2005, “Tripod Dynamics and Its InertiaEffect,” ASME J. Mech. Des., 127�1�, pp. 144–149.
�11� Tönshoff, H. K., Soehner, C., and Isensee, G., 1997, “Vision-Guided TripodMaterial Transport System for the Packaging Industry,” Rob. Comput.-Integr.Manufact., 13�1�, pp. 1–7.
�12� Xi, F., Han, W., Verner, M., and Ross, A., 2001, “Development of a Sliding-Leg Tripod as an Add-On Device for Manufacturing,” Robotica, 19�3�, pp.285–294.
�13� Wu, H., Handroos, H., Kovanen, J., Rouvinen, A., Hannukainen, P., Saira, T.,and Jones, L., 2003, “Design of Parallel Intersector Weld/Cut Robot for Ma-chining Processes in ITER Vacuum Vessel,” Fusion Eng. Des., 69�1–4�, pp.327–331.
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