On the computer solutions of kinematics analysis of linkages · the contribution shows that the proper relative joint or artic-ular accelerations can be obtained quite simply from

ORIGINAL ARTICLE

On the computer solutions of kinematics analysis of linkages

G. I. Perez-Soto • C. Crane • J. M. Rico • J. J. Cervantes-Sanchez •

M. A. Gonzalez-Palacios • L. A. Gallardo-Mosqueda •

P. C. Lopez-Custodio • M. A. Sanchez-Ruenes • A. Tadeo-Chavez

Received: 13 February 2013 / Accepted: 13 May 2013 / Published online: 18 June 2013

� Springer-Verlag London 2013

Abstract Since computer programs for solving kinematic

analyses of spatial linkages have been around for more than

45 years, it is taken for granted that those programs provide

the proper solution in any situation. This contribution

shows that this is not always the case. Thus, the contri-

bution is a reminder that computer aided analyses results

always need to be double-checked. Furthermore, the con-

tribution shows how to interpret the results obtained from a

computer program that does not provide the expected

results.

Keywords Screw theory � Spatial linkages � Computer

programs � Acceleration analysis

1 Introduction

Before the computer age, the kinematic and dynamic

analyses of kinematic chains were reduced to solve rela-

tively simple problems using mechanical calculators or

graphic methods that were inaccurate, time consuming, and

labor intensive. Thus, it is not surprising that the devel-

opment of computer programs for those tasks are at least

45 years old. The first computer programs were tailored to

solve the kinematic and dynamic analyses of a specific

linkage, the class of programs that is common homework in

today’s kinematics of machinery courses, see [11]. How-

ever, the first general purpose program appears to be

Integrated Mechanisms Program, IMP, developed by Sheth

and Uicker, [27], at the University of Wisconsin. This

program appears to be based in the seminal work by

G. I. Perez-Soto (&)

Departamento de Ingenierıa Electromecanica, Universidad

Autonoma de Queretaro, Campus San Juan del Rıo,

76807 San Juan del Rıo, QRO, Mexico

e-mail: [email protected]

C. Crane

Department of Mechanical and Aerospace Engineering,

University of Florida, Gainesville, FL 32611, USA


J. M. Rico � J. J. Cervantes-Sanchez �M. A. Gonzalez-Palacios � L. A. Gallardo-Mosqueda �P. C. Lopez-Custodio � M. A. Sanchez-Ruenes

Departamento de Ingenierıa Mecanica, Universidad de

Guanajuato, Campus Irapuato-Salamanca, 38010 Salamanca,

GTO, Mexico


J. J. Cervantes-Sanchez


M. A. Gonzalez-Palacios


L. A. Gallardo-Mosqueda


P. C. Lopez-Custodio


M. A. Sanchez-Ruenes


A. Tadeo-Chavez

Departamento de Ingenierıa Mecatronica, Instituto Tecnologico

Superior de Irapuato, 37856 Irapuato, GTO, Mexico


123

Engineering with Computers (2015) 31:11–28

DOI 10.1007/s00366-013-0321-2

Denavit et al. [8].1 The main purpose of this program was

to solve the kinematic analysis and, just after this step, to

compute the dynamic analysis. The University of Michigan

and its spin-off companies undertook the development of

similar programs, starting with Dynamic Analysis of

Mechanical Networks (DAMN) [12], followed by Dynamic

Response of Articulated Machinery (DRAM), [12], and

finally with Automatic Dynamic Analysis of Mechanical

Systems, Adams�, [21, 22], see also [4]. There are many

others general purpose kinematic and dynamic analyses

programs such as Machine Dynamics-Universal Solver

Analysis (MEDUSA), [9], Kinematic and Dynamic Anal-

ysis (KIDYAN), [5], Mechanism Analysis Procedure

(MAP), [12], Dynamic Analysis and Design of Systems

(DADS), [10], MEchanism SAtelite VEhicle Robot

Dynamic Equations (MESAVERDE), [29], among others.

For detailed information about the beginning and evolution

of computer aided kinematic and dynamic analyses the

reader is refereed to [11, 18, 19, 23].

In recent years have seen a surge of solid modeling pro-

grams such as Inventor� of Autodesk or SolidWorks� of

Dassault Systemes among others. These programs simplify

the task of designing parts, assemblies, and machines.

Moreover, recent versions of some of these solid modeling

programs can perform the kinematic and dynamic analyses of

machines as a final goal or as a preliminary step to carry out

the corresponding stress analysis via finite-element methods,

see [6, 7, 26]. Thus, the capabilities of these solid modeling

programs are similar to the dedicated kinematic and dynamic

analyses programs discussed above. In particular, Autodesk

acquired, in 2005, Solid Dynamics S. A. a french company

that developed simulation software for dynamic analysis and

integrated this software to the Inventor� platform, see [2].

Similarly, SolidWorks� incorporated as a tool for the kine-

matic analysis COSMOSMotion� of Structural Research and

Analysis Corporation, who is ‘‘powered’’ by Adams�, see [3].

Due to this trend, a practitioner with a fleeting knowledge

of kinematics and dynamics can complete tasks that were

previously reserved to engineering professionals with a rea-

sonable deep knowledge of these fields. Therefore, it is nec-

essary to ensure that the results obtained by these programs

are free from errors or misrepresentations; otherwise, indus-

trial accidents of unforeseeable consequences may occur.

This contribution will show that Adams�, one of the

leading kinematic and dynamic software programs, provides

as default results joint accelerations quantities which are not

the joint or articular accelerations that undergraduate or

graduate students, professors, and practitioners of kinematics

would expect to obtain. Moreover, these default results are

the same ones provided by SolidWorks�. In spite of these

comments, this contribution can not be regarded, by any

means, as a disqualification of the software, instead it must be

regarded as still another reminder that engineering specialists

must check the results obtained by any software. Moreover,

the contribution shows that the proper relative joint or artic-

ular accelerations can be obtained quite simply from other

information that Adams� correctly provides. An initial and

partial report on this problem was presented in spanish, [28].

Furthermore, it is shown that the while the computing engine

of ANSYS Workbench�, under the Rigid Dynamics mode,

versions 13 and 14, correctly computes the angular velocities

and accelerations of spatial chains, its postprocessor, also

displays, in at least one case, incorrect angular velocities.

A cursory review of the contents of the rest of this

contribution follows. Section 2 presents the velocity and

acceleration analyses of kinematic chains as described in

Rico et al. [25] and Gallardo [13]. Moreover Sect. 2

explains the process by which Adams� computes the ‘‘joint

angular accelerations’’; furthermore, Sect. 2 computes the

differences between the ‘‘joint angular accelerations’’

computed by Adams� and those joint angular accelerations

which kinematics professors and engineering practitioners

regard as most appropriate. Furthermore, this section shows

those cases where both computations agree. Section 3

presents two examples, a serial chain—a Fanuc manipu-

lator—, see Sect. 3.1, and a closed chain—a spherical four

bar—, see Sect. 3.2 In both cases a comparison of the

results between a dedicated program developed in Maple�

15 software, and the results obtained by ANSYS Work-

bench� 13 and 14, Inventor� 2011, Adams� 2012 and

SolidWorks� 2012 are presented2.

2 Velocity and acceleration analyses of kinematic

chains

In this section, the fundamentals of the velocity and accel-

eration analyses of kinematic chains are presented. In order to

reduce the length of the paper, only the most important results

are presented. For a complete derivation of the equations the

reader is referred to Rico and Duffy [24], Gallardo [13] or

Rico et al. [25]. Furthermore, it will be shown the way that

Adams� determines what it defines as ‘‘joint angular accel-

erations’’ of the pairs of a kinematic chain.

2.1 Velocity analysis of a single-loop linkage

Consider the rigid bodies shown in Fig. 1. The velocity

state of a rigid body B with respect to another rigid body A,

chosen as a reference frame, is given by

1 The authors are grateful to an anonymous reviewer for bringing to

our attention this early reference.

2 From now on neither the word Workbench nor the version program

will be repeated.

12 Engineering with Computers (2015) 31:11–28

123

AVB �AxB

AvOB

� �ð1Þ

where AxB is the angular velocity of the rigid body B as

observed from the rigid body A, and AvOB is the velocity of

a point O, fixed to the rigid body B, as observed from the

rigid body or reference frame A.

Consider the serial chain shown in Fig. 2, the chain has

m links where link 1 is the fixed link and adjacent links are

connected via screw pairs.3

It can be shown that for any link j ¼ 2; 3; . . .;m in the

serial chain, the velocity state of link j with respect to the

fixed link 1 is given by

1V j ¼1 V2 þ2 V3 þ � � � þj�2 Vj�1 þj�1 V j: ð2Þ

Note that when this equation is applied, the reference

points O in each of the bodies must be coincident. In terms

of the screws representing the kinematic pairs, Eq. (2) can

be written

1V j ¼ 1x21$2 þ2 x3

2$3 þ � � � þ j�2xj�1j�2$j�1

þ j�1xjj�1$ j

where ixi?1 and i$iþ1for i ¼ 1; 2; . . .; j� 1 are

respectively, the angular velocity of the screw pair and

the representation—using Plucker’s coordinates—of the

screw pair that joins links i and i ? 1. Therefore

i$iþ1 �isiþ1

isiþ1O

� �ð3Þ

where isiþ1 is a unit vector along the axis of the screw pair

and

isiþ1O � ihiþ1

isiþ1 þ isiþ1 � rO=P ð4Þ

and rO/P is the position vector of point O with respect to an

arbitrary point P along the screw axis.

Consider now the single-loop linkage shown in

Fig. 3, the linkage is formed from the serial chain

shown in Fig. 2 by rigidly joining link m with link 1.

Then, the velocity analysis of the single-loop linkage is

given by

1x21$2 þ 2x3

2$3 þ � � � þ m�2xm�1m�2$m�1 þ m�1xm

m�1$m

¼ 0:

ð5ÞEquation (5) can be written in matrix form as

Jx ¼ 0 ð6Þ

where

Fig. 1 Velocity state of a rigid body B with respect to another rigid

body or reference frame A

Fig. 2 Serial chain formed by m links

Fig. 3 Single-loop linkage

3 It is well known that prismatic and revolute pairs are special cases

of a screw pair and that complex pairs such as spherical or cylindrical

pairs can be obtained as a combination of revolute or prismatic pairs.

Thus the treatment is fully general.

Engineering with Computers (2015) 31:11–28 13

123

J ¼ 1$2 2$3 � � � m�2 $m�1 m�1$mh i

is the Jacobian matrix and

x ¼ 1x2; 2x3; . . .; m�2xm�1; m�1xmð ÞT

is the vector of articular or joint velocities. Finally, 0 is a

vector of zeros of dimension 6.

2.2 Acceleration analysis of a single-loop linkage

Consider the rigid bodies shown in Fig. 1. The reduced

acceleration state of a rigid body B with respect to another

rigid body A, chosen as a reference frame, is given by

AABR �

AaB

AaBO �A xB �A vB

O

� �ð7Þ

where AaB is the angular acceleration of the rigid body

B with respect to the rigid body or reference frame A, andAaO

B is the acceleration of a point O fixed in the rigid body

B with respect to the rigid body or reference frame A.

It can be shown that for any link j ¼ 2; 3; . . .;m in

the serial chain shown in Fig. 2, the reduced acceler-

ation state of link j with respect to the fixed link 1 is

given by

1A jR ¼ 1A2

R þ 2A3R þ � � � þ j�2Aj�1

R þ j�1A jR þ 1V2; 2V j

� �þ 2V3; 3V j� �

þ � � � þ j�2Vj�1; j�1V j� �

ð8Þ

where iViþ1; iþ1V j� �

for i ¼ 1; 2; . . .; j� 1 represents the

Lie products of the corresponding velocity states. The Lie

product of two velocity states is given by

ixji$ j; jxk

j$kh i

¼ix j

ivOj

� �;

jxk

jvOk

� ��

¼ix j � j xk

ix j � j vOk � j xk �i vO

j

� �:

In terms of the screws representing the kinematic pairs,

it follows that

1A jR ¼ 1a2

1$2 þ 2a32$3 þ � � � þ j�1aj

j�1$ j

þ 1x21$2; 2x3

2$3 þ � � � þ j�1xjj�1$ j

h i

þ 2x32$3; 3x4

3$4 þ � � � þ j�1xjj�1$ j

h i

þ � � � þ j�2xj�1j�2$j�1; j�1xj

j�1$ jh i

:

ð9Þ

Then, the acceleration analysis of the single-loop

linkage shown in Fig. 3 is given by

1AR2 þ 2AR

3 þ � � � þ m�2ARm�1 þ m�1AR

m þ 1V2; 2Vm� �

þ 2V3; 3Vm� �

þ � � � þ m�2Vm�1; m�1Vm� �

¼ 0:

ð10Þ

Rewriting Eq. (10) in terms of the screws representing

the kinematic pairs it follows that

1a21$2 þ2 a3

2$3 þ � � � þm�1 amm�1$m

þ 1x21$2; 2x3

2$3 þ � � � þ m�1xmm�1$m

h i

þ 2x32$3; 3x4

3$4 þ � � � þ m�1xmm�1$m

h i

þ � � � þ m�2xm�1m�2$m�1; m�1xm

m�1$mh i

¼ 0:

ð11Þ

Equation (11) can be written in matrix form as

Ja ¼ �$L ð12Þ

where J is the same Jacobian matrix, and

a ¼ 1a2; 2a3; . . .; m�2am�1; m�1amð ÞT ð13Þ

is the vector of articular or joint accelerations. Finally, $L is

the so-called Lie screw defined as

$L ¼ 1x21$2; 2x3

2$3 þ � � � þ m�1xmm�1$m

h i

þ 2x32$3; 3x4

3$4þh

� � � þm�1xmm�1$m

�þ � � � þ m�2xm�1

m�2$m�1; m�1xmm�1$m

h i:

The components of the vector of joint accelerations, [see

Eq. (13)], iai?1 for i ¼ 1; 2; . . .;m� 1; are the angular

accelerations of the kinematic pairs that form the kinematic

chain. These are the default results that Adams� compute

differently.

2.3 How Adams� computes the angular accelerations

Several analyses were conducted using the Adams� soft-

ware—two of these are presented in Sect. 3 Based on the

results of these analyses, it seemed apparent how Adams�

computes, by default, the angular acceleration of the

kinematic pairs. Firstly, the computations for serial open

chains will be analyzed. From Eq. (9) for i ¼1; 2; 3; . . .; j� 1; j; it follows that

1A1R ¼ 0 ð14Þ

1A2R ¼ 1a2

1$2 ð15Þ1A3

R ¼ 1a21$2 þ2 a3

2$3 þ 1x21$2; 2x3

2$3h i

ð16Þ


123

1A4R ¼ 1a2

1$2 þ2 a32$3 þ3 a4

3$4

þ 1x21$2; 2x3

2$3 þ3 x43$4

h i

þ 2x32$3;3 x4

3$4h i ð17Þ

1Aj�1R ¼ 1a2

1$2 þ2 a32$3 þ � � � þj�2 aj�1

j�2$j�1

þ 1x21$2; 2x3

2$3 þ � � � þ j�2xj�1j�2$j�1

h i

þ 2x32$3; 3x4

3$4 þ � � � þ j�2xj�1j�2$j�1

h iþ � � �

þ j�3xj�2j�3$j�2; j�2xj�1

j�2$j�1h i

ð18Þ1A j

R ¼ 1a21$2 þ2 a3

2$3 þ � � � þj�2 aj�1j�2$j�1 þj�1 aj

j�1$ j

þ 1x21$2; 2x3

2$3 þ � � � þ j�1xjj�1$ j

h i

þ 2x32$3; 3x4

3$4þh

� � �

þj�1xjj�1$ j

�þ � � � þ j�2xj�1

j�2$j�1; j�1xjj�1$ j

h i:

ð19Þ

For some unknown reasons,4 the default results

provided by Adams� computes the ‘‘joint angular

accelerations’’ as follows

1a#2

1$2 ¼ 1A2R � 1A1

R ¼ 1a21$2 � 0 ¼ 1a2

1$2 ð20Þ

2a#3

2$3 ¼ 1A3R � 1A2

R

¼ 1a21$2 þ 2a3

2$3 þ 1x21$2; 2x3

2$3h in o

� 1a21$2

n o¼2 a3

2$3 þ 1x21$2; 2x3

2$3h i

ð21Þ

3a#4

3$4 ¼ 1A4R � 1A3

R

¼ 1a21$2 þ2 a3

2$3 þ3 a43$4 þ 1x2

1$2; 2x32$3

hn

þ3x43$4iþ 2x3

2$3;3 x43$4

h io

� 1a21$2 þ2 a3

2$3 þ 1x21$2; 2x3

2$3h in o

¼ 3a43$4 þ 1x2

1$2;3 x43$4

h iþ 2x3

2$3; 3x43$4

h i

¼ 3a43$4 þ 1x2

1$2 þ 2x32$3;3 x4

3$4h i

ð22Þ

..

.

j�1a#j

j�1$j¼1AjR�1Aj�1

R

¼ 1a21$2þ2a3

2$3þ��þj�2aj�1j�2$j�1

n

þj�1ajj�1$jþ 1x2

1$2;2x32$3þ��þ j�1xj

j�1$jh i

þ 2x32$3;3x4

3$4þh

��

þj�1xjj�1$j

�þ��þ j�2xj�1

j�2$j�1;j�1xjj�1$j

h io

� 1a21$2þ2a3

2$3þ��þj�2aj�1j�2$j�1

n

þ 1x21$2;2x3

2$3þ��þ j�2xj�1j�2$j�1

h i

þ 2x32$3;3x4

3$4þh

��j�2xj�1j�2$j�1�

þ��þ j�3xj�2j�3$j�2;j�2xj�1

j�2$j�1h i

g

¼j�1 ajj�1$jþ 1x2

1$2;j�1xjj�1$j

h i

þ 2x32$3;j�1xj

j�1$jh i

þ j�2xj�1j�2$j�1;j�1xj

j�1$jh i

¼j�1 ajj�1$j

þ 1x21$2þ2x3

2$3þ��þ j�2xj�1j�2$j�1;

h

j�1xjj�1$j

ið23Þ

where iai?1# indicates the ‘‘joint angular accelerations’’

obtained by Adams� by default. Separating the so called

real part of equation (23), i.e. the first three rows, it follows

that

j�1a#j

j�1s j ¼ j�1ajj�1s j

þ 1x21s2 þ 2x3

2s3 þ � � � þ j�2xj�1j�2sj�1

� �� j�1xj

j�1s j� �

: ð24Þ

It is important to note that another interpretation of the

‘‘joint angular accelerations’’ obtained by Adams�, is

related to the computation of the Coriolis acceleration, as

follows

j�1a#j

j�1s j ¼ dj�1xj

dt

� �1

¼j�1 ajj�1s j þ 1x2

1s2 þ 2x32s3 þ � � �

�þ j�2xj�1

j�2sj�1Þ � j�1xjj�1s j

� �ð25Þ

where the subscript, 1, in the square brackets indicates the

reference system with respect to which the derivative has

4 A possible explanation is set forth in the conclusions.


123

been computed. It is important to recognize that this

derivative and its corresponding angular acceleration

magnitude j-1aj# is not the joint angular acceleration mag-

nitude as indicated by the vast majority of textbooks and

research papers, i.e. j-1aj, the relative angular acceleration

magnitude of body j relative to body j - 1.

The correct joint angular acceleration is obtained by the

same derivation as in Eq. (25), except that the reference

frame is j - 1 instead of 1.

j�1a j ¼ dj�1x j

dt

� �j�1

¼ dj�1xjj�1s j

dt

� �j�1

¼ dj�1xj

dtj�1s j ¼ j�1aj

j�1s j

ð26Þ

See for example, Ginsberg [14], page 128, Huang [15],

page 43, Rico and Duffy [24] or Rico et al. [25].

The difference between the ‘‘joint angular accelera-

tions’’ computed by Adams� is given by

j�1E j ¼ j�1a#j

j�1s j � j�1ajj�1s j

¼ 1x21s2 þ 2x3

2s3 þ � � � þ j�2xj�1j�2sj�1

� �� j�1xj

j�1s j� �

:

ð27Þ

Thus, for serial open chains, Adams� computes the

most appropriate angular accelerations for the first

kinematic pair, see Eq. (20), or when the displacements

of the serial chain are contained in the subgroup of

Schonfliess displacements5 since, in that case, see

Eq. (27), all the unit vectors isiþ1 for i ¼ 1; 2; . . .; j�1 are parallel and their vector products are zero. In all

other general cases, the ‘‘joint angular accelerations’’

provided by the default results of Adams� are

different.

Furthermore, it can be proved that the difference

between the ‘‘joint angular accelerations’’ computed by

the default results of Adams� and the most appropriate

joint angular accelerations, see Eq. (27) is perpendicular

to the most appropriate angular acceleration and, therefore

to the axis of the kinematic pair. Indeed, it can be easily

computed that

j�1s j � j�1E j ¼ j�1s j � 1x21s2 þ 2x3

2s3 þ � � ��

þj�2xj�1j�2sj�1

�� j�1xj

j�1s j� �

� ¼ 0

ð28Þ

since in the triple dot product two of the vectors involved

are equal to j�1s j.

Finally, for closed linkages, there is still another case

where the default results of Adams� also provides the

most appropriate value of the joint angular acceleration.

Consider a closed kinematic chains with j - 1 links.

This closed kinematic chain is obtained from an open

serial chain with j links, where the j-th link is rigidly

joined with the fixed link, 1. Then, from the final form

of Eq. (23),

j�1a#j

j�1$ j ¼ j�1ajj�1$ j

þ 1x21$2 þ 2x3

2$3 þ � � �h

þ j�2xj�1j�2$j�1; j�1xj

j�1$ j�:

ð29Þ

However, from the velocity analysis of a closed kinematic

chain, see Eq. (5), substituting m by j, it follows that

1x21$2 þ2 x3

2$3 þ � � � þ j�2xj�1j�2$j�1 þ j�1xj

j�1$ j ¼ 0:

ð30Þ

Therefore

1x21$2 þ 2x3

2$3 þ � � � þ j�2xj�1j�2$j�1 ¼ �j�1xj

j�1$ j:

Hence, the Lie product in the right part of Eq. (29)

involves two equal screws with opposite sign, and the

result is 0. Hence

j�1a#j

j�1$ j ¼ j�1ajj�1$ j:

Thus it can be concluded that the joint angular

accelerations of the two kinematic pairs that connect the

linkage to the fixed link are properly computed by the

default results of Adams�. This result can be extended to

multi-loop linkages without unexpected surprises.

3 Examples of acceleration analysis of kinematic chains

In this section, two examples of kinematic chains, a serial

spatial manipulator and a closed spherical linkage, are

presented. These examples compare the joint angular

accelerations obtained by using the theory presented in

Sect. 2 and programmed in Maple�, with the results

obtained employing ANSYS�, Inventor�, Adams� and

Solid Works�. The comparison shows that the default

results of Adams� differently determines the ‘‘joint angular

accelerations’’ of the kinematic pairs of a kinematic chain.

Furthermore, it is shown that using the capabilities of Solid

Works� of choosing the coordinate system with respect to

which the joint angular acceleration is represented6, it is

possible to provide a clear explanation of why the default

results of Adams� computations of the ‘‘joint angular

accelerations’’ are different from those commonly accepted

as most appropriate.

5 It should be noted that the subgroup of planar displacements, a very

important case, is contained in the Schonfliess subgroup.

6 As previously indicated, Solid Works� employs the same Solver

module of Adams�.


123

3.1 Kinematic analysis of a FANUC-200iB Serial

Manipulator

In this section the results for the inverse acceleration

analysis of the serial manipulator FANUC-200iB as an

example of an open serial chain. The results of the ana-

lytical method indicated in Sect. 2 and programmed in

Maple� software is compared with the results obtained by

using the default results of Adams�, ANSYS�, Inventor�

and SolidWorks�. Furthermore, an additional comparison

with the results obtained with SolidWorks� by choosing an

appropriate coordinate system is also added.

In order to provide a detailed comparison, the results for the

inverse acceleration analysis for two different values of time

are presented. Finally, it is verified that, unlike the ‘‘joint

angular accelerations’’ the translational accelerations of points

are properly computed by the default results of Adams�.

3.1.1 Kinematic of the serial manipulator FANUC-200iB

The FANUC-200iB manipulator has six revolute joints and

six degrees of freedom, a simplified diagram of the serial

manipulator is presented in Fig. 4a where a cylinder repre-

sents a revolute joint, all the dimensions required to perform a

kinematic analysis are shown. The coordinate systems OiX-

iYiZi correspond to the proximal variant of the Denavit-Har-

tenberg convention, see [20]. From this diagram it is possible

to obtain all the Denavit-Hartenberg parameters—the offsets,

di, the link lengths, ai, and the link angles, ai—, this infor-

mation is presented in Table 1. The parameters hi corresponds

to the angles between the planes formed by the links i - 1 and

i, for i = 1, 2, 3, 4, 5, 6, respectively. From now on,

h1 = 0h1, h2 = 1h2, h3 = 2h3, h4 = 3h4, h5 = 4h5 and

h6 = 5h6.

3.1.2 End-effector movement description

In order to perform an inverse acceleration analysis of the

FANUC-200iB manipulator, it is necessary to specify the

movement of the end-effector. In this case, a very simple

movement of the end-effector is employed. It consists of a

translational movement of the end-effector, where the point

O6, fixed in the end-effector, describes a circular path on a

plane parallel to the X - Z plane given by y = 200 mm. The

center of the circular path is given by rC = (300 mm, 200 mm,

650 mm) and the radius of the circular path is r = 50 mm,

see Fig. 4b. Thus, the position of point O6, with respect to the

origin of the coordinate system is given by

rO6=O0tð Þ ¼ 300þ 50 sinðtÞ½ �iþ 200jþ 650þ 50 cosðtÞ½ �k

ð31Þ

Table 1 Numerical values for the Denavit-Hartenberg parameters,

proximal variant

i hi(rad) di(mm) ai(mm) ai(rad)

0 – – 0 0

1 h1 350 150 -p/2

2 h2 0 250 0

3 h3 0 75 -p/2

4 h4 290 0 p/2

5 h5 0 0 -p/2

6 h6 80 – –

Fig. 4 a Diagram of the

FANUC-200iB manipulator. All

dimensions in millimeters.

b Plane of movement of point

O6


123

Thus, the initial position of point O6 is given by

rO6=O00ð Þ ¼ 300 mm i; 200 mm j; 700 mm k

� �ð32Þ

where i; j and k represent the unit vectors along the

X, Y and Z directions associated with the coordinate system

shown in Fig. 4a. In addition, the orientation of the

coordinate system, O6X6Y6Z6, with respect to O0X0Y0Z0,

is given by the constant rotation matrix. Thus the

orientation of the end-effector during the movement

remains constant as defined by

0R6 ¼0 0 1ffiffi

2p

2�ffiffi2p

20ffiffi

2p

2

ffiffi2p

20

24

35: ð33Þ

Since, in general, for a given position of the end-effector

there are up to eight different solutions for the inverse

position analysis, and the velocity and acceleration analyses

depends upon the particular solution of the position

analysis, it is necessary to indicate the desired position

solution, shown in Fig. 4b. In this case, the solution is given

by 0h1 = 0.7378150 rad, 1h2 = - 2.079407 rad, 2h3 =

0.306659 rad, 3h4 = 1.787951 rad, 4h5 = - 0.759903

rad and 5h6 = - 2.651654 rad.

3.1.3 Analytical results

The velocity and acceleration of the point O6 are the first

and second order time derivatives of Eq. (31). Hence, one

can obtain the velocity and reduced acceleration state of

the end-effector with respect to the fixed link; i. e.0VO

6 , 0AR6, for any instant of time. Thus, it is possible to

carry out, iteratively the inverse position, inverse velocity,

and inverse acceleration analysis. This analytical procedure

was programmed in the Maple� computer software. The

program provides the results for 360 evenly spaced inter-

vals of time, where time goes from 0 to 2p s. For the

movement of the end-effector, there are four real solution

sets of the inverse position analysis. The inverse position

analysis results corresponding to the desired position

solution, shown in Fig. 4b are shown in Fig. 5. Once the

inverse position analysis is solved, it is possible to set up

the equations for the inverse velocity, see Eq. (2), and the

inverse acceleration analysis, see Eq. (9).

The graphs for the angular accelerations are provided,

the first three pairs, 0a1, 1a2 and 2a3 in Fig. 6a and the last

three pairs, 3a4, 4a5 and 5a6 in Fig. 6b.

3.1.4 Software results

The same inverse position, inverse velocity and inverse

acceleration analysis were carried out using four different

programs, ANSYS�, Inventor�, Adams� and Solid-

Works�. The graphs for the joint angular accelerations,

obtained from all these programs, are given in the same

order as in the analytical results. The first graphs present

the joint angular accelerations of the first three pairs,

0a1, 1a2, 2a3 and the second graphs present the angular

accelerations of the last three pairs, 3a4, 4a5, 5a6. The

graphs are arranged as follows. Figure 6a, b for the alge-

braic computation of the angular accelerations using

Maple�, similarly Fig. 6c, d for ANSYS�, Fig. 6e, f for

Inventor�. Figure 7a, b show the magnitudes or absolute

values of the angular accelerations as computed with

Maple�, similarly Fig. 7c, d for the default results of

Adams� and Fig. 7e, f for SolidWorks� also show the

absolute values. It should be noted that the graphs corre-

sponding to the default results of Adams� and Solid-

Works� show the absolute value of the ‘‘joint angular

accelerations’’. This feature is due to the fact that in the

postprocessing phase of both programs, this is the only

readily available option. As indicated previously, it is

possible for Adams� to obtain the most appropriate joint

angular accelerations by requesting different results from

those of the default option. However, for beginners this

option is not readily available.

Figure 6, shows a very good agreement between the

results obtained by the analytical process programmed in

Maple� and those obtained by using ANSYS� and

Inventor�. Figure 7, shows a very good agreement between

the results obtained by either the default results of Adams�

and SolidWorks�7. This result is not surprising since, as

indicated in Sect. 1, SolidWorks� uses the default results

of Adams� as the tool for solving the kinematic analysis.

Finally, Figs. 6 and 7 shows that the result obtained by the

Fig. 5 Inverse position analysis results for serial manipulator FAN-

UC-200iB

7 It should be noted that SolidWorks� provide the results of angular

acceleration in deg/s2.


123

default results of Adams� and SolidWorks� shows a

sizeable difference compared with those obtained by the

analytical process programmed in Maple� or obtained by

using ANSYS� or Inventor�. For example, while Maple�,

ANSYS� and Inventor� show that 1a2 = 0 around time

t = 3 s and 2a3 = 0 around time t = 2 s, Adams� and

SolidWorks� show that none of these events happen

according with their results. Moreover, Fig. 7 also shows

significant differences between the magnitudes or absolute

values of the angular accelerations computed by Maple�

Fig. 6 Maple�, ANSYS� and Inventor� results for the inverse acceleration analysis of a FANUC-200iB manipulator


123

and those computed by the default results of Adams� or

SolidWorks�.

It is important to note that SolidWorks� has postpro-

cessing capabilities that, on the one hand, allow the user to

obtain the most appropriate solution for the joint angular

acceleration and, on the other hand, provide a decisive evi-

dence of the difference between the ‘‘joint angular

acceleration’’. SolidWorks� allows the user to decompose a

‘‘joint angular acceleration’’ on its components along a

coordinate system in which one of the axes is parallel to the

revolute axis that connects two consecutive links. For

example, Fig. 8a shows the dialog box that allows one to

decompose an angular acceleration, in this case 3a4; along a

coordinate system whose X axis is parallel to the axis of the

Fig. 7 Maple�, the default results of Adams� and SolidWorks� results for the inverse acceleration analysis of a FANUC-200iB manipulator


123

revolute that connects links 3 and 4. Figure 8(b) shows the

graphs of the three components x, y and z of 3a4: It is easy to

see that the graph of 3a4x agrees with the most appropriate

results for 3a4 provided by Maple�, ANSYS� and Inventor�.

Furthermore, it must be now obvious that the y and

z components are related to the terms indicated in Eq. (27),

and as indicated in Sect. 2 they are perpendicular to the most

appropriate angular acceleration. Finally, it must be equally

obvious that a joint angular acceleration cannot have com-

ponents perpendicular to the directions of the revolute axis.

3.1.5 Obtaining the most appropriate joint angular

accelerations using Adams�

In this section, it will be shown that using Adams�, it is

possible to obtain the most appropriate joint angular

accelerations. For that purpose, one has to make a

new request under the instrumentation toolbox of

Adams�, see [1]. Figure 9 exemplifies the request for

all the most appropriate joint angular accelerations

associated with the kinematic pairs of the FANUC-

200iB manipulator. It is important to note that only

the angular accelerations around the Z axes of each

revolute pair is requested. Similar requests can be

asked for the X and Y components of the angular

accelerations

Figure 10 shows the results of such requests. Figure 10a

presents the results for the first three revolutes, while

Fig. 10b presents the results for the last three revolutes.

Now, it is evident that these results are in complete

agreement with those obtained using Maple�, Ansys� and

Inventor�, see Fig. 6.

(a) (b)

15.0

10.0

5.0

0.0

-5.0

-10.0

Ang

ular

Acc

eler

atio

n (d

eg/s

ec**

2)

-15.00.00 1.00 2.00 3.00

Time (sec)

4.00 5.00 6.28

3α4,z

3α4,y

3α4,x

Fig. 8 a SolidWorks� dialog box used to descompose 3a4 along a coordinate system whose X axis is parallel to the axis of the revolute pair that

connects links 3 and 4. b Graphs of the components of 3a4

Z

X

Y

Fig. 9 Request for the angular

accelerations around the Z axes

of each revolute pair


123

Fig. 10 Adams� results for the inverse acceleration analysis of a FANUC-200iB manipulator through the tool create a request

Table 2 Instantaneous joint

angular acceleration comparison

All values in 10-3 : rad/s2

Remark It can be proved that

after considering the request of

results different for those of the

default option, all

disagreements between Adams�

and the rest of the programs

disappear

Angular

acceleration

Maple

analytical

ANSYS Inventor SolidWorks

component

Adams

magnitudes

SolidWorks

magnitudes

Analysis for t1 = 3.6 s

0a1 -30.5083 -30.507 -30.51 -30.5085 30.5085 30.5085

1a2 147.5411 147.54 147.54 147.5412 148.1446 148.1444

2a3 -232.8751 -232.87 -232.88 -232.8746 232.8890 232.8889

3a4 19.5509 19.552 19.55 19.5503 32.9827 32.9827

4a5 11.7299 11.731 11.73 11.7321 24.8914 24.8915

5a6 -74.3043 -74.306 -74.30 74.3027 74.3033 74.3027

Analysis for t2 = 5 s

0a1 -134.9142 -134.91 -134.91 -134.9137 134.9138 134.9137

1a2 106.9109 106.91 106.91 106.9111 107.0159 107.0159

2a3 -7.2627 -7.2659 -7.26 -7.2632 11.6523 11.6523

3a4 -33.7928 -33.795 -33.80 -33.7952 35.7158 35.7164

4a5 133.7636 133.76 133.76 133.7606 133.8098 133.8093

5a6 98.2247 98.227 98.23 98.2273 98.2272 98.2273

Fig. 11 Acceleration of the O3 point. a Analytical results. b Adams� results


123

3.1.6 Instantaneous angular acceleration comparison

In this section, the results for the inverse angular acceler-

ation analysis for two different instants of time will be

analyzed.

Table 2 presents the joint angular accelerations of all six

revolute joints obtained using the analytical procedure

programmed in Maple� and the results obtained using

ANSYS�, Inventor�, Adams� and SolidWorks�. More-

over, for SolidWorks� two different results are presented.

The column SolidWorks� component, provides the results

obtained using the decomposition indicated in Sect. 3.1.4,

while SolidWorks� magnitude, like Adams�, provides only

the magnitude of the angular acceleration.

Table 2 shows that for t1 = 3.6 s, the default results of

Adams� and SolidWorks� magnitude, show significant dif-

ferences in the values of 3a4 and 4a5. Similarly, for t2 = 5 s,

the default results of Adams� and SolidWorks� magnitude,

present significant deviations in the values of 2a3 and 3a4.

3.1.7 Traslational acceleration

Since it was found that the default results of Adams� does

not readily compute the relative joint angular accelerations

of revolute pairs, it was regarded necessary to verify the

correctness of the computations of traslational accelerations

of points that belong to the different links of the serial

manipulator. To that end, the point O3 located at the origin of

the coordinate system O30X3Y3Z3, see Fig. 4a, was chosen.

Figure 11a shows the acceleration components of the

point O3 found analytically. On the other hand, the acceler-

ation components of the same point O3 computed by the

default results of Adams� are shown in Fig. 11b. The results

show that the default configuration of Adams� computes

correctly the traslational accelerations of point O3.

Table 3 Numerical values for the Denavit-Hartenberg parameters,

proximal variant

i hi (�) di (mm) ai (mm) ai (�)

1 h1 0 0 -80

2 h2 0 0 60

3 h3 0 0 30

4 h4 0 0 60

Fig. 12 A discrepance in Ansys� postprocessing

Oi

4

3

2

1

z1z4

z3

z2

x1

x2

x3x4

60°30°

60°

-80°

Fig. 13 Diagram of the spherical four bar linkage

Fig. 14 Position analysis results for spherical four bar linkage


123

3.1.8 A discrepance in Ansys� postprocessing

Finally, this section presents a discrepancy in the post-

processing of the angular velocity analysis of the same

FANUC-200iB manipulator. Figure 12 shows the results

for the angular velocity and acceleration corresponding to

the revolute that connects links 3 and 4. Figure 12b shows

that the angular acceleration is not a constant function, and

therefore, it is impossible that the angular velocity shown

in Fig. 12a can be the zero function.

The discrepancy has been reported and it is expected

that it will be promptly resolved.

3.2 Kinematic analysis of a spherical four bar linkage

A spherical four bar linkage is presented in Fig. 13 where

the cylindrical assemblies represent a revolute joint. All the

dimensions required to perform a kinematic analysis are

shown. The coordinate systems OiXiYiZi correspond to the

proximal variant of the Denavit-Hartenberg convention,

see [20]. From this diagram it is possible to obtain all the

Denavit-Hartenberg parameters—the offsets, di, the link

lengths, ai, and the link angles, ai—. This information is

presented in Table 3. Because a spherical four bar linkage

with revolute joints on the axes Z1, Z2, Z3 and Z4 is a one

degree of freedom mechanism, a single angle must be

specified. The angle h4 is selected as the input angle, and

for the position shown in Fig. 13, is given by h4 = 0�.

In Table 3, hi corresponds to the angles between the

planes formed by the links i - 1 and i, for i=1, 2, 3, 4,

respectively; it should be noted that in this case, the link 0

correspond to the link 4. From now on, h1 = 4h1, h2 = 1h2,

h3 = 2h3 and h4 = 3h4.

3.2.1 Analytical results

In order to carry out the velocity and acceleration analyses,

it is necessary to perform the position analysis. For any

position of the input angle, 3h4, there are two solutions, for

the angles 4h1, 1h2 and 2h3. The position analysis results

corresponding to the selected solution is shown in Fig. 14.

Once the position analysis is solved, it is possible to set up

the equations for the velocity, see Eq. (5), and the accel-

eration analysis, see Eq. (11). For the velocity analysis, it is

Fig. 15 Maple�, ANSYS� and Inventor� results for the joint acceleration analysis of a spherical four bar linkage


123

assumed that the angular velocity 3x4 is known and equal

to 2 p rad/s. Similarly, for the acceleration analysis, it is

assumed that the angular acceleration 3a4 is known and

equal to 0 rad/s2.

3.2.2 Software results

The same position, velocity and acceleration analysis were

carried out using four different programs, ANSYS�,

Inventor�, the default results of Adams� and Solid-

Works�. The graphs for the angular accelerations, obtained

from all these programs, are arranged as follows. Fig-

ure 15a for Maple�, Fig. 15b for ANSYS�, Fig. 15c for

Inventor�, Fig. 16a for the magnitudes or absolute values,

Fig. 16b for the default results of Adams� and Fig. 16c for

SolidWorks�. It should be noted that the graphs corre-

sponding to the default results of Adams� and Solid-

Works� show the absolute value of the angular

acceleration. This feature is due to the fact that in the

postprocessing phase of both programs, this is the only

readily available option. As indicated previously, it is

possible for Adams� to obtain the most appropriate joint

angular accelerations by making a request for results

different from the default option. However, for beginners

this option is not readily available.

Figure 15, shows a very good agreement between the

results obtained by the analytical process programmed in

Maple� and those obtained by using ANSYS� and

Inventor�. Figure 16, shows a very good agreement

between the absolute value results obtained by either the

default results of Adams� and SolidWorks�8 but Fig. 16a

shows sizeable differences between the magnitudes or

absolute values as obtained with Maple� and the magni-

tudes obtained by either Adams� and SolidWorks�.

Finally, Figs. 15 and 16 shows that the result obtained by

the default results of Adams� and SolidWorks� shows

sizeable differences compared with those obtained by the

analytical process programmed in Maple� or obtained by

using ANSYS� or Inventor�.

It is important to note that SolidWorks� has postpro-

cessing capabilities that on the one hand allow the user to

obtain the most appropriate solution for the joint angular

acceleration and on the other hand provide a decisive proof

Fig. 16 Maple�, the default results of Adams� and SolidWorks� results for the acceleration analysis of a spherical four bar linkage

8 It should be noted that SolidWorks� provide the results of joint

angular acceleration in deg/s2.


123

of the miscalculation of the angular acceleration. Solid-

Works� allows the user to decompose a joint angular

acceleration on its components along a coordinate system in

which one of the axes is parallel to the revolute that con-

nects two consecutive links. For example, Fig. 17a shows

the dialog box that allows to decompose a joint angular

acceleration, in this case 2a3; along a coordinate system

whose Z axis is parallel to the axis of the revolute that

connects links 2 and 3. Figure 17b shows the graphs of the

three components x, y, and z of 2a3: It is easy to see that the

graph of 2a3z agrees with the most appropriate results for 2a3

provided by Maple�, ANSYS� and Inventor�.

Furthermore, it must be now obvious that the x and

y components are related to the terms indicated in Eq. (27),

and as indicated in Sect. 2 they are perpendicular to the

most appropriate joint angular acceleration. Finally, it must

be equally obvious that a joint angular acceleration can not

have components perpendicular to the directions of the

revolute axis.

3.2.3 Instantaneous angular acceleration comparison

In this section, the results for the inverse angular acceler-

ation analysis for two different instants of time will be

analyzed.

Table 4 presents the joint angular accelerations of all

four revolute joints obtained using the analytical procedure

programmed in Maple� and the results obtained using

ANSYS�, Inventor�, the default results of Adams� and

SolidWorks�. Moreover, for SolidWorks� two different

results are presented. The column SolidWorks� component,

provides the results obtained using the decomposition

indicated in Sect. 3.1.4, while SolidWorks� magnitude, like

the default results of Adams�, provides only the magnitude

of the angular acceleration.

Table 4 shows that for t1 = 0.35 s and t2 = 0.8 s, the

default results of Adams� and SolidWorks� magnitude,

present significant differences in the values of 1a2 and 2a3,

respectively.

Table 4 Instantaneous angular

acceleration comparison

All values in rad/s2

Remark. It can be proved that

after considering the request of

results different for those of the

default option, all

disagreements between Adams�

and the rest of the programs

disappear

Angular

acceleration

Maple

analytical

ANSYS Inventor SolidWorks

component

Adams

magnitudes

SolidWorks

magnitudes


3a4 0 0 -0 0 0 0

4a1 42.43721 42.437 42.43721 42.43721 42.43721 42.43721

1a2 -3.48178 -3.4818 -3.48178 -3.48178 8.97345 8.97345

2a3 -32.52036 -32.52 -32.5204 -32.5204 42.77995 42.77995


3a4 0 0 0 0 0 0

4a1 -8.56208 -8.5621 -8.56208 -8.56208 8.56208 8.56208

1a2 8.91847 8.9185 8.91847 8.91847 10.16295 10.16295

2a3 -5.98466 -5.9847 -5.98466 -5.98466 12.25051 12.25051

4

12

3

(a) (b)

5000

4000

3000

2000

1000

-1000

-2000

-30000.0 0.1 0.2 0.3 0.4 0.5

Time (sec)

0.6 0.7 0.8 1.00.9

0

Ang

ular

Acc

eler

atio

n (d

eg/s

ec**

2)

2α3,y

2α3,x

2α3,z

Fig. 17 a SolidWorks� dialog box used to descompose 2a3 along a coordinate system whose Z axis is parallel to the axis of the revolute pair that

connects links 2 and 3. b Graphs of the components of 2a3


123

4 Conclusions

The present contribution has shown that the default

results of Adams� incorrectly computes the angular

acceleration of the kinematic pairs of spatial and spherical

kinematic chains. Therefore, SolidWorks�, that employs

Adams� as the software that solves the kinematic analy-

ses, also provides incorrect results. The readers are

warned about that.

Furthermore, using the postprocessing capabilities of

SolidWorks� a simple explanation for the mistake is

found and a simple method for correcting the results is

presented. The correct results for the joint angular accel-

erations can be also obtained using Adams� by making a

request for all the proper joint angular accelerations,

unfortunately this option is not easily accessible to

beginner users of Adams� and, as far as the authors are

aware, the option is not available to users of SolidWorks�,

at least up to the version used to carry out this contribu-

tion. It should be stressed that this contribution does not

advocate one to discontinue the use of Adams� and should

be thought more as an advisory note to double check the

results of any computer simulation. It may be the case that

the Adams� developers calculated the angular acceleration

vector as the derivative with respect to ground of the

angular velocity vector between bodies j and j ? 1 and

then returned the magnitude of this vector. This magnitude

is not the relative angular acceleration about the screw

axis connecting these bodies, which is what most users

would be interested in calculating. It could also be spec-

ulated that Adams� developers chose to differently com-

pute the joint angular accelerations because, if one follows

this approach, the ‘‘joint angular accelerations’’ are addi-

tive; i.e.

0an ¼ 0a1 þ 1 a2 þ � � � þ n�1an:

As indicated, the so called real part of Eq. (5), the

angular velocities, are indeed additive; i. e.

0xn ¼ 0x1 þ 1x2 þ � � � þ n�1xn:

However, it is well known, see Kane and Levinson

[17], page 26, or Josephs and Huston [16], page 93, that

angular accelerations are not additive. As a marginal

note, these results can be thought of as a verification of

the validity of the acceleration equations developed

about 15 years ago, see [24], Rico et al. [25] and

Gallardo [13]. The correctness of these equations have

been challenged several times by anonymous reviewers

in the last 6 years.

Finally the paper also shows that sometimes the results

correctly computed by the solver engine of a program, for

example ANSYS�, can be incorrectly displayed.

Acknowledgments The authors thankfully acknowledge the support

of Conacyt, the Mexican National Council for Science and Technol-

ogy, for its support via Basic Science Project Number CB-2010-01-

156558. Moreover, the first author and the last four thank Conacyt for

their graduate studies scholarships. All the authors thank to Division de

Ingenierıas, Campus Irapuato-Salamanca, Universidad de Guanajuato,

for its continuous support. In addition, the authors thank M.Sc. Carlos

Franco, from SCC group, the authorized representant of ANSYS� in

Mexico for his willingness and openness to answer our queries and

following up with the fixing of the discrepancy.

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On the computer solutions of kinematics analysis of linkages · the contribution shows that the proper relative joint or artic-ular accelerations can be obtained quite simply from