Strojniški vestnik - Journal of Mechanical Engineering 68(2022)9, 560-570 Received for review: 2022-02-21
© 2022 The Authors. CC BY 4.0 Int. Licensee: SV-JME Received revised form: 2022-07-01
DOI:10.5545/sv-jme.2022.69 Original Scientific Paper Accepted for publication: 2022-07-11
*Corr. Author’s Address: Tianjin University, School of Mechanical Engineering, China, niyb5812@tju.edu.cn 560
Limit-protection Method for the Workspace  
of a Parallel Power Head
Ni, Y . – Lu, W. – Jia, S. – Lu, C. – Zhang, L. – Wen, Y .
Yanbing Ni
1,*
– Wenliang Lu
2
 – Shilei Jia
2
 – Chenghao Lu
2
 – Ling Zhang
2
 – Yang Wen
1
1 
Tianjin University, School of Mechanical Engineering, China 
2 
Key Laboratory of Mechanism Theory and Equipment Design of Ministry of Education, China
The end pose of a one-translation two-rotation (1T2R) parallel mechanism is a mapping of the servo motor action in the joint space. Because 
it is difficult to obtain information about the end attitude state, we have designed and implemented a simplified algorithm for determining 
the attitude of such a mechanism. The kinematic inverse solution of the robot and the modelling analysis of the workspace are carried out. 
From this, it is deduced that the length transformation of the three branch chains of the mechanism reflects the position and attitude of 
the end-motion platform. Based on this algorithm, the limit protection of a parallel power head under arbitrary configuration is realized. The 
correctness of the calculation method is verified by simulation. Finally, based on the software and hardware conditions of an existing control 
system, experimental verification is carried out. The experimental results show that the simplified algorithm can implement limit protection for 
this type of machine.
Keywords: parallel power head, modelling, workspace analysis, position and pose judgment, limit-protection
Highlights
• 	 The kinematics of the 1T2R head are analysed, and the workspace of the mechanism is obtained. 
• 	 A new simplified algorithm for position and attitude judgment is proposed. 
• 	 The new simplified algorithm of position and attitude judgment is combined with the mechanism control system to realize a 
fast limit.
• 	 The algorithm is verified with simulation and experimental data.
0  INTRODUCTION
Due to its compact structure, high stiffness to mass 
ratio, high precision, and good dynamic characteristics 
[1] and [2], the parallel power head has been of interest 
to both academia and the manufacturing industry and 
has been widely used in high-end manufacturing 
fields, such as large aircraft structural parts [3]. 
The working space is the key index for evaluating 
the performance of parallel power heads. Unlike 
tandem robots, which are driven by a series of 
connecting rods and rotating joints in series, parallel 
power heads are connected by at least two independent 
kinematic chains. Although the working space is 
relatively small, their structure is precise and compact 
with high repetitive positioning accuracy [4] and [5]. 
Workspaces can be partitioned in a number of ways, 
depending on performance requirements and selected 
parameters [6]. The factors influencing the size and 
shape of the workspace include the constraints of the 
length of the branch chain, the rotation angle, and 
the size of the revolute joint [7] and [8]. The methods 
used for the analysis of the workspace include the 
geometric, numerical [5], and discretization methods 
[9] and [10]. Gui et al. [11] proposed a reliability 
mathematical model based on random probability and 
presented a measurement and calculation method for 
the evaluation of the reliability level of mechanism 
motion. Shao et al. [12] analysed the new spatial-
planer parallel mechanism using geometric methods 
and verified that it has good accuracy and efficiency. 
Kaloorazi et al. [13] used the structural geometry 
method to determine the maximum non-singularity 
workspace of a 3-degree of freedom (3-DOF) 
parallel mechanism, and Huang et al. [9] obtained 
the workspace of the Stewart-Gough manipulator 
by a discrete method. The scientific community has 
also made many achievements in numerical method 
research. Zhang [14] et al. used the fast search method 
to calculate the workspace volume and took it as the 
optimization objective function. In addition, Gao 
and Zhang [15] and Zhu et al. [16] used the modified 
boundary search method to obtain the workspace of 
a parallel mechanism. Farzaneh-Kaloorazi et al. [17] 
used an interval analysis approach for the barrier-free 
working space of a parallel mechanism. Majid [18] 
et al. analysed the workspace of a three-prismatic-
prismatic-spheric-revolute (3-PPSR) manipulator 
using the numerical method and found that three 
regions in the workspace corresponded to the postures 
of a type of manipulator.
In order to make more reasonable and effective 
use of the existing working space and ensure the 
safe and reliable operation of the parallel power 

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)9, 560-570
561 Limit-protection Method for the Workspace of a Parallel Power Head 
head, the method of trajectory planning and space 
limitation is often adopted [19] and [20]. Khoukhi et 
al. [21] proposed a multi-objective dynamic trajectory 
planning method for the parallel mechanism under the 
constraints of task, workspace, and mechanism with the 
help of the discrete augmented Lagrangian technique. 
Reveles et al. [22] proposed a kinematic redundant 
parallel robot joint trajectory planning method using 
a feasibility map, which plans the joint trajectory 
while avoiding parallel singularities through the 
graphical evaluation of the robot pose related to four 
working modes. Dash et al. [23] proposed a numerical 
path planning method to avoid the singularity of the 
mechanism in the accessible workspace of a parallel 
robot by clustering the singularities and using the local 
routing method based on Grassmann line geometry 
to avoid the singularities. Iqbal et al. [24] discussed 
two complex control strategies: computational torque 
control (CTC) and variable structure control (VSC) 
and improved the trajectory-tracking performance of 
the robot. Manzoor et al. [25] integrated the functions 
of mechanical computer-aided design (CAD) and 
robot CAD into the same platform and achieved the 
accurate control of the robot through various three-
dimensional models in the platform. Alam et al. [26] 
considered two different methods based on sliding 
mode control (SMC) to achieve the nonlinear control 
of an elastic joint robot. This control method enables 
the robot to obtain a locally stable closed-loop system.
Because the pose of the end of the one-translation 
two-rotation (1T2R) parallel mechanism in the 
operation space is a nonlinear mapping of the motion 
of the servo motor in the joint space, the modelling 
and calculation process is complicated, the motion 
controller takes a long time. It is therefore impossible 
to realize the real-time operation to prevent exceeding 
the limit during the operation of the mechanism. 
The purpose of this paper is to propose a simplified 
algorithm for position and attitude judgment, which 
can greatly reduce the calculation amount of limit, 
and realize the limit protection of the 1T2R parallel 
power head through a combination of software and 
hardware. Firstly, the inverse kinematics analysis 
was carried out, and the mapping relationship 
between the terminal pose of the mechanism and 
each input value was constructed based on the inverse 
kinematics model. Then, according to the mechanism 
structure, scale parameters, range of motion of each 
pair, interference and other constraints, the working 
space of 1T2R power head was determined. By 
analysing and summarizing the motion rules of 1T2R 
power head, a simplified algorithm for judging the 
position and pose of 1T2R power head was obtained, 
which was used to realize the limit protection of the 
1T2R head. Finally, the algorithm was verified via 
experimentation.
1  KINEMATIC ANALYSIS 
The position inverse solution of the 1T2R head is to 
solve the rod length quantity of each branch chain 
motion joint by knowing the positional parameters 
of the end reference point of the tool providing the 
theoretical model for mechanism error analysis and 
control.
1.1  Machine Tool Profile 
As shown in Fig. 1, a 1T2R mechanism is a parallel 
mechanism with one translational and two rotational 
degrees of freedom. A 1T2R power head is composed 
of a moving platform, a static platform, and three 
RPS branch chains, in which R, P, and S represent 
revolute joints, active prismatic pairs and spherical 
joints, respectively. One end of each RPS branch 
chain is connected to the moving platform through a 
spherical joint, and the other end is connected to the 
static platform through a revolute joint. The motorized 
spindle is installed on the moving platform and the 
active prismatic pair is driven by a servo motor.
Fig. 1.  Model of 1T2R spindle head
1.2  Inverse Kinematic Solution 
The power head of the 1T2R mechanism diagram is 
shown in Fig. 2, A
i
 and B
i
 represent the centres of the 
spherical joint and the revolute joint, respectively; 
ΔA
1
A
2
A
3
 and ΔB
1
B
2
B
3
 are equilateral triangles; points 
O' and O are the geometric centres of the two triangles; 
the moving platform is represented by ΔA
1
A
2
A
3
; the 
static platform is represented by the plane of ΔB
1
B
2
B
3
; 
point P is the tip at the end of the mechanism; e is the 
distance from P to the plane of the moving platform; 
a, b are the circumcircle radii of the moving and static 
platform, respectively; l represents the initial branch 
length.

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)9, 560-570
562 Ni, Y. – Lu, W. – Jia, S. – Lu, C. – Zhang, L. – Wen, Y.
The fixed and moving coordinate systems O – xyz  and O' – x'y'z' are established at the centre O and O' of 
the fixed and moving platforms, respectively. In the 
initial position, the z' axis and z axis are perpendicular 
to the planes A
1
A
2
A
3
 andB
1
B
2
B
3
, respectively, as 
shown in Fig. 2. The x' axis and x axis are along the  
AA
32
 
 direction and BB
32
 
, respectively. The y´ axis 
and y axis are determined according to the right-hand 
rule. α and β represent rotation about the x axis and y 
axis, respectively.'
𝐴 1
 
 
𝐴 3
 
𝐵 2
 
𝐵 1
 
𝐵 3
 
𝑂 
𝑥 
𝑦 
𝑏 
𝑂 ′
 
′
 
𝑦 ′
 
′
 
𝑥 1
  
 
𝑧 1
 
𝑃 
𝑒 
𝒃 𝟏 
𝑞 1
𝒘 𝟏 
 
− 𝒂 𝟏 
𝑒 𝒘 
𝒓 𝑷 
𝛼 
𝛽 
 
𝒓 𝟏 
𝑧 
𝑧 𝑥 𝐴 2
Fig. 2.  Mechanism sketch of 1T2R mechanism
The attitude transformation matrix of O' – x'y'z' 
in the connected body coordinate system of a moving 
platform relative to O – xyz in the static platform 
coordinate system is R:
R  
 
RotR ot Rot
=
zxz
cc scsc ss cc ss
sc
,,, 
    
   











ccss sc cc cs
ss sc c
   
  
= uvw , (1)
where s = sin, c = cos, u, v, w represent the 
measurement of three coordinate axes of O' – x'y'z' 
in the static platform coordinate system O – xyz, 
respectively, and w can be used to represent the tool 
attitude vector.
ψ, θ, ϕ are the precession angle, nutation angle 
and spin angle, respectively, which are related to α 
and β:
 
 





 












arccos
sin
cosc os
atan2
coss in
, 
sin
sin
.. (2)
As shown in Fig. 2, the position vector of the 
tool point P at the end of the mechanism in the static 
platform coordinate system O – xyz is:
 r
PP PP
xyz  
T
. (3)
The following vector equation can be obtained:
 rb wa w
Pi ii i
qe i   123 ,,, (4)
where a
i
, b
i
 is the position vector of A
i
, B
i
; a
i
 = Ra
i0
; 
b
ii i
b   coss in  0
T
; a
i0
 is the measurement of
 
A
i 
in the connected coordinate system of moving 
platform, a
ii i
a
0
0   coss in 
T
; φ
i
 is the 
structural angle of ′ OA
i
 
 and OB
i
 
 relative to x' axis 
and x axis respectively,  
i
i    21 32 ; q
i
 and 
w
i
 are the length of the branch chain and the unit 
vector, respectively.
Since the revolute joint will restrict the branch 
chain from moving along the x
i
 axis, dot product x 
axis direction unit vector u
ix ii
coss in =
T
 0  at 
both ends of Eq. (4):
 ra wu
Pi ix
e   
T
0. (5)
Solve Eq. (5) to obtain
 
x
a
e
y
a
e
P
P
    
    
2
21
2
21
sinc os sins in ,
cosc os coss
 
  i in ,
.

  (6)
From Eq. (6)
 
qe
q
i
iPii Pi i
i
Pi i
i
==
= 
 ,
rbawrbs
w
rbs
 







123 ,,
 (7)
where s
i
 is the vector of the geometric centre of 
the spherical pairs pointing to the tool reference point 
in the static coordinate system, and s is the vector of 
the geometric centre of the spherical pairs pointing 
to the tool reference point in the moving coordinate 
system. 
 sR Rss
ii
ae   ,,
T
0 (8)
 
 R
i

   
   








coss in
sinc os
 
 
ii
ii
22 0
22 0
00 1
 


,
,,. i 123
 (9)
2  WORKSPACE ANALYSIS 
The constraints of the 1T2R head include its structure, 
scale parameters (a, b, l, e), motion range of each 
prismatic pair and interference. Workspace analysis 

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)9, 560-570
563 Limit-protection Method for the Workspace of a Parallel Power Head 
is used to determine the set of all reachable space 
position points of the tool reference point under the 
above constraints.
2.1  Constraint Analysis
The 1T2R power head is restricted by its own 
mechanical structure and other conditions and can only 
work within a certain space range. After analysing the 
structure of the 1T2R head, the constraint conditions 
are as listed in Table 1.
Table 1.  Restrictions of 1T2R mechanism
Constraint type Constraint value
Active prismatic pair length constraint 0.4 m ≤ q
i
 ≤0.915 m
Rotation angle constraint of revolute joint –12° ≤ θ
i
 ≤ 3°
Angle constraint of spherical joint |α
i
| ≤ 45°, |β
i
| ≤ 90°
Interference between principal axis and 
branched chain
δ
i
 ≥ 0.15 m
Interference between tool point and table z
P
 ≤ 1.250 m 
q
i min
 and q
i max
 are the maximum and minimum 
lengths of the active prismatic pairs, respectively; θ
i min
 
and θ
i max
 are the maximum and minimum values of 
each revolute joint angle, respectively; α
i min
 and β
i max
 
are the maximum values of angles α and β of each 
spherical joint, respectively; δ
i
 is the linear distance 
from point S of the spindle end to the branch chain; 
δ
i min
 is the minimum allowable actual linear distance 
between the spindle end and the branch chain. As 
shown in Fig. 3, δ
i
 can be obtained as follows:
 
ii BS iP i
i
el = wr wrwwb    . (10)
2.2  Workspace Description
A hierarchical processing approach can be used to 
divide the workspace into multiple subspaces. The 
boundary region of the subspace is then determined by 
means of the quasi-spherical coordinate search method 
[27], and the envelope surface and the stereogram of 
the workspace are described ，defining the accessible 
working space for 1T2R heads as shown in Fig. 4. It 
can be seen from the calculation results that 1T2R 
head can realize the attitude space range with a 
maximum of θ ∈ [0° 40°], ψ ∈ [0° 360°]. As we 
can see from the reachable workspace, not all regions 
can achieve the maximum nutation angle.
Fig. 4.  Reachable workspace of 1T2R
3  POSE DETERMINATION  
AND SIMPLIFICATION ALGORITHM 
By analysing and summarizing the motion rules of 
1T2R head, a simplified algorithm for judging the 
position and pose of 1T2R parallel power head is 
derived. This algorithm can realize the fast response 
of the limit protection function under the existing 
hardware configuration and is used to realize the limit 
protection of the 1T2R head.  
To facilitate calculation, the relationship between 
the coordinates of the tool reference point in the z and 
Fig. 3.  Interference position of 1T2R spindle head

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)9, 560-570
564 Ni, Y. – Lu, W. – Jia, S. – Lu, C. – Zhang, L. – Wen, Y.
the coordinates z
O'
 of the geometric centre point of the 
moving platform in the direction of z is:
 zz e
O


cos.  (11)
Thus, the input parameters of the inverse 
kinematics model of the 1T2R head are replaced by 
the coordinates of the geometric centre point O' of the 
moving platform in the z direction, the process angle 
ψ and the nutation angle θ.
Q is defined as the kinematic chain length 
of the branch chain movement; Q' is the difference 
between the kinematic chain lengths of any two 
branch chains.
3.1  Position Constraint Condition
As shown in Fig. 2, let the vector from the geometric 
centre point O of the static platform to the geometric 
centre point O' of the moving platform be o. r
P
 can be 
expressed as
 row.
P
e  (12)
Substituting Eq. (12) into Eq. (3), we obtain:
 qi
ii ii
wabo   123 ,,. (13)
By summing the above equation in order of i, 
and taking into account the geometric relation of the 
prototype a
1
 + a
2
 + a
3
 = b
1
 + b
2
 + b
3
 = 0, we obtain:
 qq q
11 22 33
3 wwwo ++ = . (14)
According to Eq. (13), the left side of the above 
equation is equal to a
1
 + a
2
 + a
3
 = b
1
 + b
2
 + b
3
 = 0; 
therefore:
 
qq q
11 22 33
11 22 33
3
www
oababa b
++
. (15)
Because |a
1
 – b
1
| + |a
2
 – b
2
| + |a
3
 – b
3
| has a 
maximum value of 6a sin(θ
max
 / 2) at the maximum 
nutation angle, the following can be obtained by 
sorting:
ab ab ab
11 22 33
62 

 
max
max
sin. a  (16)
By combining Eq. (15) and Eq. (15), and 
substituting |q
1
w
1
| + |q
2
w
2
| + |q
3
w
3
| = Q, the following 
is obtained:
 33 62 oo    Qa sin.
max
 (17)
It can be seen from Eq. (17) that, theoretically, 
the sum of kinematic chains of the 1T2R power head 
branch prismatic pair is always approximately three 
times the distance from the geometric centre point of 
the static platform to the geometric centre point of the 
moving platform.
3.2  Pose Constraint Condition
A similar calculation is used for the difference of 
length between any two branch chains.  The length 
difference between branched chains 1 and 2 is used 
as an example, which can be determined from Eq. 
(13):
 qq
11 22 1212
ww aabb   . (18)
The difference between the two sides of a triangle 
is less than the third:
 qq qq
11 22 11 22
ww ww   . (19)
Substitute Eq. (23) into, and we obtain:
 qq
11 22 1212
ww aabb   , (20)
because a
1
= aRV
1
, a
2
 = aRV
2
, b
1
= bV
1
, b
2
 = bV
2
, V
i
 = 
[cos φ
i
 sin φ
i
 0]
T
, i = 1, 2.
According to the actual parameters of the 
mechanism, a = b is substituted, and then:
    
aabb
RE TR E
1212
12
21

   
 aa () cos,  (21)
where, E is identity matrix, 
T = [cos φ
1
 – cos φ
2 
sin φ
1
 – sin φ
2 
0]
T
.
According to Eq. (1), the matrix R is related 
to the precession angle ψ, nutation angle θ and spin 
angle ϕ, and further:
 aabb
1212
21 23      af cos, .   (22)
Substituting Eq. (22) into Eq. (21), we obtain:
    qq af
11 22
21 23 ww      cos, .   (23)
According to Eq. (26), we then obtain:
      Qa f 21 23 cos, .   (24)
It can be seen from Eq. (24) that, theoretically, the 
upper bound of the length difference between any two 
branching chains of the 1T2R head is a function of ψ
 
and θ. 

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)9, 560-570
565 Limit-protection Method for the Workspace of a Parallel Power Head 
3.3  Simulation
The form of motion in which the tool makes an arc 
trajectory parallel to the plane of the static platform is 
relatively simple and easy to describe. Therefore, 
under this motion, the above conclusions are verified 
by combining the existing 1T2R parallel mechanism 
data. When the values of z
O'
 and θ remain unchanged 
at ψ ∈ [0° 360°], the length of 1T2R head branch 
chain prismatic pair is calculated as shown in Fig. 5.
Since the leg length data q
1
, q
2
 and q
3
 satisfy the 
symmetric three-phase sine quantity. The values of z
O’ 
and θ can be changed and the leg length data summed 
to obtain Table 2. The difference of the leg length data 
is taken to obtain Table 3.
Table 2.  Summation of 1T2R head branches’ length
θ = 39 θ = 29º θ = 19º θ = 9º θ = 0º max/min
1.8720
Q
max
1.879424 1.874194 1.872384 1.872018 1.872000 1.879424
Q
min
1.876457 1.873482 1.872296 1.872016 1.872000 1.872000
1.9695
Q
max
1.976447 1.971561 1.969863 1.969517 1.969500 1.976447
Q
min
1.973780 1.970920 1.969783 1.969515 1.969500 1.969500
2.0670
Q
max
2.073528 2.068943 2.067343 2.067016 2.067000 2.073528
Q
min
2.071116 2.068363 2.067271 2.067015 2.067000 2.067000
2.1645
Q
max
2.170656 2.166338 2.164826 2.164516 2.164500 2.170656
Q
min
2.168465 2.165811 2.164760 2.164514 2.164500 2.164500
2.2620
Q
max
2.267824 2.263744 2.262310 2.262015 2.262000 2.267824
Q
min
2.265824 2.263262 2.262250 2.262013 2.262000 2.262000
Table 3.  Subtraction of any two 1T2R head branches’ length
z
O'
θ = 39 θ = 29º θ = 19º θ = 9º θ = 0º
0.6240
Q'
max
0.271960 0.209725 0.140941 0.067737 0
Q'
min
-0.271960 -0.209725 -0.140941 -0.067737 0
0.6565
Q'
max
0.272005 0.209745 0.140944 0.067737 0
Q'
min
-0.272005 -0.209745 -0.140944 -0.067737 0
0.6890
Q'
max
0.272056 0.209763 0.140947 0.067737 0
Q'
min
-0.272056 -0.209763 -0.140947 -0.067737 0
0.7215
Q'
max
0.272098 0.209779 0.140950 0.067737 0
Q'
min
-0.272098 -0.209779 -0.140950 -0.067737 0
0.7540
Q'
max
0.272134 0.209793 0.140952 0.067737 0
Q'
min
-0.272134 -0.209793 -0.140952 -0.067737 0
Average value
Q'
max
0.272051 0.209761 0.140947 0.067737 0
Q'
min
-0.272051 -0.209761 -0.140947 -0.067737 0
Fig. 5.  Data distribution type of 1T2R branches’ length
According to the data of Fig. 5 and Table 2, the 
following relationship can be fit:
 QQQ ≤≤ , (25)
where, Q represents the lower bound of the length 
sum of the branch chain movement, Qz
O


3 ,
Q represents the upper bound of the length sum of 
the branch chain movement
 Qz z
OO
 

30 012251 0 015190 .. .

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)9, 560-570
566 Ni, Y. – Lu, W. – Jia, S. – Lu, C. – Zhang, L. – Wen, Y.
From the data in Table 2 and the Eq. (25), it can 
be seen that the length of the 1T2R head branch chain 
prismatic pair is always approximately equal to three 
times of z
O’
, which satisfies the Eq. (21). Therefore, 
the motion characteristic law of this form of motion 
can be extended.
By the same token, through Fig. 5 and Table 3, Q' 
and θ can be fitted as follows:
    QQ , (26)
where, ′ Q represents the upper bound of the length 
difference |Q'| of the auxiliary leg of the branch chain 
movement,  Q 0 006997 0 003748 ..  . 
It can be seen from Eq. (26) that in the case of 
the same nutation angle, the upper bound of the 
difference between the length of any two kinematic 
chains is almost the same, with only an error of orders 
of magnitude. This satisfies Eq. (24). Therefore, the 
motion characteristic law of this form of motion can 
be extended.
The above is the analysis of the motion 
characteristics of a 1T2R head and the simplified 
algorithm of pose judgment. The aim of the algorithm 
is to further divide the working space of the mechanism 
through the information of the three branch chains, so 
that the judgment of the attitude of the mechanism 
can be obtained without establishing a complex 
mapping model of the servo motor action. Within its 
workspace, the sum length of the 1T2R head branch 
chain prismatic pair is always approximately equal 
to three times the coordinate value of the geometric 
centre of the moving platform. The absolute value of 
the difference between the lengths of any two branch 
chains is always less than a specific value related to 
the nutation angle. Thus, the limit protection of 1T2R 
head can be realized simply, reliably, and efficiently.
4  LIMIT PROTECTION IMPLEMENTATION PROCESS 
Based on the above analysis results of the movement 
characteristics of a 1T2R power head, the simplified 
algorithm of position and pose judgment, the limit 
protection method is designed in combination with 
the actual mechanical structure of the 1T2R power 
head servo feed system and its control system. This 
method will adopt two methods: proactive limit and 
preventive limit. 
The active limits are all areas of the 1T2R power 
head dexterous workspace except the boundary. The 
preventive limits are to prevent the mechanism from 
exceeding the workspace boundary, i.e., the boundary 
area of the 1T2R power head dexterous workspace. 
Real-time data of the prismatic pair length of the 
branch chain can be obtained through the feedback 
signal of the encoder of the servo motor. The PLC 
program in the motion controller PMAC is used 
to sum the leg lengths of the three branches and 
calculate the difference between the lengths of any 
two branches. The results of the operation are fed 
into the motion controller register. Then, according 
to the above pose judgment algorithm, whether the 
limit area is exceeded can be determined. The specific 
realization method of the 1T2R power head limit is 
shown in Fig. 6.
Fig. 6.  Schematic diagram of limit protection process
For the active limit, according to Table 2, 
Q
min
 = 1.872000 is determined to be the lower 
trigger condition value of the active limit, and 
Q
max
 = 2.267824 is determined to be the upper trigger 
condition value of the active limit. For the preventive 
limit position, according to the data shown in Table 
3, it is determined that Q'
max
 = 1.272134 is the upper 
limit trigger condition value of the preventive limit 
position, and Q'
min
 = –0.272134 is the lower limit 
trigger condition value of the preventive limit position.

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)9, 560-570
567 Limit-protection Method for the Workspace of a Parallel Power Head 
Fig. 8.  Servo feed control system
Fig. 9.  Schematic diagram of trajectory reachable workspace
Fig. 11a shows the numerical distribution law after 
the summation of the length data of the motion track. 
It can be seen from the distribution of the summation 
result that the sum of the length is distributed within 
the interval range of 2.2514 m to 2.2521 m and has 
periodic data fluctuation with a small amplitude. As 
shown in Fig. 11b, in comparing the length of the data 
points and the result data with the threshold value of 
the over-limit setting, it is found that none of the data 
points in the trajectory exceed the limit.
5  EXPERIMENTAL ANALYSIS
The experimental platform is shown in Fig. 7, and the 
1T2R parallel power head is driven by three chains. 
Fig. 8 shows the servo feed control system. The 
real-time length of three branches can be obtained 
through the code disk of the branch servo motor. 
The trajectory is an approximately circular trajectory 
parallel to the plane of the static platform, as shown 
in Fig. 9. According to the scale parameters (a, b, l, 
e) mentioned above, after consulting the machine 
tool operation manual, it is found that collision 
interference occurs easily when the nutation angle is 
40°, so the nutation angle θ is selected as 30°. The 
motion position of the tool reference point P at the 
end of the mechanism in the z direction is 1.1687 m, 
the nutation angle θ is 30°, and the precession angle 
ψ is varied from 0° to 360°. The length data of each 
chain obtained during the experiment are shown in 
Table 4. Fig. 10 shows the variation rule of the length 
of the branch chain corresponding to the trajectory.
Fig. 7.  1T2R parallel power head
Table 4.  Experimental kinematic chain length data
t [s] 0.0000 0.0221 0.0443 0.0664 0.0885 0.1107 0.1328 0.1549
q
1
 [m] 0.6269 0.627 0.627 0.627 0.627 0.627 0.627 0.6271
q
2
 [m] 0.8121 0.8121 0.8118 0.8114 0.811 0.8106 0.8101 0.8097
q
3
 [m] 0.8123 0.8128 0.8132 0.8136 0.814 0.8144 0.8149 0.8153
t [s] 0.1771 0.1992 0.2214 0.2435 0.2656 0.2878 0.3099 …
q
1
 [m] 0.6271 0.6271 0.6271 0.6271 0.6271 0.6272 0.6272 …
q
2
 [m] 0.8093 0.8089 0.8084 0.8080 0.8076 0.8071 0.8067 …
q
3
 [m] 0.8157 0.8161 0.8165 0.8169 0.8173 0.8177 0.8181 …
t [s] 35.9922 36.0143 36.0365 36.0586 36.0807 36.1029 36.1250 36.1471
q
1
 [m] 0.627 0.627 0.627 0.627 0.627 0.627 0.627 0.627
q
2
 [m] 0.8129 0.8126 0.8125 0.8125 0.8125 0.8125 0.8125 0.8125
q
3
 [m] 0.8121 0.8124 0.8125 0.8125 0.8125 0.8125 0.8125 0.8125

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)9, 560-570
568 Ni, Y. – Lu, W. – Jia, S. – Lu, C. – Zhang, L. – Wen, Y.
Fig. 10.  Branches’ length variation law
Fig. 12a shows the numerical distribution of 
the leg length difference of any two branch chains. 
It can be seen from the distribution of the difference 
results that the leg length has a periodic fluctuation 
with a fluctuation range of 0 m to 0.2164 m, and the 
distribution form of the difference results of different 
branch leg lengths is the same except for the phase 
differences. As shown in Fig. 12b, by comparing the 
result data of the leg length difference of data points 
with the threshold value of the over-limit setting, it can 
be seen that none of the data points in the trajectory 
exceed the limit.  
Through the above simulation verification of the 
simplification algorithm of pose judgment, it is proved 
that the simplification algorithm of pose judgment 
can realize real-time judgment of the terminal pose 
state of the mechanism during the action process of 
a 1T2R power head. The result of pose judgment of 
this algorithm is found to be accurate. The active limit 
and preventive limit set according to the simplified 
algorithm of position and pose judgment can realize 
the limit protection function of 1T2R mechanism 
accurately and reliably and ensure the safe and reliable 
operation of the mechanism.
a)            b) 
Fig. 11.  Branches’ length summation distribution of data points;  
a) numerical distribution of the summation, and b) extra-limit judgment of summation
a)           b) 
Fig. 12.  Distribution of branches’ length difference; a) numerical distribution of the leg length difference,  
and b) extra-limit judgment of difference of leg length

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)9, 560-570
569 Limit-protection Method for the Workspace of a Parallel Power Head 
6  CONCLUSION 
(1) This paper presents a simplified algorithm for the 
determination of the position and pose of 1T2R 
power head. Compared with the pose judgment 
method using the inverse solution model, it is 
estimated to be 99.42 % faster with no need to 
upgrade the original system. The simplified 
algorithm is easy to implement, efficient and 
reliable.
(2) The real-time limit protection of a 1T2R power 
head can be realized simply and reliably with the 
help of the simplification algorithm of position 
and pose judgment. Based on the software and 
hardware conditions of the existing numerical 
control system, this method can effectively 
solve the real-time limit protection problem of 
parallel machine tools by combining software 
and hardware and improve the operation safety of 
such topological machine tools.
7  ACKNOWLEDGEMENTS
The authors would like to acknowledge the financial 
supported from the National Natural Science 
Foundation of China (Grant No.51575385) and 
National Key Research and Development Program 
of China (Grant No.2019YFA0709004). The authors 
also acknowledge Key Laboratory of Mechanism 
Theory and Equipment Design of Ministry of 
Education and School of Mechanical Engineering the 
in Tianjin University for providing the experimental 
environment. 
8  REFERENCES
[1] Guo, J.Z., Wang, D., Fan, R., Chen, W.Y. (2017). Design 
and workspace analysis of a 3-degree-of-freedom parallel 
swivel head with large tilting capacity. Proceedings of the 
Institution of Mechanical Engineers, Part B: Journal of 
Engineering Manufacture, vol. 231, no. 10, p. 1838-1849, 
DOI:10.1177/0954405415607782.
[2] Ni, Y., Zhou, H., Shao, C., Li, J. (2019). Research on the error 
averaging effect in a rolling guide pair. Chinese Journal of 
Mechanical Engineering, vol. 32, art. ID. 72, DOI:10.1186/
s10033-019-0386-y.
[3] Cheng, G., Gu, W., Yu, J.-l., Tang, P. (2011). Overall structure 
calibration of 3-ucr parallel manipulator based on quaternion 
method. Strojniški vestnik - Journal of Mechanical Engineering, 
vol. 57, no. 10, p. 719-729, DOI:10.5545/sv-jme.2010.167.
[4] Ni, Y., Zhang, Y., Sun, K., Wang, H., Sun, Y. (2016). Interpolation 
control algorithm for a three-RPS parallel spindle head. 
Proceedings of the Institution of Mechanical Engineers, Part 
I: Journal of Systems & Control Engineering, vol. 230, no. 7, p. 
661-671, DOI:10.1177/0959651816645687.
[5] Baizid, K., Ćukovic, S., Iqbal, J., Yousnadj, A., Chellali, R., 
Meddahi, A., Devedžić, G., Ghionea, I. (2016). IRoSim: 
Industrial robotics simulation design planning and optimization 
platform based on CAD and knowledgeware technologies. 
Robotics and Computer-Integrated Manufacturing, vol. 42, p. 
121-134, DOI:10.1016/j.rcim.2016.06.003.
[6] Kumar, V. (1992). Characterization of workspaces of parallel 
manipulators. Journal of Mechanical Design, vol. 114, no. 3, p. 
368-375, DOI:10.1115/1.2926562.
[7] Xu, P., Cheung, C.-F., Li, B., Ho, L.-T., Zhang, J.-F. (2017). 
Kinematics analysis of a hybrid manipulator for computer 
controlled ultra-precision freeform polishing. Robotics and 
Computer-Integrated Manufacturing, vol. 44, p. 44-56, 
DOI:10.1016/j.rcim.2016.08.003.
[8] Macho, E., Altuzarra, O., Amezua, E., Hernandez, A. (2009). 
Obtaining configuration space and singularity maps for parallel 
manipulators. Mechanism & Machine Theory, vol. 44, no. 11, 
p. 2110-2125, DOI:10.1016/j.mechmachtheory.2009.06.003.
[9] Huang, C.K., Tsai, K.Y. (2015). A general method to determine 
compatible orientation workspaces for different types of 6-DOF 
parallel manipulators. Mechanism & Machine Theory, vol. 85, 
no. p. 129-146, DOI:10.1016/j.mechmachtheory.2014.11.011.
[10] Zhang, D., Gao, Z. (2012). Optimal kinematic calibration of 
parallel manipulators with pseudoerror theory and cooperative 
coevolutionary network. IEEE Transactions on Industrial 
Electronics, vol. 59, no. 8, p. 3221-3231, DOI:10.1109/
TIE.2011.2166229.
[11] Cui, G., Zhang, H., Zhang, D., Xu, F. (2015). Analysis of the 
kinematic accuracy reliability of a 3-DOF parallel robot 
manipulator. International Journal of Advanced Robotic 
Systems, vol. 12, no. 2, p. 15-26, DOI:10.5772/60056.
[12] Shao, J., Chen, W., Fu, X. (2015). Position, singularity and 
workspace analysis of 3-PSR-O spatial parallel manipulator. 
Chinese Journal of Mechanical Engineering, vol. 28, p. 437-
450, DOI:10.3901/CJME.2015.0122.018.
[13] Kaloorazi, M.H.F., Masouleh, M.T., Caro, S. (2015). 
Determination of the maximal singularity-free workspace of 
3-DOF parallel mechanisms with a constructive geometric 
approach. Mechanism & Machine Theory, vol. 84, p. 25-36, 
DOI:10.1016/j.mechmachtheory.2014.10.003.
[14] Zhang, D., Wei, B. (2017). Interactions and optimizations 
analysis between stiffness and workspace of 3-UPU robotic 
mechanism. Measurement Science Review, vol. 17, no. 2, p. 
83-92, DOI:10.1515/msr-2017-0011.
[15] Gao, Z., Zhang, D. (2011). Workspace representation and 
optimization of a novel parallel mechanism with three-
degrees-of-freedom. Sustainability, vol. 3, no. 11, p. 2217-
2228, DOI:10.3390/su3112217.
[16] Zhu, C., Guan, L., Han, J., Wang, L. (2009). A new resolution 
of workspace problem of parallel machine tool. IEEE 
International Conference on Automation and Logistics, p. 
1002-1007, DOI:10.1109/ICAL.2009.5262566.
[17] FarzanehKaloorazi, M.H., Masouleh, M.T., Caro, S. (2017). 
Collision-free workspace of parallel mechanisms based on an 
interval analysis approach. Robotica, vol. 35, no. 8, p. 1747-
1760, DOI:10.1017/S0263574716000497.
[18] Majid, M.Z.A., Huang, Z., Yao, Y.L. (2000). Workspace analysis 
of a six-degrees of freedom, three-prismatic-prismatic-

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)9, 560-570
570 Ni, Y. – Lu, W. – Jia, S. – Lu, C. – Zhang, L. – Wen, Y.
spheric-revolute parallel manipulator. International Journal 
of Advanced Manufacturing Technology, vol. 16, p. 441-449, 
DOI:10.1007/s001700050176.
[19] Rouhani, E., Nategh, M.J. (2015). Workspace, dexterity 
and dimensional optimization of microhexapod. Assembly 
Automation, vol. 35, no. 4, p. 341-347, DOI:10.1108/AA-03-
2015-020.
[20] Wan, J., Yao, J., Zhang, L., Wu, H. (2018). A weighted gradient 
projection method for inverse kinematics of redundant 
manipulators considering multiple performance criteria. 
Strojniški vestnik - Journal of Mechanical Engineering, vol. 64, 
no. 7-8, p. 475-487, DOI:10.5545/sv-jme.2017.5182.
[21] Khoukhi, A., Baron, L., Balazinski, M. (2009). Constrained 
multi-objective trajectory planning of parallel kinematic 
machines. Robotics and Computer-Integrated Manufacturing, 
vol. 25, no. 4-5, p. 756-769, DOI:10.1016/j.rcim.2008.09.002.
[22] Reveles, R.D., Pamanes, G.J.A., Wenger, P. (2016). 
Trajectory planning of kinematically redundant parallel 
manipulators by using multiple working modes. Mechanism 
and Machine Theory, vol. 98, p. 216-230, DOI:10.1016/j.
mechmachtheory.2015.09.011.
[23] Dash, A.K., Chen, I.-M., Yeo, S.H., Yang, G. (2005). Workspace 
generation and planning singularity-free path for parallel 
manipulators. Mechanism and Machine Theory, vol. 40, no. 7, 
p. 776-805, DOI:10.1016/j.mechmachtheory.2005.01.001.
[24] Iqbal, J., Ullah, M.I., Khan, A.A., Irfan, M. (2015). Towards 
sophisticated control of robotic manipulators: An experimental 
study on a pseudo-industrial arm. Strojniški vestnik - Journal 
of Mechanical Engineering, vol. 61, no. 7, p. 465-470, 
DOI:10.5545/sv-jme.2015.2511.
[25] Manzoor, S., Islam, R.U., Khalid, A., Samad, A., Iqbal, J. (2014). 
An open-source multi-DOF articulated robotic educational 
platform for autonomous object manipulation. Robotics and 
Computer-Integrated Manufacturing, vol. 30, no. 3, p. 351-
362, DOI:10.1016/j.rcim.2013.11.003.
[26] Alam, W., Mehmood, A., Ali, K., Javaid, U., Alharbi, S., 
Iqbal, J. (2018). Nonlinear control of a flexible joint robotic 
manipulator with experimental validation. Strojniški vestnik - 
Journal of Mechanical Engineering, vol. 64, no. 1, p. 47-55, 
DOI:10.5545/sv-jme.2017.4786.
[27] Mahmoodi, A., Sayadi, A.. (2015). Six-Dimensional space 
expression of workspace of six-DoF parallel manipulators 
using hyper spherical coordinates (HSC). Advanced Robotics, 
vol. 29, no. 23, p. 1527-1537, DOI:10.1080/01691864.2015
.1076345.