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# RicMonk: A Three-Link Brachiation Robot with Passive Grippers for Energy-
Efficient Brachiation

Shourie S. Grama1∗, Mahdi Javadi1∗, Shivesh Kumar1, Hossein Zamani Boroujeni1
and Frank Kirchner1,2 This work has been supported by the M-RoCK
(FKZ01IW21002) and VeryHuman (FKZ01IW20004) projects funded by the German
Aerospace Center (DLR) with federal funds from the Federal Ministry of
Education and Research (BMBF) and is additionally supported with project funds
from the federal state of Bremen for setting up the Underactuated Robotics Lab
(201-342-04-2/2021-4-1).1Shourie S. Grama, Mahdi Javadi, Shivesh Kumar,
Hossein Zamani Boroujeni, Frank Kirchner are with Robotics Innovation Center,
DFKI GmbH, 28359 Bremen, Germany (E-mail: firstname.lastname@dfki.de).
Corresponding author: Shourie S. Grama∗ The authors contributed equally to the
work2Frank Kirchner is also with AG Robotik, University of Bremen, 28359
Bremen, Germany.The authors acknowledge the support of Dr. Melya Boukheddimi,
Shubham Vyas, and Rohit Kumar.

###### Abstract

This paper presents the design, analysis, and performance evaluation of
RicMonk, a novel three-link brachiation robot equipped with passive hook-
shaped grippers. Brachiation, an agile and energy-efficient mode of locomotion
observed in primates, has inspired the development of RicMonk to explore
versatile locomotion and maneuvers on ladder-like structures. The robot’s
anatomical resemblance to gibbons and the integration of a tail mechanism for
energy injection contribute to its unique capabilities. The paper discusses
the use of the Direct Collocation methodology for optimizing trajectories for
the robot’s dynamic behaviors and stabilization of these trajectories using a
Time-varying Linear Quadratic Regulator. With RicMonk we demonstrate
bidirectional brachiation, and provide comparative analysis with its
predecessor, AcroMonk - a two-link brachiation robot, to demonstrate that the
presence of a passive tail helps improve energy efficiency. The system design,
controllers, and software implementation are publicly available on
GitHub111The open-source implementation is available at
https://github.com/dfki-ric-underactuated-lab/ricmonkhttps://github.com/dfki-
ric-underactuated-lab/ricmonk and the video demonstration of the experiments
can be viewed in accompanying video..

###### Index Terms:

Underactuated robots, biologically-inspired robots, education robotics.

##  I Introduction

Brachiation, a mode common among primates like long-armed gibbons, involves
swinging between tree branches using their arms. Gibbons exhibit astonishing
agility, showcasing their ability to brachiate at speeds reaching up to
$25\text{\,}\mathrm{km}\text{\,}{\mathrm{h}}^{-1}$. This locomotion mode holds
potential for diverse applications such as agriculture surveillance, forest
exploration, and biomimetic design. Robots capable of brachiating and walking,
known as Multi-locomotion robots (MLR), offer intriguing opportunities for
study and implementation [1], [2], [3]. Brachiation includes “Slow
Brachiation”, where the robot swings between branches without a free-flight
phase, and “Fast Brachiation” or “Ricochetal Brachiation” involving dynamic
free-flight phases [4]. This unique locomotion bears similarity to walking and
running, relying on full-body coordination for balance and propulsion. The
adoption of underactuated control for brachiation exploits their dynamics
despite challenges posed by nonlinear control.

Figure 1: RicMonk performing brachiation maneuvers

Research in brachiating mechanisms has a longstanding history. An early study
involved simulating and analyzing a single cycle of motion for a five degrees
of freedom (DOF) brachiation robot [5]. Subsequent studies concentrated on
simpler designs, typically comprising two or three links, and offered
experimental validation for the proposed algorithms [6], [7], [8]. These
designs exhibited favorable power-to-weight ratios, enabling efficient and low
mechanical cost momentum generation [9]. Early studies introduced acrobot-type
underactuated robots equipped with actuated grippers, emphasizing heuristic
methods for behavior generation [10], [11], [12]. The motion along a flexible
rope offers energy-efficient advantages by including the cable dynamics in the
robot’s model as a compound double pendlum [13]. Experimental validation by
taking advantage of Time-Varying Linear Quadratic Regulator (TVLQR) and Sum-
of-Squares (SOS) optimization highlights the importance of the energy optimal
model-based controllers [14]. Similarly, a three-link robot design focused on
discontinuous brachiation motion, optimizing design parameters for energy
efficiency is described in [7]. The role of the tail in exciting the robot’s
natural frequency is a crucial factor in achieving ricochetal transverse
brachiation on the real system [15]. A comprehensive overview of the previous
research on brachiation-type mobile robots from an energy point of view is
presented in [16]. The common feature among all these brachiation robots is
the presence of active grippers, which introduces additional complexities to
the system design, and maintenance and energy expenditure. AcroMonk is the
first underactuated two-link robot capable of energy-optimal continuous
brachiation with passive grippers over a horizontally-laid ladder bar [17].
Even though the robot is capable of an unlimited number of brachiation
maneuvers in experiments, it is only able to perform robust forward
brachiation. This paper introduces RicMonk, a three-link underactuated robot
with passive grippers capable of energy-optimal bidirectional brachiation as
shown in Fig. 1. Finally, we present a comparative analysis with AcroMonk by
emphasizing the tail’s role in enhancing energy-efficient brachiation using
TVLQR, and the gripper improvement. In the spirit of [17], [18], [19], and
[20], the project is open-sourced222 https://github.com/dfki-ric-
underactuated-lab/ricmonkhttps://github.com/dfki-ric-underactuated-lab/ricmonk
to help researchers and provide them with a low-cost kit to encourage the
study of underactuated robotics. The hardware test demonstrations are
available and can be viewed in the accompanying video.

#### Organization

Section II addresses the mechatronics design of the RicMonk. Trajectory
generation for energy-optimal brachiation is presented in Section III. Section
IV details the stabilization methods utilized for robot control. Results are
presented in section V and finally, a conclusion is provided in Section VI.

##  II Mechatronics

The process of Mechatronic integration for a robotic system involves
incorporating mechanical design and essential electrical components to ensure
its operational functionality, capability, and safety. As envisioned in the
preceding generation of the RicMonk, the AcroMonk [17], the design is intended
to remain compact and wireless and needs to be made of readily available and
low-cost materials. Utilizing a single actuator in AcroMonk was a deliberate
decision, but it resulted in a trade-off. The price paid for this simplicity
is evident in its inability to perform multiple backward brachiation maneuvers
robustly, as the arms could not be commanded independently to release the bar
without the risk of falling. Furthermore, AcroMonk cannot dynamically to build
up momentum like a monkey. The design of RicMonk takes these shortcomings into
account and will be discussed in detail in the following sections. To address
these shortcomings, RicMonk is equipped with two actuators and a tail that
mimics a monkey’s body, as depicted in Fig. 1. The subsequent sections provide
detailed explanations of both mechanical and electrical system designs.

Figure 2: Exploded view of the RicMonk

###  II-A Design strategy

Incorporating a compact design, RicMonk meets the demands for portability and
agility, while its lightweight 3D-printed modular components allow for easy
adjustment of the robot’s inertial parameters. RicMonk adopted the arm and
gripper design from AcroMonk. The tail was carefully designed to reduce wiring
and tangling issues while providing a symmetrical housing for the integration
of electrical and computing components. Moreover, it acts as the point of
connection for the housings of both motors, while each arm is directly linked
to its corresponding motor shaft. The exploded view of RicMonk along with the
labeling of the components is presented in Fig. 2. An efficient gripper design
is crucial in providing a stable passive pivotal joint as the robot swings.
AcroMonk consists of a passive gripper with a cylindrical groove of radius
$11.5\text{\,}\mathrm{mm}$. Employing such a gripper for RicMonk, which weighs
nearly twice as much as AcroMonk (due to the extra actuator and tail), induces
instability, vibrations, and undesired motion during brachiation. To resolve
this, the cylindrical groove was replaced by a conical one, which has a larger
radius of $13\text{\,}\mathrm{mm}$ and a smaller radius of
$11.5\text{\,}\mathrm{mm}$. These values are obtained by considering the
center of mass offset from the hooking surface. This enhanced stability, and
also increased torque requirement for releasing the arm from the support bar.

###  II-B Electrical implementation

RicMonk is a three DOF underactuated brachiation robot equipped with two
Quasi-Direct-Drive motors [21] that allow independent arm movement. With a 6:1
gear ratio, these quasi-direct drives soffer the advantage of delivering high
torque and being back-drivable with low friction. The motor controller boards
enable communication with the actuators via Pi3hat [21], ensuring reliable
communication and incorporating an onboard Inertia Measurement Unit (IMU)
(shown in Fig. 2). State estimation utilizes IMU measurements and actuator
data to determine the state of the passive DOF with respect to the vertical
plane, denoted as $q_{1}$ in Fig. 3. A 3000 mAh Lithium polymer (LiPo) battery
is integrated as the power source to satisfy the portability design demand.
For safety purpose, a Radio Communication Transmitter Protocol (TX protocol)
is employed to incorporate a relay for turning off motors during erratic
behavior.

##  III Behavior State Machine

Javadi et al. [17] proposed a state machine to ensure robust and continuous
forward or backward brachiation while accounting for potential failures.
RicMonk also employs a similar state machine that integrates gripper-related
heuristics with a swing phase to generate a maneuver. The swing motion is
associated with a sequence of transitions among three fixed configurations,
referred to as atomic behaviors. These points are determined by hanging or
double support configuration of the robot known as Zero-to-Front (ZF), Zero-
to-Back (ZB), Front-to-Back (FB), and Back-to-Front (BF). This approach
enables uninterrupted robust brachiation even amidst disruptions and also
simplifies trajectory optimization. For swing behavior generation, a fixed-
base robot model is used as illustrated in Fig. 3.

Figure 3: Fixed base model of the robot

In this model for RicMonk, a virtual joint connecting the support arm to the
ladder bar at $\mathbf{(y_{c},z_{c})}$ is depicted as a revolute joint
(indicated by the dotted circle in Fig. 3). The arms are distinguished by
different colors, with blue denoting the support arm and green for the swing
arm. The support arm holds the bar and forms a passive revolute joint during
maneuvers, while the swing arm can move freely to catch the target bar.
Gripper-related heuristics involve the actions of releasing and catching the
front and back bars, which are abbreviated as Front Release (FR), Back Release
(BR), Front Catch (FC), and Back Catch (BC). A complete motion cycle involves
a coordinated sequence of actions performed by the swing arm, including the
release, swing maneuver, and catch phases. Combining these sequences and
alternating between support and swing arms in the cyclic motion results in bi-
directional brachiation as illustrated in Fig. 4. For example, executing FR
$\rightarrow$ FB $\rightarrow$ BC $\rightarrow$ switch arms $\rightarrow$ FR
$\rightarrow$ FB $\rightarrow$ BC results in two cycles of backward
brachiation. The following subsections provide a detailed explanation of swing
behavior generation and highlight the heuristics with two actuators for
RicMonk.

Figure 4: Illustration of bi-directional brachiation maneuver

###  III-A Gripper Heuristics

To catch and release from the bars with passive grippers, it is necessary to
apply torques to the arms in appropriate directions. The colored arrows in
Fig. 4 illustrates the force directions for front and back release of the
swing and support arm. For the back release, the support arm torque
($4.3\text{\,}\mathrm{N}\text{\,}\mathrm{m}$) is higher than the swing arm
torque ($2.3\text{\,}\mathrm{N}\text{\,}\mathrm{m}$), whereas for the front
release, the situation is reversed, with the support arm torque
($2.2\text{\,}\mathrm{N}\text{\,}\mathrm{m}$) being lower than the swing arm
torque ($6.3\text{\,}\mathrm{N}\text{\,}\mathrm{m}$).

The swing arm needs to grab the bar from above for FC, while the gripper needs
to grab it from below for BC. To obtain the initial state for the trajectory
optimization of these maneuvers, iterative release experiments are conducted
and the robot states are collected. The mean value for BR is
$\mathbf{x}_{0}^{\text{BR}}=[-0.59,0.68,-2.39,-0.37,-0.32,1.76]$ with
$\boldsymbol{\sigma}^{\text{BR}}_{0}=[0.02,0.03,0.05,0.24,0.27,0.6]$.
Similarly, for FR the mean value is
$\mathbf{x}_{0}^{\text{FR}}=[0.57,-0.57,2.76,0.68,-1.47,4.24]$ with
$\boldsymbol{\sigma}^{\text{FR}}_{0}=[0.01,0.03,0.03,0.42,0.29,1.8]$.
Statistical analysis suggests that $\mathbf{x}_{0}^{\text{BR}}$ and
$\mathbf{x}_{0}^{\text{FR}}$ are sufficiently reliable initial states for
trajectory optimization of BF and FB respectively.

###  III-B Trajectory Optimization for Swing Behavior

The presence of an onboard battery underscores the importance of energy-
efficient brachiation while respecting dynamics, actuation, and task space
limitations. This can be formulated as an optimal control problem, involving
the minimization of an objective function within a finite time frame while
respecting specified constraints. Trajectory optimization for the swing
behavior spans the period between the release and catch phases. It begins
right after the release action and continues until just before the swing arm
grabs the bar. The state of the robot,
$\mathbf{x}=[\mathbf{q},\dot{\mathbf{q}}]$, consists of the generalized
position vector ($\mathbf{q}=[q_{1},q_{2},q_{3}]\in\mathbb{R}^{3}$) and its
first order time derivative. The input vector of the robot,
$\mathbf{u}=[u_{1},u_{2}]\in\mathbb{R}^{2}$, consists of the torque input
values for the actuated joints that control $q_{1}+q_{2}$ and $q_{3}$. The
dynamics of the system are given by Eq. 1, where $\mathbf{M}(\mathbf{q})$ is
the mass-inertia matrix of the robot,
$\mathbf{C}(\mathbf{q},\dot{\mathbf{q}})$ is the matrix that describes the
Coriolis forces, $\tau_{g}(\mathbf{q})$ is the gravity vector, and
$\mathbf{B}$ is the actuator selection matrix.

$\mathbf{M}(\mathbf{q})\ddot{\mathbf{q}}+\mathbf{C}(\mathbf{q},\dot{\mathbf{q}})\dot{\mathbf{q}}=\tau_{g}(\mathbf{q})+\mathbf{B}\mathbf{u}$
(1)

The formulation of the mathematical problem for trajectory optimization is
presented in Eq. 2. The cost function contains state and input regularization
terms, where the weighting matrices $\mathbf{K}$ and $\mathbf{R}$ penalize
high joint velocities and input torques. Eq. 2 is the set of first-order
ordinary differential equations that captures the system dynamics as shown in
Eq. 1.

$\aligned&\min_{\mathbf{x},\mathbf{u}}\int_{0}^{T}(\dot{\mathbf{q}}^{T}\mathbf{K}\dot{\mathbf{q}}+\mathbf{u}^{T}\mathbf{R}\mathbf{u})\
dt\\\ \mathrm{subject\ to}:$ (2)

& ˙x = f(x, u)
g(x, u, t) ≤0
x(t_0) = x_0
x(t_f)  defined using  c_f^* All the inequality constraints like position,
velocity or torque limits, and collision constraints are represented by Eq. 2.
The initial state ($\mathbf{x}_{0}$) of the robot for the ZF and ZB maneuver
is zero and the robot builds up energy to reach the final state. For BF and FB
maneuvers, the initial state of the robot is obtained based on state
information from iterative release experiments with the robot. As previously
mentioned, both FB and BF swings commence after a release behavior, which
means the initial state is not static as in ZB and ZF. As mentioned in Section
II, the connection of the tail to arms results in a non-serial kinematic
coupling. Due to this, the inverse geometry cannot directly provide the
robot’s final state $\mathbf{x}(t_{f})$, which is necessary for the trajectory
optimization as stated in Eq. 2. The methodology to obtain the
$\mathbf{x}(t_{f})$ includes usage of the desired final cartesian position and
velocity of the gripper of the swing arm, i.e.
$\mathbf{c}_{f}^{*}=[y^{*},z^{*},\dot{y}^{*},\dot{z}^{*}]$. At each iteration,
the solver calculates the forward kinematics to obtain the task space
coordinates of the swing arm $\mathbf{c}_{f}$ as shown in Fig. 3, based on the
state’s decision variable and tries to minimize the error compared to
$\mathbf{c}_{f}^{*}$.

Taking advantage of the Direct Collocation method by Drake [22] toolbox, the
trajectories are optimized for the swing phases using the Sparse Nonlinear
OPTimizer (SNOPT) [23]. Hyperparameters for each of the maneuvers are detailed
in Table I, with $N$ representing the number of collocation points and
$\hat{t}_{f}$ initial guess period in seconds. For optimal tracking of the
trajectory, it is important to have smooth trajectories with minimal jerk,
acceleration, and torque consumption. Considering the system’s behavior, it is
observed that the robot is unable to efficiently track trajectories when
absolute jerk and acceleration exceed
$3000\text{\,}\mathrm{rad}\text{/}{\mathrm{s}}^{3}$ and
$200\text{\,}\mathrm{rad}\text{/}{\mathrm{s}}^{2}$ respectively. Trajectory
profiles were carefully analyzed to ensure the compatibility of the obtained
trajectories with the real system. In the initial iterations, the optimization
problem is warm-started using a straight-line trajectory that connects the
initial and final states with a first-order hold over an interval extending
from zero to an estimated final time ($\hat{t}_{f}$ in seconds). After several
iterations and utilizing agreeably smooth previously generated trajectories to
warm-start the optimization problem, optimal trajectories were obtained for
each of the maneuvers. The hyperparameters are carefully chosen to prevent
impulsive peaks in the torque or joint velocity profiles.

TABLE I: Hyperparamters for trajectory optimization problem Parameter | ZB | ZF | BF | FB
---|---|---|---|---
N | 40 | 60 | 40 | 35
$\hat{t}_{f}$ | [0, 3.0] | [0, 3.2] | [0, 4] | [0, 1.5]
$\mathbf{c}_{f}^{*}$ | [-0.345, 0.0, 0.83, 0.75] | [0.35, 0.025, 0.74, -1.4] | [0.36, 0.03, 0.9, -1.2] | [-0.35, 0.0, 0.83, 0.75]
$\mathbf{R}$ | diag[100, 7000] | diag[520, 440] | diag[6000, 5000] | diag[120, 120]
$\mathbf{K}$ | diag[0, 100, 100] | diag[0, 200, 240] | diag[0, 500, 200] | diag[0, 94.5, 108]
|  |  |  |

The BF and FB trajectories generated could not be tracked by the controller,
given they have an initial velocity and lower torque usage. The inability to
successfully track trajectory points to the sim-to-reality gap. Several
factors contribute to widening this gap, of which motor parameters like
friction, damping, and rotor inertia play a major role. Even with perfect
identification of these parameters, current solvers tend to disregard them
when solving the optimization problem. Consequently, there is always a need
for some heuristics to fill the sim-to-reality gap based on the behavior of
the system. We gained inspiration by observing the behavior of brachiation
masters like monkeys and gibbons. These animals consistently aim above the
bar, allowing for a brief moment of descent due to gravity. Similarly, when
approaching a handhold from below, they first make contact with the surface
and then grasp the bar from beneath. A similar heuristic approach is applied
to enable RicMonk to successfully perform brachiation. It assumes a final
configuration above the bar to facilitate falling over it, which also holds
when approaching the bar from below.

##  IV Trajectory Stabilization

Trajectory stabilization is crucial for all robots to ensure trajectory
tracking and it has been implemented in RicMonk to help the robot perform
brachiation motion. This section describes the model-based TVLQR that is
employed to stabilize the optimal trajectories for the RicMonk.

The controller linearizes the nonlinear dynamics shown in Eq. 2 using first-
order Taylor expansion in error coordinates that result in the state space
vector system (Eq. 3).

$\gathered\dot{\Bar{\mathbf{x}}}(t)\approx\mathbf{A}(t)\Bar{\mathbf{x}}(t)+\mathbf{B}(t)\Bar{\mathbf{u}}(t)\\\
\text{with $\Bar{\mathbf{x}}=\mathbf{x}-\mathbf{x_{0}}$,
$\Bar{\mathbf{u}}=\mathbf{u}-\mathbf{u_{0}}$}$ (3)

TVLQR minimizes a time-varying cost function, $J(t)$ (Eq. 4), as it optimizes
a varying control input, $\mathbf{u}(t)$. The matrices $\mathbf{Q}(t)$,
$\mathbf{R}(t)$, and $\mathbf{Q}_{f}(t)$ serve as weighting matrices crucial
for stabilizing the trajectories. Specifically, $\mathbf{Q}(t)$ and
$\mathbf{R}(t)$ are responsible for weighting the state and control input,
respectively, while $\mathbf{Q}_{f}(t)$ is the weighting matrix for the final
state. The optimal time-varying feedback matrix $\mathbf{K}(t)$ is obtained by
solving the time-varying Riccati equation [24].

$J=\mathbf{\bar{x}}^{T}(t)\mathbf{Q}_{f}\mathbf{\bar{x}}(t)+\int_{0}^{t_{f}}\left(\mathbf{\bar{x}}^{T}(t)\mathbf{Q}\mathbf{\bar{x}}(t)+\mathbf{\bar{u}}^{T}(t)\mathbf{R}\mathbf{\bar{u}}(t)\right)\,\,dt$
(4) $\mathbf{u}(t)=\mathbf{u}(t)-\mathbf{K}(t)\Bar{\mathbf{x}}(t)$ (5)

Mismatch between the mathematical model and the robot may affect the tracking
performance for the TVLQR on the real system. Approximation of highly non-
linear systems could also add challenges to the controls. However, TVLQR has
the capability for real-time error compensation using previously obtained
optimal control gains. The TVLQR hyperparamters $\mathbf{Q}$ and $\mathbf{R}$
for various maneuvers are summarized in Table II and
$\mathbf{Q}_{f}=\mathbf{I}_{6\times 6}$ for all maneuvers.

TABLE II: TVLQR hypermeters for the atomic behaviours Maneuver | $\mathbf{Q}$ | $\mathbf{R}$
---|---|---
ZB | diag[1, 1, 15, 1, 2, 1] | diag[8, 8]
ZF | diag[1, 1, 15, 1, 2, 1] | diag[5, 5]
BF | diag[1, 1, 17, 1, 2, 0.2] | diag[20, 20]
FB | diag[1, 1, 15, 1, 2, 1] | diag[20, 20]
|  |

TVLQR performed well with the ZB and ZF trajectory. However, to enable RicMonk
to perform BF and FB maneuvers, heuristics were considered in trajectory
optimization as mentioned in the Section III using TVLQR.

##  V Experimental Validation

All the atomic behaviors were realized on the real system with trajectory
stabilization provided by the TVLQR. These basic maneuvers were stitched
together to enable the robot to perform multiple brachiation motions. In Fig.
5 RicMonk performs three consecutive backward brachiation maneuvers, namely, a
ZB maneuver followed by two FB maneuvers. Additionally, Fig. 9 provides plots
representing the desired (indicated with superscript *) and measured
positions, velocities, and input torques. The plots demonstrate reasonable
tracking of desired trajectories resulting in successful brachiation.

Figure 5: Snapshots of RicMonk performing multiple backward brachiation
cycles. First row corresponds to a ZB maneuver (left to right), while second
and third rows correspond to consecutive FB maneuvers (left to right). Green
and red dots indicate start and end points respectively

[b]0.32  [b]0.32  [b]0.32

Figure 6: Generalized position plot Figure 7: Generalized velocity plot Figure
8: Optimum input plot Figure 9: Plots for the multiple backward brachiation
maneuvers - one ZB maneuver, and two FB maneuvers in succession. Vertical
dotted lines separate swing phases and grasp and releases

###  V-A Robustness Tests

The controller’s robustness in executing multiple brachiation maneuvers is
investigated through several tests. Firstly, it is tested with the addition of
extra weight, up to $200\text{\,}\mathrm{g}$ is loaded on the tail (symmetric
about the sagittal plane). Secondly, its performance is evaluated when
$170\text{\,}\mathrm{g}$ is loaded on one of the arms (asymmetric about the
sagittal plane). Finally, the robustness is examined in the presence of a
$520\text{\,}\mathrm{g}$ disturbance placed in the robot path. In the
scenarios mentioned above, the controller effectively stabilized the
trajectories, enabling RicMonk to perform multiple brachiation maneuvers in
two out of three trials. Fig. 10 illustrates the robot performing three
forward brachiation maneuvers as it carries $80\text{\,}\mathrm{g}$ on one of
the arms, i.e. asymmetric loading, and $100\text{\,}\mathrm{g}$ on the tail.
It also faces significant disturbances.

Figure 10: Snapshots of RicMonk performing multiple brachiation motions in the
presence of mass uncertainty that is loaded asymmetrically. First row
corresponds to a ZF maneuver (left to right), while second and third rows
correspond to consecutive BF maneuvers (left to right). In row 2, second and
third images illustrate the robot arm stuck due to a disturbance in motion.
Green and red dots indicate start and end points respectively

###  V-B Comparitive Analysis

AcroMonk is a minimalist brachiation robot, lightweight
($1.6\text{\,}\mathrm{kg}$) and highly portable. Its underactuated design
employs a single actuator, enabling robust forward brachiation, but it
struggles with backward motion and limited momentum due to its lightweight
build. RicMonk, a three-link brachiation robot with a body-like structure,
weighs $3.3\text{\,}\mathrm{kg}$, offering portability and agility. Featuring
two actuators, it achieves versatile forward and backward brachiation,
benefitting from its tail structure for enhanced energy input. A comparative
analysis in terms of standard deviations for FR
($\boldsymbol{\sigma}^{\text{FR}}_{0}$) and BR
($\boldsymbol{\sigma}^{\text{BR}}_{0}$) between AcroMonk and RicMonk,
indicates larger values for RicMonk. However, it’s crucial to approach these
values objectively, which stems from several factors. Firstly, RicMonk weighs
twice as much as AcroMonk, and the torque used for front and back release in
RicMonk is nearly three times greater than that in AcroMonk. Moreover, RicMonk
possesses the capability to independently apply torque to each arm, a feature
that is missing in AcroMonk. Cost of Transport (CoT) [25], [26] is a
dimensionless measure for energy efficiency, allowing comparison across sizes
and structures. Formulated as Eq. 6, energy input $E$, mass $m$, distance
traveled $d$, and acceleration due to gravity $g$.

$CoT=\dfrac{E}{mgd}$ (6)

A lower CoT value indicates greater energy efficiency within a given system.
Table (III) compares the total energy (TE) consumed by the AcroMonk and
RicMonk as they perform five consecutive forward brachiation (BF) maneuvers
(in joules), their CoT (dimensionless), and the time of transport ($t$) for
the five maneuvers in seconds.

TABLE III: Comparitive analysis | TE (J) | CoT |  $t$ (s)
---|---|---|---
AcroMonk | 8.9547 | 0.3355 | 11
RicMonk | 15.1947 | 0.2760 | 17
|  |  |

The table highlights several key insights. Though AcroMonk consumes a lot less
energy in total, RicMonk has a lower CoT, indicating higher energy efficiency.
However, these findings are limited to the trajectories that have been
optimized and stabilized, and other trajectories might have different
characteristics. Also, CoT is one of many perspectives to compare energy
efficiency among systems.

##  VI Conclusion and Outlook

This paper addressed the mechatronic design, generation, and stabilization of
optimal trajectories, and experimental validation of RicMonk, a three-link
underactuated brachiation robot, with passive grippers. RicMonk is portable,
modular, and easily reproducible. The presence of a tail structure and two
actuators help RicMonk to perform multiple brachiation maneuvers in both
forward and backward directions and this is a novelty in literature. This also
presents a notable advantage when compared to its predecessor, AcroMonk. The
presence of the tail also improves the energy efficiency as shown by the
comparison of cost of transport. We are working on the use of a floating-base
model to optimize trajectories for ricochetal brachiation. Additionally,
brachiation over irregularly placed bars and online trajectory optimization
are challenges that lie ahead. Further, the gripper design may be improved to
require a lower amount of torque to perform release action. Enabling RicMonk
to perform ricochetal brachiation is a very exciting outlook.

## References

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