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Parametric Synthesis of Compliant Joints for
Impact-Robust Shaftless Leg Mechanisms
Egor A. Rakshin1, Dmitriy V. Ogureckiy1, Ivan I. Borisov1,2,
and Sergey A. Kolyubin1
Abstract
This paper describes a novel parametric optimization procedure for three flexure cross hinges
(TFCH) integrated into multi-link leg mechanisms with closed-loop kinematics. Despite advantages
such as compliance, no need for joint lubrication, light weight and cost-efficiency, such shaftless
mechanisms have not been widely used, especially in the field of dynamic locomotion, also because
their design is challenging and barely studied. Using a morphological computation approach, we
have optimized the TFCH geometry to achieve the desired joint stiffness using frequency analysis,
ensuring safe and stable hopping under external perturbations. We combined rigid body dynamics
with lumped stiffness model and finite element modeling using the SPACAR toolbox to simulate var-
ious designs within our optimization pipeline. To illustrate the efficiency of the resulting designs, we
built a prototype and conducted a series of full-scale experiments with ramp jumps whose trajectories
were recorded by a motion capture system. The experiments showed that TFCH can be effectively
integrated into leg mechanisms, providing benefits such as impact robustness, energy recuperation,
and the ability to work in extreme conditions.
1 Introduction
Robots require fast motions, forceful contacts, impacts mitigation, compliance with safety require-
ments, etc. Meeting these requirements should not affect the robot’s ability to move dynamically in a
complex environment with unpredictable interaction with physical objects. Robot’s hardware and soft-
ware have to prevent from damaging the robot itself, surrounding environment, and humans. To ensure
the safety and reliability of robots when working in unstructured environments, the passivity property
must be taken into account [1].
Passivity can be implemented at either the software or physical level. Software solutions such as
impedance control, while proven to be robust and effective, necessitate accurate position and force sen-
sors to control a robot’s state and simulate passivity [3]. Passivity can be achieved at the physical level
using elastic elements. The physical properties of an elastic or flexible element allow it to store poten-
tial energy when deformed. This enables it to amplify the generated power and make use of low-power
actuators. Furthermore, it can release the stored energy in a powerful burst and has an unlimited band-
width [4].
This paper presents a co-design method to solve the elasticity allocation problem for a linkage mech-
anism with closed kinematics. We believe that our method is versatile enough to be employed in the
synthesis of various types of robots. Nonetheless, in the context of this study, we have chosen to apply
the proposed method specifically to the synthesis of a hopping robot. This choice has been made in order
to validate the efficacy and effectiveness of our approach.
*This work was supported by the Analytical Center for the Government of the Russian Federation (IGK
000000D730321P5Q0002), agreement No. 70-2021-00141.
1Egor A. Rakshin, Dmitriy V. Ogureckiy, Ivan I. Borisov, and Sergey A. Kolyubin are with the Biomechatronics
and Energy-Efficient Robotics (BE2R) Lab, ITMO University, Saint Petersburg, Russia e-mail: {earakshin,
s.kolyubin}@itmo.ru
2Ivan I. Borisov is also with the SberRoboticsCenter, Moscow, Russia
Figure 1: CAD render of a hopping robot from [2] but equipped with elastic joints only, i.e. TFCH, for
impact robustness, energy recuperation, and torque peak mitigation
E
G
F
D
K
=
diag([
])
kB
Lumped springs mechanism
Compliant
mechanism
O
A
P
B
D
C
E
G
F
1
2
3
1
2
3
4
5
0 1 2 3 40 1 2 30 1 2 0 1 2 3 4
6
Series
Elastic
Actuator
(a) (b) (c)
PP
OO
A
A
B
B
C
mg
D
1
2
P
O
A
C
3
4
B
D
1
2
W
b
H
O
A
DoF = 2DoF = 1DoF = 2 DoF = 2
L
t
kAkO
kOkAkBkD
kD
6
5
αdes
run
P
y
0
Ψ
0
x0
yP
Ψ
P
xP
Figure 2: The proposed design: (a) mechanism and (b) compliant joints synthesis, (c) motion trajectory
planning and control
To provide the required stiffness, the elastic elements can be combined into a compliant joint (CJ)
and there are various solutions for shaping the geometrical parameters and topology of CJs [5]. In this
study we utilize CJs to replace rotational bearings, adapt to external forces, eliminate backlash, get rid
of lubrication, get the desired stiffness while maintaining geometric and stress constraints [6]. To verify
the feasibility of the proposed method, we built a prototype of a hopping robot equipped with CJs in
constrained joints (Fig. 1), replacing conventional bearings.
1.1 Related Work
Flexible links and CJs can be used as a source of compliance to compensate robot weight [7]; to
withstand high dynamic loads during touchdown phase of locomotion [8], [9]; to use elastic components
as a passive source of torques [10]. Series elastic actuators (SEA) [11] can be used for resonance-based
locomotion tasks [12], and parallel elastic actuators (PEA) with drive couplings are capable of storing
energy, recovering it in a single explosive impact [13].
When the shape, geometry, and arrangement of elastic bodies in CJs are unknown, topology opti-
mization algorithms can be applied. The goal is to determine the most efficient structure that satisfies
given criteria, such as desired rotational and longitudinal stiffnesses [14], uniform stress distribution
across all deformable bodies [15], or the highest strain energy density [16].
The biggest disadvantage of topology optimization is that the method is computationally expensive
and does not ensure convergence. A method of structural-parametric synthesis of flexible joints with
desired stiffness and precise motion based on modular “building” blocks is given in [17]. The topologies
of the blocks are specified, so the optimization problem is reduced to finding the best combination of
these blocks and their geometry.
1.2 Contribution & Overview
CJs must be able to handle large deformations and distributed stresses, have minimal displacement
of the center of rotation, and have enough stiffness in the longitudinal direction to avoid collapsing
under load, CJs should be easily fabricated using additive manufacturing techniques for rapid prototyping
and have minimal subsequent machining. The three flexure cross hinge topology (TFCH) fulfills these
requirements [6].
Within this paper, only TFCH is considered for the sake of brevity and to verify the proposed method.
In the context of this work, the main goal is to improve robot adaptability and potentially energy effi-
ciency. Consequently, we introduce the novel contribution of this paper: a design method for dynamically
locomotive robots with compliant joints, demonstrated through the following statements:
• The co-design method to solve the elasticity allocation problem for a linkage mechanism with
closed kinematics under dynamic forceful interaction
• The three flexure cross hinge topology (TFCH) can be used to provide safe interaction with un-
structured environment and energy efficiency through potential energy recuperation.
The rest of the paper is organized as follows: the following section 2 describes our proposed pipeline
in detail. The 3 section presents a description of the prototype we developed and the physical experiment
we conducted. Conclusions and suggestions for future work are presented in the 4 section.
2 Proposed Method Outline
The design procedure consists of several steps (Fig. 2): (1) synthesis of the mechanism, i.e., op-
timization of the topology and linkage geometry, (2) CJ allocation involving finite element modeling
(FEM), (3) motion planning and actuator control, and (4) final prototyping to verify the obtained re-
sults. We call the problem of finding the optimal configuration of flexible joints and their stiffness the
stiffness allocation problem.
CJs do not operate as an ideal rotational joint due to the parasitic center of rotation displacement
during deformation. Nevertheless, a simplified model of a closed chain leg with rigid joints was used in
the dynamics simulation for faster modeling without calculating large deflections of elastic bodies. We
used MATLAB Simscape Multibody as a simulation modeling tool.
Since CJs cannot perform unbounded rotation, we need closed-loop kinematics to limit the possible
rotation angles. This paper focuses on the synthesis of CJs rather than topological optimization of closed-
loop chain kinematics. Thus, we used the previously proposed method to optimize the links geometric
parameters, mass and elasticity distribution for a closed-loop leg mechanism of a hopping robot [2].
The SPACAR toolbox for MATLAB environment was chosen for modeling elastic joints because
it allows high-fidelity modeling, calculates the resonant frequencies of the entire compliant mecha-
nism [18], and requires less computational resources compared to CAE systems.
We applied global non-gradient optimization for CJs geometry synthesis and motors trajectory plan-
ning. Due to nonconvex objective function, we used genetic algorithm from Global Optimization Tool-
box.
The pipeline can be used to create mechanisms for grasping, wearable rehabilitation devices, ex-
oskeletons, robot arms and legs, and other closed kinematics linkage mechanisms.
For the hopping robot, we consider only two dynamic motions: jumping on a place and forward
hopping. To simplify the task of motion planning, we use parametric sine wave as a position references
for motors. With the two sine waves, we can manipulate the signal parameters to obtain the necessary
trajectory to perform the desired dynamic behavior.
2.1 Rigid-body closed-chain mechanism synthesis
2.1.1 Topology of the basic mechanism
The topology of the leg with closed chain kinematics and optimization procedure was taken from our
previous study [2]: the five-bar mechanism with two degrees of freedom (DoF). The five-bar mechanism
is structurally the simplest topology of closed chain kinematics with two degrees of freedom, and besides
we can compare the modification with the previous result. This subsection relies on contribution of [2].
According to the Chebychev–Gr¨
ubler–Kutzbach criterion, DoF of a planar mechanism can be calcu-
lated as:
W=3N−2p(1)
where Wis number of DoF, Nis a number of moving links, and pis a number of lower kinematic pairs
such as revolute or prismatic joints.
If we extract pfor W=2, we get the following:
p=3
2N−1,p∈Z,(2)
where Nhas to be an even number. If we exclude the pair [N,p] = [2, 2] since it is an open chain
mechanism, the smallest number set we can get is [N,p] = [4, 5], which means a five-bar closed chain
mechanism.
The parametric synthesis of link lengths was performed such that one motor performs sinusoidal
wave motion (main motor) and the second motor (mode motor) is designed to act as a source of recon-
figuration of the mechanism by changing the contact point trajectory.
Despite the presence of CJs capable of partially dampening shock loads, SEA was added to the main
motor. This choice is justified because the CJs with high longitudinal stiffness do not guarantee complete
external shocks decoupling from the motor and increase backdrivability [19].
2.1.2 Desired trajectory
The task is to get the desired angle of attack α(Fig. 2, a) and then obtain the initial angles of the
passive revolute joints [O,A,B,D]where CJs have to be placed.
For the chosen mechanism topology and link lengths, the feasible smallest angle of attack for jumping
is αdes
run =75◦. To obtain the desired trajectory it is necessary to solve the forward kinematics (FK)
problem in the form of
qd=f(qi),(3)
where dependent and independent coordinates qd,qiform the holonomic constraints Nc=0.
Firstly FK was simplified to a four bar mechanism by fixing link BD and using trigonometric trans-
formations (Fig. 2, a). Then, links DC and CB were included. To calculate the actual angle of attack for
the jump motion αact
run, the inverse kinematics (IK) problem must be solved. To simplify the calculation
we reduce IK to the following optimization problem:
min
qact
iP(qdes
i)−P(qact
i)2
,(4)
where Pis a toe position vector in frame ΨPrelated with inertial frame ΨO.
Initialization SPACAR
solver
Synthesis
criteria met
Optimization
procedure
Simscape
solver
Start Pmech
0
no
Pmech
opt
yes
Kopt
Prun
0 P jump
0
Prun
opt P jump
opt
Motion planning
criteria met
Optimization
procedure
End
no
yes
(a) (b)
Figure 3: Optimization procedure: (a) CJs synthesis and stiffness matrix Kextraction, (b) motors motion
planning for jumping and hopping behavior
2.5 3 3.5 4 4.5 5 5.5
0.5
1
1.5
2.5 3 3.5 4 4.5 5 5.5
5
10
15
20
25
2.5 3 3.5 4 4.5 5 5.5
5
10
15
20
Figure 4: The relationship between stiffness, strain, and stress on frequency
2.2 Compliant mechanism co-design
In this step, we focus on integrating CJs as components of a compliant mechanism, optimizing
resonant frequencies, and developing effective motion planning strategies with lumped stiffnesses.
2.2.1 Compliant mechanism
Parameters of the TFCH with two symmetric side (s) leafsprings and one central (c) are
PT FCH = [Lc,Ls,Hc,Hs,tc,ts,bc,bc,W]
APT FCH <b
tc,s=0.4n,n∈Z,
(5)
where L,H,t,b,Ware the leaf spring length, height, thickness, width, and hinge width, respectively, in
the undeformed state (see Fig. 2, bfor an illustration of the geometrical parameters of the TFCH).
To constrain the size of the hinge and to ensure that the springs have clearances so that they do not
interfere with each other, we introduce a linear inequality, where Aand bare the constraint matrix and
vector, respectively. Since we produce CJ using FDM 3D printing and the nozzle has a fixed diameter of
0.4 mm, we constrain the thickness using integer constraints.
Since the mechanism has four CJs [O,A,B,D], total optimization vector of the mechanism is
Pmech = [PT FCH
O,PT FCH
A,PT FCH
B,PT FCH
D]
| {z }
36×1
.(6)
2.2.2 Joints synthesis
To operate on compliance of the mechanism, we can use stiffnesses of the CJs (Fig. 2, b). However,
this approach is impractical because the CJs motion is not circular and obtaining only the rotational
stiffness would be incorrect. Another method is to obtain the resonant frequency of the entire mechanism
by solving the eigenvalue problem [6]. This yields a list of mode shapes, in which the smallest one is the
frequency of leg motion of interest. The task is to bring the frequency to the desired value. CJs that are
too stiff result in a “statue” mechanism where the motors cannot move the links. On the other hand, CJs
that are too weak cannot support the weight of the robot and collapse after a touchdown.
The movement of the flexible leg is influenced by the stiffness of its joints, which are limited in their
range of rotation due to yield constraints. For the TFCH topology, the limiting range of motion was
chosen between −45◦and 45◦[17]. To objectify the frequency selection, we first analyze the maximum
joint motion and adjust the frequency so that it satisfies the constraint criteria. Because calculation of
the falling object at the impact varies on the contact model, a static force equivalent to weight of robot
with mass m=0.5 kg with multiplication factor of 2 is applied to the simulation model in SPACAR. By
applying this load, we set the strain and stress limits to obtain the operating ranges of the CJs.
The procedure of the optimization is shown at Fig. 3. During initialization, we set the robot geometry,
initial parameters, and optimization constraints.
SPACAR calculates the stiffness matrix Kfrom given parameters vector Pmech. The synthesis criteria
are:
min
Pmech −fleg(Pmech),
s.t.σmax ≤[σ],
qCJ ∈[max(qdes)±qso ft ],
(7)
where fleg is the frequency of a mode shape of interest, σmax,[σ]are the maximum stress and yield
stress respectively, qCJ ,max(qd es),and qso ft are current deflection of CJ, maximum deflection, and soft
constraint respectively. Results of synthesis procedures are shown in Fig. 4. A frequency of 2.63 Hz was
chosen since it ensures large deflection under desired motion range.
2.2.3 Motion planning
On the next step the stiffnesses are transferred to the motion planning procedure (Fig. 3). Here, the
mechanism with lumped springs (Fig. 2, c) is modeled in MATLAB Simscape Multibody. Since the
toe position (contact point) is known through the FK problem, we can create a motor control sequence.
This paper examines the use of simple constant and sinusoidal reference position control, as opposed
to utilizing complex control algorithms and motion planning strategies. When moving, the angles of
the motors change in a sinusoidal pattern. However, for jumping, the reference motor angle remains
constant. The mode motor temporarily fixes the leg, providing the desired angle of attack, while the
main motor with SEA performs the jumping movements.
The parameter vectors to be tuned for running and jumping behaviors are the following:
Prun = [Amain,Amod e,Fmain,Fmode,Bmain,Bmode,ϕmod e],
Pjump = [Amain,Fmain,Bmain,Cmode ],
(8)
where A,F,B,ϕ,and Care amplitude, frequency, bias, phase shift of sine waves, and constant value
respectively.
Objectives for the running and jumping behaviors:
min
Prun −Xend ˜
V
Eend
,
min
Pjump −˜
h,
(9)
where Xend,˜
V,Eend ,˜
hare the total distance traveled, mean longitudinal speed, total energy consumed,
and mean jumping height respectively.
3 Design
3.1 Physical prototype
A physical prototype was fabricated to verify the proposed co-design method (Fig. 5, a). We used
a laser cutter to facilitate the assembly processes and fabrication of body parts from polyoxymethylene
(d)(b)
(a)
(c)
Figure 5: To validate the proposed co-design method, we constructed a prototype and carried out a series
of full-scale experiments: (a) examination of the manufactured leg prototype at rest, (b) testing a ramp
jump, (c) forward jumping, and (d) touchdown after a kick
Body
Leg
Body
Leg
(a) (b) (с)
Figure 6: Experimental data for different locomotion scenarios include: (a) forward jumping, (b) ramp
jumping, and (c) jumping in a place. Red and blue lines indicate the torques for the main and mode
motors respectively, while orange and violet indicate the energy characteristics
(POM) sheets. 3D printers were used to fabricate the guides and CJs from PETG and PA6 nylon, re-
spectively. The choice of nylon is justified by good mechanical properties of this polymer under elastic
deformations. The body parts were connected to each other using hooks and fastening clips. The com-
ponents of the CJs are manufactured as prismatic parts without the need for additional supports. This
design decision was made due to the challenges of printing nylon supports and the potential for damag-
ing the CJs when removing them. The CJs’ parts are fixed via dovetail guides and are held in place by
frictional forces. The total mass of the robot is 530 grams. A toe (contact point) is made of rubber-like
TPU A95 plastic to increase surface adhesion. Steel and other elastic metals possess thinner hysteresis
regions, leading to lower material damping compared to nylon. This translates into enhanced energy
preservation capabilities. Since this study is at the prototype level, nylon was chosen as a polymer that
is more convenient in terms of rapid prototyping and availability.
We used Dynamixel AX-18A as actuators because of their availability and good power characteris-
tics. The motors are controlled using Arduino Nano Every and a 74LVC2G241 interface converter for
communication.
3.2 Virtual Experiment
We utilized the rotational stiffness and minimum frequency of the compliant mechanism as the out-
puts of SPACAR simulation model. The springs in Simscape Multibody have been modeled to accurately
Table 1: Different scenarios features
Scenario Distance travelled, V,˜
h,Efficiency
m m/s m
Jumping forward 4.22 0.26 0.045 CoT = 5.3
Ramp jump 4.03 0.24 0.035 CoT = 5.8
Jumping in a place - - 0.04 ˜
E= 2.1, J
represent their equilibrium position, stiffness, and damping characteristics.
The DAE solver was used in the simulations, and numerical differentiation tolerances were increased
and adjusted to ensure the holonomic constraints converged sufficiently. A penalty-based contact model
was employed to simulate the interaction between the foot’s toe and the ground. This approach was
utilized to accurately capture the behavior of this contact interaction in the simulation.
3.3 Real Experiment
The physical prototype is attached to a 1.15 meter long rod with a spherical workspace. One side is
screwed to the robot and the other side is fixed on the base [20]. Three types of experiments were per-
formed: (1) jumping forward, (2) ramp jumping with a slope of β=8◦(3) and jumping in a place (Fig. 5).
Fig. 6 shows experimental data for different locomotion scenarios include trajectories, torque values,
and power consumption. We used OptiTrack with 6 cameras to capture the real-time body and leg
positions of the robot. The average residual calibration error was 0.36 mm over a ray length of 7.5 m.
The operation of the motors was recorded by measuring current and angular position. Torque, velocity,
and energy characteristics were calculated based on these parameters. Table 1 provides a summary of
the observed characteristics, where CoT denotes the cost of transport metric.
Note that the average height in the jumping scenario with the motor running in fixed mode is lower
than in the jumping scenario with two motors running synchronously. For the 115 measured jumps in
place, the average energy expended per jump is 2.1 J.
4 Discussion
We conducted an investigation on the integration of three flexure cross hinges (TFCH) into closed-
kinematics, shaftless compliant, and impact-robust linkage mechanisms that work under forceful dy-
namic interaction. A novel parametric optimization procedure for TFCH has been developed and tested
for a case study on hopping robot design. It starts from topology, kinematics, and reference trajectory
optimization for basic rigid-body closed-kinematics mechanism with further transition to structure with
compliant joints and its numeric optimization combining lumped-parameters and finite-element models.
The result is the desired TFCH geometry, the optimal joint stiffnesses calculated for that geometry, and
the motion frequency supplied as a reference for the motion control system. Field tests demonstrate the
high robustness of the prototype to unpredictable external disturbances.
Despite a basic motion planning algorithm and the absence of feedback from the robot to the environ-
ment, the robot achieves proficient jumping and is capable of moving with a lateral velocity in the sagittal
plane of up to 0.53 m/s. This impressive feat is accomplished with a relatively low cost of transport of
1.3. To realize energy-efficient locomotion, the resonant frequency needs to be calculated considering
the mass and stiffness of the robot. Thus, for the payload scenario, we had to calculate the stiffness of
CJs.
In the near future, our goal is to create an enhanced control system and a faster, more precise CJ
model in order to improve energy efficiency and lower transportation costs, and shrink the reality gap
between the simulation model and the physical prototype. Although the current small-size prototype
is 100 grams heavier than our previous design with bearings, we expect a better weight ratio for new
shaftless design can be achieved with up-scaling. The integration of flexible joints operating in elastic
deformation mode provides motion with low intramolecular friction. At the same time, polymers such as
nylon have higher viscous deformation characteristics than steel, so this results in higher energy losses.
For these reasons, in future research we will focus on developing larger prototypes and utilizing TFCHs
made of metal.
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