NaVARo II: A New Parallel Robot With Eight Actuation Modes

Abstract

This article presents a new variable actuation mechanism based on the 3-RPR parallel robot. This mechanism is an evolution of the NaVARo robot, a 3-RRR parallel robot, for which the second revolute joint of the three legs is replaced by a scissor to obtain a larger working space and avoid the use of parallelograms to operate the second revolute joint. To obtain a better spatial rigidity, the leg mechanism is constructed by placing the scissors in an orthogonal plane to the displacement. Unlike the first NaVARO robot, the kinematic model is simpler because there is only one solution to the inverse kinematic model. Surfaces of singularity can be calculated and presented in a compact form. The singularity equations are presented for a robot with a similar base and mobile platform.

Full text

Proceedings of the ASME 2018 International Design Engineering Technical Conferences &
Computers and Information in Engineering Conference
IDETC/CIE 2018
August 26-29, 2018, Qu´
ebec, Canada
DETC2018-85949
NAVARO II, A NEW PARALLEL ROBOT WITH EIGHT ACTUATION MODES
Damien Chablat
CNRS, Laboratoire des Sciences du Num´
erique de Nantes
UMR CNRS 6004, 1 rue de la No ¨
e,
44321 Nantes
Email: damien.chablat@cnrs.fr
Luc Rolland
School of Engineering and Computing
University of the West of Scotland,
Paisley, Scotland, UK
Email: luc.rolland@uws.ac.uk
ABSTRACT
This article presents a new variable actuation mechanism
based on the 3-RPR parallel robot. This mechanism is an evo-
lution of the NaVARo robot, a 3-RRR parallel robot, for which
the second revolute joint of the three legs is replaced by a scissor
to obtain a larger working space and avoid the use of parallel-
ograms to operate the second revolute joint. To obtain a better
spatial rigidity, the leg mechanism is constructed by placing the
scissors in an orthogonal plane to the displacement. Unlike the
first NaVARO robot, the kinematic model is simpler because there
is only one solution to the inverse kinematic model. Surfaces of
singularity can be calculated and presented in a compact form.
The singularity equations are presented for a robot with a similar
base and mobile platform.
INTRODUCTION
A major drawback of serial and parallel mechanisms is the
inhomogeneity of kinetostatic performance in their workspace.
For example, dexterity, accuracy and stiffness are generally poor
in the vicinity of the singularities that may appear in the work-
ing space of these mechanisms. For parallel robots, their inverse
kinematics problem often has several solutions, which can be
called “working mode” [1]. However, it is difficult to achieve
a large workspace without singularity for a given working mode.
Therefore, it is necessary to plan a change path of the working
mode to avoid parallel singularities [2, 3]. In such a case, the
initial trajectory would not be followed.
One solution to this problem is to introduce activation re-
dundancy, which involves force control algorithms [4]. Another
approach is to use the concept of joint coupling as proposed by
[5] or to select the articulation actuated in each leg in relation
to the placement of the end-effector, [6], as emphasized in this
article.
To solve this problem, a first variable actuation mechanism
(VAM) was introduced in 2008 [7], called NaVARo for Nantes
Variable Actuation Robot. This mechanism has eight actuation
modes and is based on a 3-RRR parallel robot with either the
first or second revolute joint actuated. As this mechanism has
eight solutions to the inverse kinematic model, the determination
of singularities and separation according to the current working
mode is very difficult algebraically [8]. In addition, the volume
swept by the robot’s legs is large and parallelograms reduce the
workspace. A framework has been developed to pilot the proto-
type of this robot. The main problem comes from the position
of the position sensor on the motor, which is not always con-
nected to the robot base. Additional sensors may separate as-
sembly modes, but a slight slip in the couplings may disturb the
location of the mobile platform [9].
The aim of the article is to propose a new mechanism, based
on the 3-RPR parallel robot for which singularities are easier
to calculate for all actuation modes. The outline of this article
is as follows. The first section presents the architecture of the
NaVARo II robot with its eight actuation modes. The second
section presents the study of kinematics and presents the alge-
braic equations of parallel singularities as well as the limits of
the workspace. Depending on these limits, the singular surfaces
are reduced to present only the singularities in the workspace.
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FIGURE 1. Isometric view of NaVARo II
Mechanism architecture of the NaVARo II
The VAM concept was examined in [5, 6]. They derived a
VAM from the architecture of the 3-RPR planar parallel manip-
ulator by actuating either the first revolute joint or the prismatic
joint of its legs. The same concept was introduced in [7] based
on the 3-RRR and a first prototype was created in [10].
The new 3-RPR robot concept with variable actuation is
shown in Fig. 1. The use of scissors makes it possible to limit
the space requirement during movements in contrast to previous
designs and improves rigidity. The number of scissors can be
optimized according to the possible height of the mobile plat-
form, the desired stiffness or the desired maximum length of the
equivalent prismatic joint [11–13].
This mechanism can be represented on a projection like all
3-RPR mechanisms. The pose of the mobile platform is deter-
mined by the Cartesian coordinates (x,y) of the operating point
Pexpressed in the basic frame Fband the angle α, i.e. the an-
gle between the reference frames Fband Fp(Figure 2). To il-
lustrate this article, the following dimensions have been fixed,
A1A2=A1A3=A2A3=90, B1B2=B1B3=B2B3=30, ε=π/3
and 8 ≤ρi≤59 for i=1,2,3.
A new transmission system has been developed and installed
in each branch of the NaVARo II so that the manipulator can eas-
ily switch from one actuation mode to another. As for NaVARo
I, it consists of one motor with a shaft connected to two clutches
to make the pivot connection between the base and the leg or to
make a prismatic joint to operate the scissor. Figure 3 shows a
actuation diagram of the NaVARo II. This system can be consid-
ered as a double clutch and contains: (i) an electric motor (pink),
(ii) a main shaft (pink), (iii) a base (blue), (iv) the first axis of the
leg (green), and (v) two electromechanical clutches (red) which
connects to the shaft of the first revolute joint (orange) or to the
A1A2
A3
B
B
B
1
2
3
x
y
a
y’ x’
P(x,y)
q
1
q
2
q
3
r
2
r
3
r
1
Fb
Fp
e
FIGURE 2. 3-RPR with variable actuation
Sensor 1
Sensor 2


Motor Main shaft
Base
Mobile
platform
Clutch 1
Clutch 2
Leg
Shaft
FIGURE 3. The NAVARO II transmission system
prismatic link shaft (yellow). Two position sensors give the an-
gular position of the leg relative to the base (sensor 1) and the
length of the prismatic joint via the ball screw position (sensor
2). The values of the two sensors, combined with the joint limits,
allow us to know the current assembly mode of the robot.
The actuation modes are slightly different from the NaVARo
I. Each transmission system has four actuation schemes, that are
defined thereafter:
1. None of clutches 1 and 2 are active. The main shaft can
move freely in relation to the base. In this case, neither the
pivot joint nor the prismatic joint is actuated. The leg can
move freely, i. e. θior ρiare passive, i=1,2,3.
2. Clutch 1 is active while clutch 2 is not. The first leg axis
(green) is driven by the rotation of the motor shaft. In this
case, the angle θiis active while ρiis passive, i = 1,2,3.
3. Clutch 2 is active while clutch 1 is not. The first leg
joint is free but the rotation of the motor shaft leads to a
displacement of the slider, which activates the scissor. In
this case, the θiis passive and ρiis active, i=1,2,3.
4. Both clutches 1 and 2 are active. Both joints cause a
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TABLE 1. The eight actuating modes of the 3-RRR VAM
Actuating mode number active joints
1RPR1-RPR2-RPR3θ1,θ2,θ3
2RPR1-RPR2-RPR3θ1,θ2,ρ3
3RPR1-RPR2-RPR3θ1,ρ2,θ3
4RRP
1-RPR2-RPR3ρ1,θ2,θ3
5RPR1-RPR2-RPR3θ1,ρ2,ρ3
6RRP
1-RPR2-RPR3ρ1,ρ2,θ3
7RPR1-RPR2-RPR3ρ1,θ2,ρ3
8RPR1-RPR2-RPR3ρ1,ρ2,ρ3
synchronized rotation and translation motion. The end of
the leg will make a spiral motion.
The latter actuation mode differs from the NaVARo I. Only the
second and third actuation modes are used in our study. Thus,
NaVARo II has eight actuation modes, as shown in Table 1.
For example, the first actuation mode corresponds to the 3-RPR
mechanism, also referred to as the RPR1-RPR2-RPR3mecha-
nism, since the first revolute joint (located at point Ai) of its leg
are actuated. Similarly, the eighth actuation mode corresponds
to the 3-RPR manipulator, also known as the RPR1-RPR2-RPR3
mechanism, since the prismatic joints of its leg are actuated.
Kinematic modeling of the NaVARo II
In this section, we present the kinematic model that is com-
monly used to define geometrically singular configurations, then
the constraint equations, the workspace boundaries and surfaces
that define the singularity loci.
Kinematic modeling
The velocity ˙
pof point Pcan be obtained in three different
forms, depending on which leg is traversed, namely,
˙
p=˙
θiE(bi−ai) + ˙
ρi
bi−ai
||bi−ai|| +˙
αE(p−bi)(1)
with matrix Edefined as
E=0−1
1 0 (2)
Thus, p,bi,aiare the position vectors of points P,Aiand Bi,
respectively, and ˙
αis the rate of angle α.
To compute the kinematic model, we have to eliminate the
idle joints θior ρias a function of the actuation mode. For ˙
θi, we
have to dot-multiply by
hi= (bi−ai)T(3)
and for hi=˙
ρiby
Ebi−ai
||bi−ai||.(4)
The kinematic model of the VAM can now be cast in vector
form, namely,
At =B˙
qwith t= [˙
p˙
α]Tand ˙
q= [ ˙q1˙q2˙q3]T(5)
with ˙
qithus being the vector of actuated joint rates where
˙qi=˙
θiwhen the first revolute joint is driven and ˙qi=˙
ρiwhen the
prismatic joint is driven, for i=1,2,3. Aand Bare respectively,
the direct and the inverse Jacobian matrices of the mechanism,
defined as
A=

h1h1E(p−b1)
h2h2E(p−b2)
h1h3E(p−b3)

(6)
B=diag[ρ1ρ2ρ3](7)
The geometric conditions for parallel singularities are well
known in the literature for the first and eighth actuation modes.
For the first actuation mode, it is when the lines Li, normal to
the axis (AiBi)are intersecting at one point, see Fig. 4. For the
eighth actuation mode, it is when the lines Mi, passing through
the axis (AiBi)are intersecting at one point, see Fig. 5. For the
other modes, it is just necessary to consider either the Lior Mi
lines according to the actuated joints, i.e. Liwhen the ith revolute
joint is actuated and Miwhen the ith prismatic joint is actuated.
Constraint equations
To maintain the symmetry of the robot, the position of
the end-effector is in the center of the mobile platform. The
constraint equations for all actuation modes can be written by
traversing the closed loops of the mechanism. Equations 8-11
describe the two closed loops and equations 12-13 define the po-
sition and orientation of the mobile platform.
ρ1Cθ1+30Cα−ρ2Cθ2−90 =0 (8)
ρ1Sθ1+30Sα−ρ2Sθ2=0 (9)
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A1
B1
B2
B3
A2
A3
L2
L3
L1
FIGURE 4. Example of singular configuration for the first actuation
mode when the lines L1,L2and L3intersect at one point
A1
B1B2
B3
A2
A3
M2
M3
M1
FIGURE 5. Example of singular configuration for the eighth actua-
tion mode when the lines L1,L2and L3intersect at one point
ρ1Cθ1+15(Cα−√3Sα−3)−ρ3Cθ3=0 (10)
ρ1Sθ1+15(Sα+√3Cα−3√3)−ρ3Sθ3=0 (11)
x−ρ1Cθ1−15Cα+5√3Sα=0 (12)
y−ρ1Sθ1−15Sα−5√3Cα=0 (13)
with Cα=cos(α),Sα=sin(α),Cθi=cos(θi),Sθi=sin(θi)for
i=1,2,3. To make these equations algebraic, we use a substi-
tution of all trigonometric functions as well as the square root
function with
√3=S3 and S32=3
cos(β) = Cβand sin(β) = Sβfor any angles β.
We obtain a system with 11 equations, four for loop closures,
two for the position and orientation of the mobile platform, four
FIGURE 6. Minimum and maximum lengths of the scissors
for trigonometric functions and one for the square root function
with 14 unknowns. In this case, the manipulation of equations
is not trivial and powerful algebraic tools must be used like the
Siropa library implemented in Maple [14, 15].
Workspace boundaries
If the revolute joints have no limits, the boundary of the
workspace is given by the minimum and maximum extension of
the scissors as shown in Fig. 6. The minimum value of ρiper-
mits to avoid serial singular configuration where ρi=0. Using
the constraint equations and ranges limits of prismatic joints
(8 ≤ρi≤59), we find six surface equations that describe the
boundary of the working space. These limits mean that there is
no collision between the legs and the mobile platform.
10√3(xSα−yCα) +
x2−30xCα+y2−30ySα−3181 =0 (14)
(−10Cαy+10Sα(x−90))√3+ (30x−2700)Cα+
x2+y2+30ySα−180x+4919 =0 (15)
(20Cαy+ (−20x+900)Sα−90y)√3+
x2+y2−90x−2700Cα+4919 =0 (16)
10√3(xSα−yCα) +
x2−30xCα+y2−30ySα+236 =0 (17)
(−10Cαy+10Sα(x−90))√3+ (30x−2700)Cα+
x2+y2+30ySα−180x+8336 =0 (18)
(20Cαy+ (−20x+900)Sα−90y)√3+
x2+y2−90x−2700Cα+8336 =0 (19)
One of the functions of the Siropa library is to display surfaces
that can be limited by inequality equations. The surfaces in blue,
red, green represent the minimum and maximum limits of leg
one, two and three, respectively in Fig. 7. The projections onto
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FIGURE 7. Workspace of the NaVARo II in isometric view and three
projections onto the planes (xy), (xα) and (yα)
the planes (xy), (xα) and (yα) are used to estimate the main di-
mensions of the workspace. A cylindrical algebraic decompo-
sition (CAD) can also be performed to have a partition of the
workspace for each actuation mode [16].
Singular configurations
From the constraint equations, it is possible to write the de-
terminant of matrix A. These determinants depend on the posi-
tions of the mobile platform and the positions of the passive and
active joints. Only an elimination method like Groebner’s basis
can successfully obtain the representation of singularities in the
Cartesian workspace. Note that for only the first and eighth ac-
tuation modes, the determinant of Ais factorized to form two
parallel planes (for the eighth actuation mode, an unrepresented
singularity exists for α=π). The equations of the singularities
for the eight actuation modes are given in the Appendix. As there
is only one working mode, the equations of these surfaces are
simpler than for the NAVARO I robot for which it is not possible
to simply describe these equations.
Figures 8 and 9 show all singularities for the eight actuation
modes without and with the joint limit conditions. As none of
them are superposed, it is possible to completely move through
the workspace by choosing a non singular actuation mode for
any position of the mobile platform. The same motion planning
algorithm introduced for NaVARo I can be used to select the
actuation mode able to avoid singular configurations [17].
FIGURE 8. The singularity surfaces for the eight actuation modes
FIGURE 9. The singularity surfaces for the eight actuation modes in
the robot workspace
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FIGURE 10. Singularity surfaces for actuation modes 1
FIGURE 11. Singularity surfaces for actuation modes 2
FIGURE 12. Singularity surfaces for actuation modes 3
FIGURE 13. Singularity surfaces for actuation modes 4
FIGURE 14. Singularity surfaces for actuation modes 5
FIGURE 15. Singularity surfaces for actuation modes 6
FIGURE 16. Singularity surfaces for actuation modes 7
FIGURE 17. Singularity surfaces for actuation modes 8
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Figures 10-17 represent the singularities of each actuation
mode with on the left the singularities without joint limits and on
the right the one included in the workspace. The three actuation
modes where a single prismatic joint is actuated are similar by
a rotation of 120 degrees (Figs. 11-13). Similarly, the actuation
modes where only one revolute joint actuated are also similar
(Figs. 14-16).
Conclusions
In this article, a new version of the NaVARo robot was in-
troduced. Thanks to the change of actuation mode, the entire
Cartesian workspace can be used. By eliminating the parallelo-
grams that allowed the first NaVARO robot to have an actuation
on the second pivot joint, the Cartesian workspace is larger. In
addition, it is possible to place sensors on both actuated joints
of each leg to locate the mobile platform when we solve the di-
rect kinematic problem. The use of scissors makes it possible
to have a greater rigidity in the transverse direction of the robot
movement as well as a variation of displacement which can be in-
creased according to the number of bars. Unlike the NaVARo I,
which is based on a 3-RRR robot, the NaVARo II is based on the
architecture of the 3-RPR, which has only one solution with the
inverse kinematic model for any actuation mode. This property
allows a complete writing of singularity equations whereas for
the robot 3-RRR, in the literature, these equations can be written
only for a given orientation of the mobile platform. Future works
will be carried out to evaluate the stiffness of the robot based on
the size and the number of scissors and the number of solutions
to the direct kinematic model to determine if there are actuation
modes for which the robot is cuspidal.
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Appendix
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Actuation mode 1
−3(−30√3y+x2+y2+1800Cα−90x−300)(Cα−1/3) = 0 (20)
Actuation mode 2
((−240x+5400)yC2
α+ ((120x2−120y2−5400x)Sα+360y(x−50))Cα+
(360y2+1800x)Sα+120y(x+45))√3+ (−120x2+120y2+16200x)C2
α+
((−240x+16200)ySα+ (−4x+180)y2−
4x3+540x2−18000x)Cα−4y(x2+y2−90x+8550)Sα−60x2−180y2−5400x=0 (21)
Actuation mode 3
(−160y(x+45)C2
α+ ((80x2−80y2+7200x−648000)Sα−4y(x2+y2−180x−600))Cα+
(4x3+4xy2−720x2+30000x+108000)Sα+80y(x−90))√3+
(−240x2+240y2+21600x)C2
α+ (−480y(x−45)Sα−4(x−90)(x2+y2−180x−600))Cα−
4y(x2+y2−180x+7500)Sα+120x2−120y2−21600x+972000 =0 (22)
Actuation mode 4
(−160y(x−135)C2
α+ ((80x2−80y2−21600x+648000)Sα+4x2y+4y3−34800y)Cα+
(−4x3+360x2+ (−4y2+2400)x+360y2−108000)Sα+80xy)√3+ (240x2−240y2−21600x)C2
α+
(480y(x−45)Sα−4x3−4xy2+34800x)Cα+ (−4x2y−4y3+2400y)Sα−120x2+120y2=0 (23)
Actuation mode 5
((1680xy −70200y)C2
α+ ((−840x2+840y2+70200x)Sα−12x2y−12y3−2100y)Cα+
(12x3+12xy2−95100x)Sα−840xy +32400y)√3+ (840x2−840y2−210600x+6804000)C2
α+
((1680xy −210600y)Sα−36x3+3240x2+ (−36y2−6300)x+3240y2+1323000)Cα−
36y(x2+y2−7925)Sα+180x2+1020y2+97200x−6902000 =0 (24)
Actuation mode 6
((80x2−80y2−7200x+324000)C2
α+ (160y(x−45)Sα+360y2−54000)Cα−
360y(x−45)Sα−60x2+20y2+5400x−156000)√3−
4y(x2+y2−90x+8100)Cα+4(x−45)(x2+y2−90x)Sα+5400y=0 (25)
Actuation mode 7
(480y(x−45)C2
α+ ((−240x2+240y2+21600x)Sα+4y(x2+y2−180x+8100))Cα−
4(x−90)(x2+y2−180x)Sα−240y(x−45))√3+ (1944000 −240x2+240y2−21600x)C2
α+
(−480y(x+45)Sα−324000 −12x3+2160x2+ (−12y2−97200)x)Cα−12y(x2+y2−180x)Sα−
240y2+32400x−936000 =0 (26)
Actuation mode 8
−Sα(30√3y−x2−y2−1800Cα+90x+300) = 0 (27)
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