Locomotion Optimization and Manipulation Planning of a Tetrahedron-Based Mobile Mechanism with Binary Control
Liu et al. Chin. J. Mech. Eng.
Locomotion Optimization and Manipulation Planning of a Tetrahedron-Based Mobile Mechanism with Binary Control
Ran Liu 0
Yan‑An Yao 0
Wan Ding 2
Xiao‑Ping Liu 0 1
0 School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University , Beijing 100044 , China
1 Department of Systems and Computer Engineering, Carleton University , Ottawa, ON K1S 5B6 , Canada
2 Department of Mechanism Theory and Dynamics of Machines, RWTH Aachen University , 52072 Aachen , Germany
Locomotion and manipulation optimization is essential for the performance of tetrahedron‑ based mobile mechanism. Most of current optimization methods are constrained to the continuous actuated system with limited degree of freedom (DOF), which is infeasible to the optimization of binary control multi‑ DOF system. A novel optimization method using for the locomotion and manipulation of an 18 DOFs tetrahedron‑ based mechanism called 5‑ TET is proposed. The optimization objective is to realize the required locomotion by executing the least number of struts. Binary control strategy is adopted, and forward kinematic and tipping dynamic analyses are performed, respectively. Based on a developed genetic algorithm (GA), the optimal number of alternative struts between two adjacent steps is obtained as 5. Finally, a potential manipulation function is proposed, and the energy consumption comparison between optimal 5‑ TET and the traditional wheeled robot is carried out. The presented locomotion optimization and manipulation planning enrich the research of tetrahedron‑ based mechanisms and provide the instruction to the successive locomotion and operation planning of multi‑ DOF mechanisms.
Tetrahedron‑ based mobile mechanism; Binary control; GA; Locomotion optimization; Manipulation planning
Terrestrial mobile robots mainly include wheeled,
tracked, legged, hybrid, snake-like, and spherical robots
], which have different adaptability and locomotion
modes. Different from conventional robots that actuated
by motors inside to hold a constant shape, some
mechanisms realize locomotion and manipulation functions
by shifting shapes with the change of the linkages .
In recent years, some linkage structures constructed by
basic or special geometry configurations, such as
Euclidean polyhedron, have been proposed. Conformable
tetrahedrons are the simplest space-filling form in the same
way triangles are the simplest plane-filling facets [
tetrahedron-based mobile mechanism has been
Tetrahedron-based mobile mechanism is a class of
hyper-redundant robots, which has multiple DOFs and
kinematic redundancy [
]. Different from wheeled,
tracked, and legged robots, hyper-redundant
mechanisms arise from internally induced deforming. Since the
advantage of fault tolerance [
], it is superior for
operation in highly constrained environments, for instance, in
uneven terrain with some simple operation tasks, such
as nuclear reactor cores, toxic abandoned factories, and
many other conceivable environments [
Several tetrahedron-based robots have been designed,
and the emphasis mainly on building novel structures
to produce feasible gait patterns. Based on the
mathematical models of 4-TET, 8-TET, 12-TET Walkers
and tetrahedron worm built by Abrahantes et al. [
the choreographed gaits of these robots were designed
according to the geometric relationships of the struts.
For a novel tensegrity duct robot constructed with two
linked tetrahedrons [
], the main focus is also on the
design of climbing gaits. Through the structure design
and kinematic analysis of a steering crawling tetrahedron
robot which links a pushing element on one of its four
], the slope crawling gait was presented.
However, the research on successive path planning of such
linkage-based mobile mechanism is relatively rare. And
its successive motion much depends on complex control
system. A light source tracking 1-TET designed by Yu
and Nagpal [
] realize its rolling motion by control the
self-adapting system contains amounts of sensors. And a
spine tensegrity robot simulated its successive rolling by
using central pattern generators (CPGs) [
Since the large redundancy of multi-DOF robots, the
key issue of the realization of successive locomotion lies
on the inverse kinematics, which is a nonlinear
problem that has multiple solutions [
]. A two-linked
tetrahedron robot adopted force density method to solve
the inverse kinematics [
] for the minimum of elastic
potential energy contained. In Ref. [
], a GA was used
to solve the inverse kinematics of a redundant robot,
which mainly focuses on finding the best solution among
multiple solutions with the objective of minimum joint
Tetrahedron-based robots are always over-constrained,
which requires to embed large extension ratio actuators
to acquire large deformation and high mobility [
there are few actuators meet this special requirement,
researchers have explored using ordinary extension ratio
actuators to design modular reconfigurable robotic
systems with high environmental adaptability [
of the modular reconfigurable robots called Tetrobot 
that can be reconstructed as tetrahedron module,
octahedral module, six-legged walker, and tetrahedron-based
manipulator. Each of these four constructions can meet
the respectively given tasks. Another modular
reconfigurable robot [
] has three reconfigurable constructions,
including an adaptive gripper, a modular tetrahedron
robot, and a modular hexahedron robot. However, the
three constructions can only meet its corresponding task.
If there were only one construction that could meet more
than one task, it would be very desirable [
In previous work, we reported a closed-loop
mechanism consisted of two tetrahedron units with two DOFs
that could realize rolling locomotion on the ground
]. And a pneumatic driving tetrahedron
mechanism with multi-DOF was proposed [
]. In this paper,
we focused mainly on the locomotion optimization
and manipulation planning of the tetrahedron-based
mobile mechanism constructed as 5-TET. A binary
control strategy [
] using binary actuators is
adopted to simplify the control system. A GA based on
binary codes is developed to optimize the locomotion
of 5-TET by altering minimum actuators, which further
simplifies the control of the mechanism. Double
control simplification provides a reference for multi-DOF
inputs combination strategy.
This paper is organized as follows. The structure and
locomotion of 5-TET are described in Section 2.
Section 3 illustrates the optimal locomotion based on
kinematic and dynamic analyses. Section 4 presents a
potential operation function of 5-TET. Energy
consumption is analyzed Section 5. And the optimal locomotion
paths and manipulation gaits are simulated Section 6.
2 Structure and Locomotion Description
2.1 Structure Description
Geometrically, 5-TET is comprised of 18 identical struts
and 8 nodes, which is a fully symmetrical spatial
geometry with isotropy feature. Mechanically, substituting
prismatic joints for each of the 18 struts and using spherical
joints connect the struts end to end, which make up the
5-TET mobile mechanism with 18 DOFs, corresponding
to each of the 18 prismatic actuators (see in Figure 1).
2.2 Binary Control and System Implementation
Binary control is one of the control concept applied to
multi-DOF mechanisms [
]. In this concept, tens or
hundreds of binary actuators are embedded in a
structure, which is analogous to the digital computer
replacing the analog computer. These digital mechanisms can
perform precise, discrete motions without the need of
sensing, complex electronics or feedbacks, so the control
elements of such devices are simple [
]. Since the binary
control only has signals of 0 and 1 that correspond to off
and on, the control operation of such devices is also
Taking the advantages of quick response, large
strength-weight ratio, and lower control complexity,
pneumatic cylinders are selected as the binary actuators
for 5-TET. Pneumatic cylinder has only ‘min/max’ length,
which correspond to the binary codes of ‘0/1’. However,
since the binary behavior, the feasible configurations of
binary mechanisms would decrease. To remedy this
limitation, it is preferred to assemble several modules in the
architecture to realize a quasi-continuous mobility [
For mobile robot, the integration design concept is
a good choice. Since the embedding of pneumatic
elements, the implementation of 5-TET calls for the
pneumatic power supply, which is a dominant bottleneck
in practical of pneumatic mobile applications. Many
researchers have been studied on this problem, and
new advances in portable pneumatic power source are
]. For example, a portable pneumatic
supply called Dry Ice Power Cell, which could provide
sustaining 0.42 MPa gas for about 218 NL with portable
]. The achievements from these researches can be
used directly to equip 5-TET to realize mobile function,
so do other elements.
According to the parameters of existing pneumatic
actuators with the consideration of the global dimension
of 5-TET, choose the specifications of the pneumatic
cylinders in Table 1. Each actuator weighs 0.5 kg. The two
extreme states are 360 mm and 460 mm. Here, the stroke
is chosen as small as possible to verify the 5-TET is
independent of large extension ratio. Accordingly, the nodes
are designed with the radius of 30 mm, whose mass
center locates in its geometry center and equals to 0.3 kg.
The implemental system of integration design concept
mainly contains the 5-TET mechanism, the pneumatic
system (including a pneumatic power supply, 18
electric solenoid valves and some pneumatic tube), a control
unit with battery, a wireless unit and a remote computer.
5-TET is an executive mechanism that activated by
electronic solenoid valves when the control unit receives the
optimal binary codes from the computer by the wireless
For the arrangement of these elements, the portable
pneumatic supply would be suspended in the basic
tetrahedron A-BCD from nodes A to D by spring with
appropriate stiffness that could absorb shock in the rolling.
And the valves, control units and other elements are
integrated to the other four nodes of E to H. However, this
leads to the position change of the center of mass (CM).
Thus counterweight is used to be added on the nodes to
stay the initial CM locates in the center of 5-TET. Then
connect the system with pneumatic tube in neat
formation. Figure 2 shows the integration design concept of
2.3 Locomotion Feasibility of 5T‑ET
In the condition of ground support, as described in
Figure 3, 5-TET remains in three-point support as ΔDEG.
Through controlling the corresponding pneumatic
cylinders, 5-TET can roll in three possible directions on
condition that the projection of the CM statically or the
Zero Moment Point (ZMP) dynamically of the
mechanism locates beyond the support area. In Figure 3, 5-TET
has three rolling directions with two types of rolling axes
from Direction I to III. Obviously Direction I and II have
the symmetric motion, which are grouped. Further, with
Electric solenoid valves
Pneumatic power supply
successive and logical planning on cooperates of the
actuators, 5-TET can realize omnidirectional rolling.
Note that since the mass of telescopic strut is uniform
and all the surfaces have the same friction coefficient,
the two nodes of the tipping axis bear the same
friction. Thus, if the length of tipping axis changes, it would
be stretched symmetrically with respect to the center of
the pneumatic cylinder. For example, based on the initial
state of 5-TET, if the tipping axis is DE, and the tipping
would happen with the extension of strut DE.
Considering that nodes D and E bear the same friction, strut DE
will be extended symmetrically with respect to its own
center, as described in Figure 4. This feature has a great
significance in the locomotion of 5-TET.
For 5-TET, in any static stable state, the support
triangle surface is composed of three of the eight nodes. And
one of the three nodes is chosen from four nodes of the
basic tetrahedron A-BCD, while the other two are chosen
from the four external nodes E, F, G and H. But during
the moving, the support area of 5-TET is always
changing, which deeply varies the kinematic analysis.
Fortunately, 5-TET is a fully symmetric configuration that
has isotropic feature [
], so that it has identical
structures with different support nodes. Using this significant
feature, the complexity in kinematic analysis would be
3 Motion Gait Optimization and Optimal Path
For mobile mechanism, it is meaningful to find the
optimal input of the required locomotion. To improve the
efficiency and accuracy of the optimization, an analytical
kinematic model based on geometric relationship rather
than coordinate transformation was built. Then based on
the tipping dynamic analysis, the motion gaits of a single
step and successive locomotion were optimized.
3.1 Geometric‑Based Forward Kinematics
For tetrahedron-based mechanism, it is complicated
and time-consuming to analyze the forward kinematics
with coordinate transformation, which directly affects
the efficiency of the locomotion optimization. However,
based on a fixed coordinate system, the analytical
solution could be quickly obtained by the proper use of the
geometrical relationship of the mechanism.
For 5-TET, the fixed Cartesian coordinates O-XYZ
is built. See in Figure 5, node D is chosen to be the
origin, axis Y is along the direction of vector DB, axis Z is
perpendicular to the surface of ΔBCD just points to
the center of tetrahedron A-BCD, and the XOY plane is
coplanar with ΔBCD. The lengths of all the 18 pneumatic
cylinders are expressed as a matrix with 18 lines and 1
column as Length = [AD, AC, AB, DC, BC, DB, ED, EC,
EA, GD, GC, GB, HD, HA, HB, FA, FC, FB]T = [a1, b1, c1,
a2, b2, c2, a3, b3, c3, a4, b4, c4, a5, b5, c5, a6, b6, c6]T.
In this coordinates, the positions of all nodes of 5-TET
could be obtained by geometric calculation. It is obvious
that for the basic tetrahedron A-BCD, three out of the
four nodes can be positioned easily as follows:
BOP= BOx, BOy, BOz
= (0, c2, 0)T,
DOx, DOy, DOz
= (0, 0, 0)T.
The rest is node A. According to geometric thinking,
a new position analysis method based on the relations
between sides and angles is derived to analyze the forward
Figure 4 Extension principle of the struts
kinematics. For example, the position coordinates of node
A is wanted, and the positions of nodes B, C, D and the
lengths of sides AB, AC, AD are all known, then the key
could be solved by a ternary quadratic group, as Eq. (4)
shown. And the results are described in Eq. (5).
(Ox − DOx)2 + (Oy − DOy)2 + (Oz − DOz)2 = a2,
(AAOx − COx)2 + (AAOy − COy)2 + (AAOz − COz)2 = b121, (4)
(AOx − BOx)2 + (AOy − BOy)2 + (Oz − BOz)2 = c12,
a21 −AO x2 −AO y2
Similarly with node A, node H has special geometric
relationships with nodes A, B and D, which make the
position coordinates of node H be solved by another
ternary quadratic group, as Eq. (6) shown.
(OH x − DOx)2 + (OH y − DOy)2 + (OH z − DOz)2 = a2,
(OH x − BOx)2 + (OH y − BOy)2 + (OH z − BOz)2 = c52, (6)
(OH x − AOx)2 + (OH y − AOy)2 + (OH z − AOz)2 = b2.
The position coordinates of node H are obtained as
shown in Eq. (7):
− AOz·OH z − R1
= aAO25x+2cc222−c522AOx .
− 2OAOxzR2R12 + 2RR32
R2 = OAOxz22 + 1,
Where R1 = b52 − a25 − Ox2 − Oy2 − Oz2 + 2AOy · OH y,
A A A
R12AOz2 − 4R2 4O1x2 +OH y2 − a25 .
The same goes for the rest nodes E, F and G, which
could also obtain its position coordinates from the
corresponding ternary quadratic groups according to nodes A,
C, D, nodes A, B, C, and nodes B, C, D, respectively.
3.2 Dynamics of the Tipping Motion
The tipping motion of 5-TET can be divided into three
phases as before tipping, tipping, and contact with the
ground. Taking tipping over axis EG by extending struts
AD and BD as example, the tipping process is shown in
The first phase is to manipulate the controlled nodes to
generate the tipping motion. The actuating forces from
DA and DB are exerted on nodes A and B during this
phase while the base nodes are placed on the ground. The
second phase comprises a tipping motion when the base
node D leaves the ground and the new node C falls down
to the ground. And 5-TET turns into the third phase after
node C hits the ground. Assume that all the actuators
are locked in this process, in other words, 5-TET can be
regarded as a rigid body during the tipping motion.
3.2.1 Before Tipping Phase
Dynamically, the critical condition of the tipping is that
the ground support force equals to zero, which
corresponds to that the CM goes on the tipping axis in
Before tipping motion, the ground support force keeps
in positive. The free body diagram in this phase is shown
in Figure 7. By applying Euler’s moment equation, shown
in Eq. (8), with respect to the tipping axis of EG, the
relation of the nodes are obtained as Eq. (9).
Mx = I α,
mAdAy + mBdBy − mC dCy + mDdD + mF dFy + mH dHy g
− RDzdD = mA P¨ AzdAy − P¨ y dz + mB P¨ z dy − P¨ y dz
A A B B B B
+ mC P¨ Cy dCz − P¨ Cz dCy + mDP¨ DzdD
+ mF P¨Fy dFz + P¨ z dy
+ mH P¨ Hz dHy − P¨ Hy dHz ,
where RzD is the z component of RD. And the tipping
motion starts when RzDbecomes negative.
θ˙CM = P˙ CM .
equations before and after the lifting off can be expressed
ri × miP˙ i = rCM × mCMP˙ CM,
The critical tipping condition of 5-tetrahedon is shown
in Eq. (10), which reveals that the tipping motion relies
on the displacement as well as the acceleration of the
mAgdAy + mBgdBy − mC gdCy
+ mDgdD + mF gdFy + mH gdHy
1 − mA P¨ AzdAy − P¨ AydAz − mB P¨ BzdBy − P¨ y dz
dD − mC P¨ Cy dCz − P¨ Cz dCy − mDP¨ DzdD
− mF P¨Fy dFz + P¨Fz dFy − mH P¨ Hz dHy − P¨ Hy dHz
Figure 8 shows the values of RzD and PCMx in different
accelerations of actuators, which correspond to the
tipping conditions in dynamics and kinematics, respectively.
The figure expresses that the CM exceeds the tipping axis
EG before the support force RzDbecomes to zero. And the
reason of this result is that the actuating forces from DA
and DB not only react on nodes A and B, but also on D.
3.2.2 Tipping Phase
For 5-TET can be regarded as a rigid body during the
tipping motion, the system was converted to an inverted
pendulum at a pivot joint on the ground, as Figure 9
The initial angular velocity and the new dynamic
parameters can be calculated using the law of
conservation of angular momentum. If we assume that the
angular momentum is conserved during the instantaneous
time of lifting off node D, then the angular momentum
where ri denotes the position vector of the node from the
pivot point (PE + PG)/2. The initial linear velocity P˙ CM
can be gotten by rearranging Eq. (11). Then the initial
angular velocity of an equivalent system can be obtained:
The dynamic equation of this system can be expressed
as a second order nonlinear equation with initial
mCMlC2Mθ¨ + mCMglCM cos θ = 0,
lCM(0) = lCM0 , θ (0) = θCM0 , θ˙(0) = θ˙CM0 .
By setting θ˙ = ddθt = s, θ¨ = s˙ = ddθs ddθt = ddθs s, the
relation between the angular velocity and rotation angle is
obtained by differential equations in Eqs. (15), (16).
θ¨ = − lCgM cos θ ,
ds s = − g cos θ ,
sds = −
cos θ dθ ,
1 s2 = 1 θ˙2 = − g sin θ +
2 2 lCM
12 θ˙C2M0 + g
sin θCM0 .
The angular velocity and acceleration of 5-TET during
the tipping phase are shown in Figure 10.
3.2.3 Contact with the Ground Phase
In the touchdown phase, the ground is modeled as a
spring and damper system as shown in Figure 11.
Assuming that the reaction force is R, the dynamic equation of
the soft contact model is expressed in Eq. (17):
kix 0 0 δix cx 0 0 δ˙ix
R = kδ + cPδ = 0 kiy 0z δizy + 0 cy 0 δ˙˙izy ,
0 0 ki δi 0 0 cz δi
where, k is the spring matrix, c is the damping matrix,
and δi is the micro displacement in contact moment.
Reaction force is proportional to δi and δi, and it becomes
to zero when the node leaves off the ground.
According to dynamic analysis, 5-TET stays in unstable
state in the first two phases, while it is stable in contact
with the ground phase.
DE, DG and EG, respectively. Thus, 5-TET is capable of
rolling successively by binary codes.
] mechanism is a popular concept in
parallel mechanism which has great potential in
practical applications with advantages of simple and compact
constructions, easy control and low cost. Taking these
advantages into the analysis of multi-DOF 5-TET, every
step tipping is regarded as the rolling of a limited-DOF
mechanism. The objective of the optimization is defined
as executing the minimum number of struts. The motion
optimization model of 5-TET was set up.
X = [x1, x2, …, x18]Τ = [AD, AC, AB, DC, BC, BD, ED,
EC, EA, GD, GC, GB, HD, HA, HB, FA, FC, FB]Τ;
min f(X) = ∑ 18
i=1|xi – xi0|;
gi (X) = xi (xi − 1) = 0, (i = 1, 2, …, 18);
g19(X) < 0;
g20(X) < 0.
Where, design variables xi (i = 1, 2, …, 18) is the
present state of actuators that could make 5-TET roll to the
specific direction, while xi0 (i = 1, 2, …, 18) is the
former state. The constraint g19(X) < 0 is the condition that
the CM goes beyond the current support area, while
g20(X) < 0 is the constraint that the CM falls in the new
Note that, during the rolling of 5-TET, the position
coordinates of every node change constantly, which
increases the difficulty of successive motion analysis.
Here, we adopted CM projection on the support planar
instead of using coordinates transformation to simplify
the kinematic model. Thus, the 19th and 20th constraints
were derived by whether the CM projection and the rest
node locate on the same side of the tipping axis. If they
were, the cross product between vectors of the two nodes
to CM projection and the cross product between vectors
of the two nodes to the rest node would be the same sign,
otherwise opposite. The derived process is described as
g19(X) = N1CM1 × N2CM1 = (COM1 P − NO1 P) × (COM1 P − NO2 P) ,
N1N3 × N2N3 (NO3 P − NO1 P) × (NO3 P − NO2 P)
g20(X) = − NN1C1NM32′ ×× NN22CNM3′ 2 = − (CO(NOM32′ PP −− NNOO11 PP)) ×× ((CNOOM3′ P2P−−NONO2P2P)) .
Tipping time t/ s
b Value of PCMx (static)
3.3 Motion Gait Optimization for the First Step Rolling
Based on the isotropic feature of 5-TET, no matter which
support area is the initial one, the same result will be
gotten. In Figure 3, the initial support area is given as ΔDEG.
The rolling feasibility analysis infers that 5-TET could
statically tip towards all of the three directions over edge
Where CM1 is the CM projection on current support
area, CM2 is the CM projection on the new support area,
N1 is one node of the tipping axis, N2 is the other node of
the tipping axis, and N3 is the rest node of current
support area, while N3′ is the new node that would touch on
According to the requirement of the optimization
model, and considering binary thinking is also applied in
], a GA combined with binary codes is developed
to optimize the motion of 5-TET. Where, the
crossover and mutation rate are determined by the traditional
selection method trial-and-error based on the principle
of crossover is expected to be higher while mutation is
expected to be lower [
], which respectively equal to 0.2
and 0.02. According to the genome length of 18, assumed
that the population number is 500 and the maximum
generation equals to 50. In GA, the implementation of
the objective function and constraints are realized within
the fitness function,which is a raw measure of the
solution value [
]. For fitness function has no requirement
for continuity in the derivatives, virtually any cost
function can be selected [
]. Here, to get rapid calculation,
the fitness function is defined as Fit = M – f (X), M is the
maximum possible value of objective function that equals
Considering that axis DE and DG are in mirror
position, which means these two tipping directions have
isotropic feature, so that tipping in Direction I and III are
3.3.1 Tipping in Direction I Over Edge DE
The tipping step to Direction I over edge DE is optimized,
the optimal result is f (X*) = 3. There are four different
combinations, described as Code 1 to 4:
The optimal inputs combination of 5-TET is based
on the fewest number of executed pneumatic
cylinders between two adjacent steps. And fitness function is
responsible to find the combinations. The convergence
processes of the four optimal solutions are shown in
Figure 12. The comparison curves demonstrate that in
different inputs combinations, the optimal results are all
identical. And the best fitness value is 15.
Taking the instability before tipping and the
stability after tipping into consideration, Code 1 is chosen as
the representative of the optimal codes. Figure 13
elaborates the tipping process over edge DE with Code 1.
With the optimal inputs, the projection of CM on the
support plane is beyond the support area of ΔDEG, and
the mechanism tips over edge DE until node H touches
on the ground, then ΔDEH comes to be the new support
3.3.2 Relationship between Direction I and II
Considering that the initial structure of 5-TET with
support area of ΔDEG, edge DE and DG are in mirror
positions, so the pneumatic cylinders that respectively related
Figure 13 Tipping process of 5‑ TET over edge DE with optimal inputs
to DE and DG are arranged mirrored as well. Here, the
correspondence is given in Table 2.
According to Table 2, four optimal inputs combinations
of tipping to Direction II were gotten as Codes 5 to 8:
Since Code 1 is the representation of tipping in
Direction I, Code 5 is chosen to represent the tipping in
Direction II. The construction of 5-TET with Code 5 is shown
in Figure 14. The CM of the mechanism is beyond the
initial support area ΔDEG, and 5-TET will tip over axis
DG until node H hits on the ground, and the next
support area is ΔDGH.
3.3.3 Tipping in Direction III over Edge EG
The third tipping direction of 5-TET mechanism is
asymmetric with the first two. For the first step rolling by tipping
over edge EG, the optimal result is f (X*) = 2. And there are
also four different inputs combinations as follows:
Code 9 [
Code 10 [
Code 11 [
Code 12 [
The optimal result indicates that 5-TET could tip to
Direction III by actuating only two struts. The fitness
curves are described in Figure 15, and the fitness
functions are all convergent to 16. Code 10 is taken as the
representative code of tipping in Direction III, and the
tipping process is described in Figure 16. The
projection of CM on the ground is beyond the support area of
ΔDEG, then the mechanism tips over edge EG, and node
C is the new support node that would make up the new
support area of ΔCEG.
3.4 Motion Optimization for Successive Rolling
It is recognized that multi-DOF mechanisms have the
advantage of high mobile flexibility while the complexity
of the movement would simultaneously exist. Therefore,
the motion simplification and optimization are significant,
especially for successive movement. Based on binary
control, 5-TET can tip to all the three directions of the initial
support area. For isotropic feather, 5-TET can realize
successive locomotion with binary control. Assume that the
mechanism can not roll back to the last step. Thus, 5-TET
has three possible tipping directions only in the first step
rolling while it would change to two from the second step.
Based on nodes combination principle of triangle support
area (one from nodes A to D, another two from E to H)
of 5-TET, there are 24 (C41 × C42 = 24) possible support
triangles obtained by permutation and combination rules.
But the real number is only 12 for linkages interference in
actual, the possibilities are shown in Table 3.
Figure 14 Tipping process of 5‑ TET over DG with optimal inputs
b Code 5: stable state
Liu et al. Chin. J. Mech. Eng. (2018) 31:11
20 25 30 35 40 45 50
d Code 12
Figure 16 Tipping process of 5‑ TET over EG with optimal inputs
Based on the optimization of single step rolling of
5-TET, a successive moving optimization algorithm is
developed. The objective of the algorithm is to reach
the target by executing the least pneumatic cylinders
between every two adjacent steps, which requires every
step moves with limited-DOF [
]. The tipping axis of
each step depends on the intersection of the line segment
between target and CM projection on the ground.
5-TET can automatically judge the tipping direction
by the tipping rule, and the new touchdown node is also
shown automatically according to Table 3.
For the special rolling mode, the mechanism not only
has the motion of straight going, but also turning motion.
3.4.1 Straight Path
The targets used to achieve the straight rolling of the
algorithm were distributed in four quadrants as follows:
(1500 mm, 2000 mm);
(− 1500 mm, 2000 mm);
(− 1500 mm, − 2000 mm);
(1500 mm, − 2000 mm).
The optimal inputs and motion paths are obtained, as
Figure 17 shown, where the straight motion paths are expressed
by the real support triangles. And the detailed optimal inputs
are elaborated in Figure 18. In straight successive motion, the
5-TET rolls to the target with at most 5 executed pneumatic
cylinders between every two adjacent steps.
3.4.2 Turning Path
According to nodes combination principle of triangle
support area, the turning motion can be divided into two
cases. Case I turns around nodes A, B, C or D; Case II
around nodes E, F, G or H.
Case I: Based on the initial support area ΔDEG, the
turning center is node D. According to the rolling algorithm,
the targets were given with the turning radius of 100 mm:
Target 1 (0 mm, − 100 mm);
Target 2 (− 100 mm, 0 mm);
Target 3 (0 mm, 100 mm).
And the program drew the turning paths as Figure 19.
There are two modes of Case I turning. The first is a
normal path whose turning angle of a cycle is less than 360
degrees. However, the second is a special situation that
the mechanism just turns a round, which indicates that
the mechanism has position self-reset ability. The detailed
optimal inputs of case I turning are elaborated in Figure 20,
which reveals that the 5-TET turns in Case I executed at
most 5 pneumatic cylinders between each two steps.
Case II: Considering 5-TET has isotropic feature, thus,
node E and G are in a group. Here, assume node G as the
turning center. According to the demand of the
successive motion algorithm, the target points were given with
the turning radius of 200 mm, as follows:
Target 1 (200 mm, 0 mm);
Target 2 (0 mm, 200 mm);
Target 3 (− 200 mm, 0 mm);
Target 4 (0 mm, − 200 mm).
And the program drew the turning path as Figure 21
shown. The detailed optimal inputs of the turning path
are elaborated in Figure 22. Similarly with Case I, 5-TET
turns a cycle of Case II by executing at most 5 alternative
pneumatic cylinders between each two steps.
From the detailed elaboration of straight paths and
turning paths, as Figures 17, 19, and 21 shown, 5-TET
can move to the target by executing at most 5
pneumatic cylinders between every two adjacent steps,
which indicates that the 5-TET is a 18 DOFs
mechanism in total while it is a 5 DOFs mechanism in every
step of movement. This feather simplifies the inputs
control and program of the rolling, which makes the
mechanism have effective foundations for inputs
4 Optimization-Based Manipulation Planning
For 5-TET structure, there are four external nodes E, F, G
and H. And based on the support principle, there are only
two of the four external nodes supported on the ground,
while the other two TETs are free and unused. For this
structural characteristic, we consider to integrate two
separate tetrahedrons as a two-finger gripper to
manipulate the object. See in Figure 23, the support nodes D, E
and G act as the fixed base, the two flying nodes F and H
together act as the end-effector of the manipulator.
For gripper manipulators, the gripping range of the
end-effector is a major parameter, which has effects on
the capacity of gripper manipulation [
]. For 5-TET
Support nodes of A, B, C, D
Support nodes of E, F, G, H
Triangle support area
b Quadrant I
End(1500, −2000) H
d Quadrant IV
−2000 −1500 −1000 −500
c Quadrant III
mechanism, the gripping range manifests as the variable
range of the dihedral angle of H-BA-F. Here, in order to
guarantee a stable gripper, nodes F and H should be fully
symmetric with strut AB. That makes the variable range
of distance between F and H equivalent to the variable
range of the dihedral angle H-BA-F. In the initial state of
5-TET, the gripper F and H are at a certain distance of
600 mm. However, to realize the gripper action, the
dihedral angle H-BA-F should open and close based on the
initial state, which would be represented by the distance
between nodes F and H. The optimization model is set up
to get the inputs of the gripper manipulation of 5-TET.
Note that, in the optimization model, the number of state
change struts is determined as even numbers to maintain
symmetry. The optimal model of the gripper is described
X = [x1, x2, …, x18]T = [AD, AC, AB, DC, BC, BD, ED,
EC, EA, GD, GC, GB, HD, HA, HB, FA, FC, FB]T;
min f (X);
gi(X) = xi (xi −1) = 0, (i=1, 2, …, 18);
g19(X) = ∑ 1i=81|xi − xi0| = j, (i=1, 2, …, 18; j=1, 2, …, 18);
(OP − PC2MP)(EOP − PC2MP)
g20(X) = − D
(OP − GOP)(EOP − GOP)
(OP − PC2MP)(GOP − PC2MP)(DOP − PC2MP)(GOP − PC2MP)
(EOP − DOP)(GOP − DOP)(DOP − EOP)(GOP − EOP)
Where g20(X) is the constraint of the stability. And
similar with the optimal parameters of motion planning,
the crossover and mutation rates respectively equal to 0.2
and 0.02, population number is 500, and the maximum
generation is 50. The fitness function is Fit = M – f (X), M
is the maximum possible value of the objective function.
4.1 Manipulation Planning of Opening Wide
Based on the optimization model of gripper, the opening
wide manipulation optimization model is set up. Where
f (X) = − (OH x − FOx)2+(OH y − FOy)2 + (OH z − FOz)2, and
M = 600. Then the opening wide manipulation was
optimized, and the optimal result is f (X*) = 846.3602, the
number of the lengths altering of pneumatic cylinders is
4, described as Code 13: [
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 1, 1, 1, 0, 1
]. The fitness function and the stable state
mathematical model are depicted as Figure 24. The
projection of CM on the ground has been in the support area
of ΔDEG to remain the stability of the overall unit, and
the final coordinates of the CM is [–34.2946, 125.7310,
00 1 2 3
4.2 Manipulation Planning of Turning Down
For the optimization manipulation model of turning
down, f (X) = (OH x − FOx)2+(OH y − FOy)2 + (OH z − FOz)2.
Here, assume that nodes A, B, F and H are coplanar,
which means the dihedral angle H-BA-F is at the
maximum. And the longest distance between F and H can be
easily obtained, lFHmax = 846.64 mm, rounded to M =
The optimal result of turning down manipulation is
expressed as f (X*) = 110.9518. The number of
alternative pneumatic cylinders is four as well. The input code of
turning down is obtained as Code 14: [
0, 0, 1, 1, 0, 0, 0, 0,
0, 0, 0, 0, 1, 0, 0, 0, 1, 0
]. The fitness function and the
stable state mathematical model are shown in Figure 25. The
coordinates of the CM is [50.1528, 100.3049, 220.4200].
During the change of the input, the ground projection
4.3 Implementation of Gripper Manipulation
According to the two ultimate gripper states of 5-TET,
the intermediate gripper steps could be planned based
on binary control strategy as well as the gripper principle
that the symmetry of nodes F and H. Take ΔDEG as the
initial support area as well, the gripper manipulation was
planned, as Figure 26 shown. There are 11 manipulate
steps which was made up by 10 different constructions of
the binary control 5-TET, the corresponding inputs are
detailed in Figure 26(b). And, ΔDEG is the support
triangle of all the 11 steps. But, in the first eight steps, the
support nodes D, E and G were fixed, while in the ninth step,
nodes E and G moved closer to each other, and in the last
step, nodes E and G moved even closer (in Figure 26(c)).
However, in the whole process of gripper manipulation,
the CM projection of 5-TET on the ground was always
in the support area, and the majority of CM projections
were in the mid position of the support area, which
maintain it stable during gripper manipulations. The planned
manipulation workspace is described in Figure 26(d).
In gripper manipulation mode, the number of the
executed pneumatic cylinders between every two adjacent
steps is no more than 4, so that 5-TET mechanism is an
18 DOFs mechanism in total while it is a 4 DOFs
mechanism in every step manipulation of gripper. And taking
rolling motion into consideration, 5-TET is a 5 DOFs
mechanism with rolling locomotion mode and gripper
b Mathematical model
5 Energy Consumption Analysis
The 5-TET realizes locomotion and manipulation
functions by deformation, and the CM of the mechanism
frequently changed by driving actuators, so that the
energy and locomotion efficiency may be lower than that
of traditional mobile mechanisms, such as wheeled and
Taking wheeled robot as example to be compared with
5-TET on energy consumption and locomotion
efficiency. The total weight of the robot is set to be the same
with 5-TET as 12 kg. And the moving path is set as the
same trajectory of going straight from the origin (0, 0) to
the target (1500, 2000) in Quadrant I corresponding to
Figure 17(b), a total of 2.5 m.
E −200 0 200400500
−300−2X0-0ax0is/2m00m400 500 −300 Y-axis/ mm
E −200 0 200400500
−300−2X0-0ax0is/2m00m400500−300 Y-axis/ mm
Figure 26 Motion planning of gripper manipulation
5.1 The Wheeled Robot
For the wheeled robot, when it goes forward, it has four
forces as gravity G, traction force Ftraction, ground friction
Ffriction and air resistance Fair, the force diagram is shown
in Figure 27.
Assume that this wheeled robot moves at a constant
speed, then the acceleration of the system equals to 0.
According to Newton’s Second Law, the resultant force F
equals to 0, as Eq. (19) shown. Here the air resistance is
negligible, so that Ftraction = –Ffriction. And Ffriction = fg =
fmg = 0.6 × 12 × 9.8 = 70.56 N. Here, f represents the
friction coefficient between automobile tires and dry
asphalt pavement that equals to 0.6.
F =ma = 0,
F =Ftraction + Ffriction + Fair + G.
For the robot goes for 2.5 m, so that the energy
consumption is just equal to the work of traction, as Eq. (20)
shown, it is 176.40 J.
Wwheeled = Ftraction × s = 70.56N × 2.5m = 176.40J.
5.2 The 5T‑ET Mechanism
During the straight going in Quadrant I, there are totally
35 executed cylinders, and the overall compressed gas
consumption is calculated in Eq. (21):
V = π r2 ×
For Dry Ice Power Cell, the specific energy of dry ice
is 11.63 W·h/kg [
]. 430 g dry ice could produce 42 L
gas at pressure of 0.42 MPa [
], thus the specific energy
of gas carbon dioxide is 4.29×105 W·s/m3. According to
the energy consumption calculation method proposed by
Zhang and Cai [
], the energy consumption of 5-TET
that goes along the path of Figure 17(a) actuated with Dry
Ice Power Cell is shown in Eq. (22), as 317.38 J.
1 − β
4.29 × 105 × 0.7034
1 − 5%
= 317.38J. (22)
Where α1 represents the specific energy of carbon
dioxide, β is the gas leakage rate, take an average value as 5%.
For compressed air, which uses the air compressor with
the power of 550 W and the output flow of 40 L/min, the
specific energy is obtained in Eq. (23), it is 8.25×105 W·s/
m3. So that the energy consumption of compressed air is
shown in Eq. (24), it is 610.85 J.
α2 = Pc =
Where α2 represents the specific energy of compressed
air, Pc is the power of air compressor, Qc is the output
flow of air compressor.
From the results of energy consumption analyses, the
traditional wheeled robot is the most energy-efficient.
And 5-TET supplied by Dry Ice Power Cell consumes less
energy than that supplied by compressed air, where the
energy-saving rate reaches to approximately 48%.
Moreover, the flexibility and stability are the individual
characters of this mechanism. With the frame-like
structure, 5-TET can contain the barriers to overcome it
easily, as shown in Figure 28. Meanwhile the objective of the
optimization is that achieve certain functions by driving
minimum number of actuators, which to some extent
reduces the energy consumption.
6 Simulations of Successive Gait and Gripper
To verify the feasibility of 5-TET, the dynamic simulation
of locomotion and manipulation were carried out. 5-TET
has a total of 18 DOFs, 18 pneumatic cylinders were used
to drive the 18 prismatic joints, so that the spherical
joints are passive joints.
Figure 29 shows the rolling path of 5-TET along a
straight line from (0 mm, 0 mm) to (1500 mm, 2000 mm).
During this process, the pneumatic cylinder BC is locked,
and the 17 remaining cylinders are driven alternately
Figure 27 Force diagram of wheeled robot
Figure 28 The 5‑ TET rolls on the rough ground
Figure 29 Rolling along a straight line with the target of (1500 mm, 2000 mm)
Object Object Object
according to the optimal inputs combinations already
obtained (see in Figure 18(b)) to control the rolling
directions. Figure 30 shows the turning paths of Case I, the
optimal inputs are shown in Figure 20. In Figure 30(a),
the 5-TET turned around with three steps, but the
support area of the last step and the initial has an overlap. In
Figure 30(b), the 5-TET turned around with three steps,
then practically reset to its initial position. The turning
path of Case II is shown in Figure 31, corresponding to
the optimal inputs in Figure 22. Case II has more detailed
division about the interval of turning degrees (Additional
Furthermore, according to the optimal codes in
Figure 26(b), the manipulation function of 5-TET was
simulated, as Figure 32 shown. The gripper object was set
between node F and H. Due to the location and size of
the object, 5-TET gripped it after four attempts, which
correspond to steps 8–11 shown in Figure 26(a).
(1) An optimization method based on a developed GA
is proposed to optimize the motion of binary
control 5-TET mechanism. The 5-TET currently
realizes its locomotion by executing at most 5
pneumatic cylinders between each two steps, rather than
using multiple actuators as before.
(2) A potential manipulation function of the
mechanism that operated with two non-support TETs
synergistically is presented based on the proposed
(3) The energy consumptions of the 5-TET and the
traditional wheeled robot are compared. Though the
wheeled robot is more energy-saving, the 5-TET
using portable supply of Dry Ice Power Cell saves
approximately 48% energy than that supplied by
general air compressor.
(4) Dynamic simulations are carried out to validate the
proposed algorithm. As a result, the dynamic
simulation trajectories consistently match with that of
the mathematic analyses, which indicates that the
optimization is effective both on the locomotion
and manipulation planning. Through the
multiterrain optimization and the whole system design in
the next step, it will prospectively have better
performance in more complex environments.
Additional file 1 Simulation animation of successive gait and gripper
Ran Liu born in 1991, is currently a PhD candidate at School of Mechanical,
Electronic and Control Engineering, Beijing Jiaotong University, China. She received
her bachelor degree from Hebei University of Engineering, China, in 2013. Her
research interests include mechanisms and mobile robotics.
Yan‑An Yao born in 1972, is currently a professor at School of Mechanical,
Electronic and Control Engineering, Beijing Jiaotong University, China. He received
his PhD degree from Tianjin University, China, in 1999. His main research inter‑
ests include mechanisms and mobile robotics.
Wan Ding born in 1987, is currently a postdoctoral fellow at Department
of Mechanism Theory and Dynamics of Machines, RWTH Aachen University,
Germany. He received his PhD degree in 2015 from Beijing Jiaotong University,
China. His research interests include mechanisms and mobile robotics.
Xiao‑Ping Liu born in 1970, is currently a professor at School of
Mechanical, Electronic and Control Engineering, Beijing Jiaotong University, China. He is
a Canada Research Chair Professor at Department of Systems and Computer
Engineering, Carleton University, Canada. He received his PhD degree from the
University of Alberta, Canada. His research interests include intelligent systems
RL carried out gait optimization studies, participated in the dynamic analysis
and drafted the manuscript. YY completed the mechanism design and
modeling, participated in the gripper function planning. WD presented the
kinematic analysis, participated in the dynamic simulation. XL participated
in the improved genetic algorithm. All authors read and approved the final
Supported by National Science‑ Technology Support Plan Projects of China
(Grant No. 2015BAK04B00), and 2015 Sino‑ German Postdoc Scholarship
Program (Grant No. 57165010).
The authors declare that they have no competing interests.
Ethics approval and consent to participate
Springer Nature remains neutral with regard to jurisdictional claims in pub‑
lished maps and institutional affiliations.
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