# Design and comparative analysis of force balanced 2 degree of freedom planar robot manipulator concepts

Yash Vyas, Matteo Bottin, Marco Tognon, Silvio Cocuzza

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Yash Vyas, Matteo Bottin, Marco Tognon, Silvio Cocuzza. Design and comparative analysis of force balanced 2 degree of freedom planar robot manipulator concepts. Journal of Mechanisms and Robotics, In press, pp.1-17. 10.1115/1.4069221. hal-05190785

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# Design and comparative analysis of force balanced 2 degree of freedom planar robot manipulator concepts

A design methodology for the high level analysis of inherently force balanced manipulators is presented. It is applied for the analysis and comparison of three closed chain, two degree of freedom, planar robotic manipulator concepts. These designs are based on four bar and five bar linkages, and modeled at an abstracted component level in terms of an extendable link profile, counter masses, and additional attachments with mass/inertia. The methodology optimizes the balanced design counter masses within attachment position constraints. Different geometric parameter combinations for each concept in terms of workspace and total mass with balancing modifications are assessed. The balanced and unbalanced designs for the optimal parameters for each concept are then simulated using multibody dynamics software for different trajectories, with measurement of the reactions forces/moments and joint torques. Through this analysis, the various factors and tradeoffs that influence the design and balance optimization aspects for these force balanced robot manipulator concepts are demonstrated.

Keywords: force balancing, dynamic balancing, robotic manipulation

**Yash Vyas<fnref n="1" />**

Department of Industrial Engineering,
University of Padova,
via Venezia 1, Padova (PD) 35131, Italy
email: yashjanardhan.vyas@phd.unipd.it

**Matteo Bottin**

Department of Industrial Engineering,
University of Padova

**Marco Tognon**

Univ Rennes, CNRS, Inria,
IRISA - UMR 6074, F-35000 Rennes, France

**Silvio Cocuzza**

Department of Industrial Engineering,
University of Padova

## 1 Introduction

Robotic manipulators are widely used in applications such as industrial manufacturing [1] and mobile robotics [2] such as wheeled robots [3] and aerial robots [4]. Manipulators can interact with the environment, either passively through sensors or actively through end effectors that can push, pull, and grip objects. The structure of the manipulator plays a crucial role in its movement accuracy, since the compliance of the joints and links depends on the load applied to the structure [5].

Operating robot manipulators with increased precision, dexterity, and robustness is a research problem approached from many different perspectives. On one hand, a variety of control theory approaches are under continual development [6]. On the other hand, different mechanical designs apart from conventional serial chain linkages such as the Delta robot [7] have been proposed for improved manipulation.

Robot manipulators typically exert reaction forces and moments on the ground fixed or mobile base. These have two sources: 1) static, such as gravity acting on the links, and 2) dynamic, from the motion of the links. Dynamic forces and moments can cause vibrations at high accelerations, instability in the case of a mobile robot, and high reactions at the base of a ground-fixed robot. Furthermore, actuators must compensate for static forces and moments for all configurations, increasing energy expenditure.

Force and moment compensation can be addressed by balancing about the center of mass (CoM). The reduction of gravity or static forces and moments about the manipulator CoM is called gravity compensation/balancing and static balancing respectively [8]. Reduction of dynamic forces and about the CoM is called force balancing, and of the dynamic moments is called moment balancing [9]. A mechanism that is both force and moment balanced is called dynamically balanced.

Through mechanical design, balanced robot manipulators can reduce the reactions on the base, thereby facilitating higher accelerations and payloads [10]. Balancing has also been addressed

using control methods, particularly in space manipulation [11] and humanoid robotics [12].

Mechanical design approaches for balancing mechanisms include adding springs [13], counter masses, counter inertias, or additional linkage mechanisms [14]. These can be passively coupled to the system [15], or actively controlled [16]. The inherent balancing approach mathematically derives balance conditions for the geometric and dynamic parameters of the whole mechanism in the range of motion determined by kinematic constraints [17]. This approach has been used to design various types of balanced parallel mechanisms [18] and principal vector designs [19], some of which have been validated with experimental prototypes [20, 21]. Moreover, if underactuated robots are balanced, they become differentially flat [22, 23] and achieve controllability [24] even at high speeds [25]. Zero reactions to manipulator base can also be obtained through kinematic redundancy and optimal motion planning techniques, as demonstrated in the case of aerial manipulators [26, 27] and in space robotics [28, 29].

As inherent balancing of complex high DOF mechanisms is often difficult or results in infeasible combinations of geometry and mass/inertia, simple low degree of freedom (DOF) mechanisms are often synthesized to form higher DOF manipulators. This approach of synthesis has been explored theoretically and demonstrated in practice with experimental validation [30, 31].

In particular, the inherent force balance conditions of four and five bar linkage closed chain parallel mechanisms were derived by Berkof and Lowen [18] as well as van der Wijk [17]. The synthesis of four-bar 1-DOF balanced linkages was shown as a method to build higher-DOF mechanisms by Wu and Gosselin [32]. These mechanisms require lower mass and inertial additions for balancing compared to an open chain manipulator, as well as having other practical benefits of closed chain mechanisms such as improved stiffness [33] and less CoM displacement due to the actuating motors being closer or aligned along the fixed mount location.

The synthesis of mechanical design and robotic control for balanced manipulators has seldom been addressed in the literature. In [34] the authors designed a manipulator for force balanced and par-

<footnote n="1">¹Corresponding Author.
Version 1.18, July 29, 2025</footnote>
tially moment balanced space applications, and reactionless path planning to extend the mechanical design to full dynamic balanced performance, with some restrictions to workspace. An analysis of the impact of kinematic parameters on dynamic balancing behavior was presented in [35]. The dynamic balance as well as the PID control of a five bar linkage was presented in [36]. A dynamically balanced parallelogram architecture manipulator was proposed and simulated by Baron et. al. [37].

However, in all of these aforementioned papers, only a single set of design parameters was considered, and the impact on the kinematics and dynamics of the manipulator from changing the design parameters was not explored. Furthermore, the mass and inertia were calculated directly without considering the implications on mechanical design at the component level.

The direct comparison of various balanced design concepts with each other is also relatively unexplored in the literature. Previous research by Wijk presented a simple analysis of various methods for dynamic balancing of a 1 DOF [14] and 2 DOF serial manipulator [15]. However, in practice these designs are not feasible to implement in a robotics application as the mass/inertia or space increase is too high. Parallel (also known as closed chain) manipulator designs are much more suitable for balancing due to their lower mass / inertia increase [17], however, a direct comparison of balanced parallel manipulators from both the mechanical design and robotics application perspective is missing in the literature.

This paper analyzes and compares three different concepts within the scope of force balanced, 2 DOF planar manipulators designed from inherently balanced four and five bar linkage mechanisms [17]. We focus on 2 DOF as these form the simplest mechanisms for which low variable closed form balance solutions have already been derived. These concepts can therefore be easily synthesized into higher DOF redundant mechanisms. Theoretical studies show that closed loop designs contain lower mass additions when balanced [38]. These mechanisms are therefore highly suitable for industrial manipulation applications, where repeated motions with high precision are required and joint fatigue should be reduced.

Furthermore, a method of optimally force balancing a high-level design with minimal counter mass additions within geometric constraints is proposed. Various combinations of counter mass positions and geometric parameters are considered for total mass and also manipulator workspace, which is an important consideration in robot manipulation applications. The optimal parameters for mass and workspace in each concept is then validated for reaction forces and torques in three simulated scenarios using multi-body physics modeling. Balanced and unbalanced versions of each concept are modeled and simulated, demonstrating both the reduction in reaction forces and torques from balancing, and also allowing for a direct comparison between the concepts to assess the tradeoffs in selecting a design for robotics applications.

This paper is organized in the following sections as follows. First, a background of the theory of inherent balancing for four and five bar linkages is provided in Section 2. This is followed in Section 3 by presentation of the design concepts, and an explanation of the method for high-level balanced manipulator design, as well as the inverse kinematics to generate joint velocities for given end-effector trajectories. The results (Section 4) present various combinations of geometric parameters and counter mass positions for total mass and workspace. The optimal parameters for feasible counter mass positions are then simulated for various trajectories, comparing both the original unbalanced and the balanced manipulator designs to validate and demonstrate improvements from force balancing.

## 2 Background Theory

### 2.1 Force Balanced Four-Bar Linkage.
A four bar linkage (4BL) is a common 1-DOF mechanism used in many industrial applications, consisting of four links of which 3 form a closed loop, with both loop ends attached to a fixed link. The end effector

is attached to a coupler link and its motion profile with respect to the actuated joint angle $\theta_1$ is determined by the geometric and kinematic relationships of the mechanism.

To find the inherent force balance equations, van der Wijk [17] derives the linear momentum of a 4BL as an open chain of 3 links with a loop closure constraint. The kinematics of the 4BL are shown in Figure 1 and defined here, where index $i = \{1, 2, 3\}$ refers to moving links and $i = 4$ is the fixed link that rotates the entire mechanism:

* A, B, C are the positions of joints 1, 2 and 3, while D is the loop closure joint position.

* $\theta_i$ is the generalized coordinate for the revolute joint motion, and represents the degree of freedom of joint $i$. $\theta_4$ is the angle of the fixed link taken from A. These are defined with respect to the inertial frame x-axis.

* Link $i$ is moved by joint $i$ for $i = \{1, 2, 3\}$. Its length $l_i$ is defined as the distance from its origin joint $i$ to the next joint $i + 1$. For link 4, we define $l_4$ as the length of the fixed bar from A to D.

* The mass of each link is $m_i$. Its center of mass (CoM) is defined as a vector oriented at $\theta_i$ from its joint origin position, where $e_i$ and $f_i$ are the components along and perpendicular to the link.

* The center of mass of the four bar linkage is at S, and its vector relative to joint 3 is $e_S$ (along) and $f_S$ (perpendicular) to link 4 at angle $\pi+\theta_4$. Note that although $\theta_4$ is defined with respect to A, the center of mass position is calculated from D as it simplifies the linear momentum balance derivation.

The centre of mass position of each link $r_i$ and mechanism $r_S$ is calculated from A in inertial frame coordinates [17]:

$$r_1 = \begin{bmatrix} e_1 \cos \theta_1 - f_1 \sin \theta_1 \\ e_1 \sin \theta_1 + f_1 \cos \theta_1 \end{bmatrix} \qquad (1)$$

$$r_2 = \begin{bmatrix} l_1 \cos \theta_1 + e_2 \cos \theta_2 - f_2 \sin \theta_1 \\ l_1 \sin \theta_1 + e_2 \sin \theta_2 + f_2 \cos \theta_2 \end{bmatrix} \qquad (2)$$

$$r_3 = \begin{bmatrix} l_1 \cos \theta_1 + l_2 \cos \theta_2 + e_3 \cos \theta_3 - f_3 \sin \theta_1 \\ l_1 \sin \theta_1 + l_2 \sin \theta_2 + e_3 \sin \theta_3 + f_3 \cos \theta_3 \end{bmatrix} \qquad (3)$$

$$r_4 = \begin{bmatrix} (l_4 - e_4) \cos \theta_4 + f_4 \sin \theta_4 \\ (l_4 - e_4) \sin \theta_4 + f_4 \cos \theta_4 \end{bmatrix} \qquad (4)$$

$$r_S = \begin{bmatrix} (l_4 - o_1) \cos \theta_4 + o_2 \sin \theta_4 \\ (l_4 - o_1) \sin \theta_4 + o_2 \cos \theta_4 \end{bmatrix} \qquad (5)$$

The closed loop constraint equations are the following:

$$l_1 \cos \theta_1 + l_2 \cos \theta_2 + l_3 \cos \theta_3 - l_4 \cos \theta_4 = 0 \qquad (6)$$

$$l_1 \sin \theta_1 + l_2 \sin \theta_2 + l_3 \sin \theta_3 - l_4 \sin \theta_4 = 0$$

Next, the linear momentum $L_S$ of the mechanism CoM is de-
rived from the kinematics:

$$
\begin{align}
L_S &= m_1 \dot{r}_1 + m_2 \dot{r}_2 + m_3 \dot{r}_3 + m_4 \dot{r}_4 \nonumber \\
&= \left[ -(m_1 e_1 + m_2 l_1 + m_3 l_1) \sin \theta_1 - m_1 f_1 \cos \theta_1 \atop (m_1 e_1 + m_2 l_1 + m_3 l_1) \cos \theta_1 - m_1 f_1 \sin \theta_1 \right] \dot{\theta}_1 + \nonumber \\
&\quad \left[ -(m_2 e_2 + m_3 l_2) \sin \theta_2 - m_2 f_2 \cos \theta_2 \atop (m_2 e_2 + m_3 l_2) \cos \theta_2 - m_2 f_2 \sin \theta_2 \right] \dot{\theta}_2 + \nonumber \\
&\quad \left[ -m_3 e_3 \sin \theta_3 - m_3 f_3 \cos \theta_3 \atop -m_3 e_3 \cos \theta_3 - m_3 f_3 \sin \theta_3 \right] \dot{\theta}_3 + \nonumber \\
&\quad \left[ -m_4(l_4 - e_4) \sin \theta_4 + m_4 f_4 \cos \theta_4 \atop m_4(l_4 - e_4) \cos \theta_4 + m_4 f_4 \sin \theta_4 \right] \dot{\theta}_4 - \nonumber \\
&\quad \left[ -m_{tot}(l_4 - o_1) \sin \theta_4 + m_{tot} o_2 \cos \theta_4 \atop m_{tot}(l_4 - o_1) \cos \theta_4 + m_{tot} o_2 \sin \theta_4 \right] \dot{\theta}_4, \tag{7}
\end{align}
$$

where $m_{tot} = m_1 + m_2 + m_3 + m_4$, the total mass of the mechanism.
The time derivative of the closed loop equations (6) are:

$$
\begin{align}
-l_1 \sin \theta_1 \dot{\theta}_1 - l_2 \sin \theta_2 \dot{\theta}_2 - l_2 \sin \theta_3 \dot{\theta}_3 + l_4 \sin \theta_4 \dot{\theta}_4 &= 0 \tag{8} \\
l_1 \cos \theta_1 \dot{\theta}_1 + l_2 \cos \theta_2 \dot{\theta}_2 + l_3 \cos \theta_3 \dot{\theta}_3 - l_4 \cos \theta_4 \dot{\theta}_4 &= 0. \notag
\end{align}
$$

(8) is factored for $\sin \theta_2 \dot{\theta}_2$ and $\cos \theta_2 \dot{\theta}_2$ and substituted into (7):

$$
\begin{align}
L_S &= \left[ -(m_1 e_1 + m_2(1 - \frac{e_2}{l_2})l_1) \sin \theta_1 - (m_1 f_1 - m_2 \frac{f_2}{l_2} l_1) \cos \theta_1 \atop (m_1 e_1 + m_2(1 - \frac{e_2}{l_2})l_1) \cos \theta_1 - (m_1 f_1 - m_2 \frac{f_2}{l_2} l_1) \sin \theta_1 \right] \dot{\theta}_1 \nonumber \\
&\quad + \left[ (m_3(l_3 - e_3) + m_2 \frac{e_2}{l_2} l_3) \sin \theta_3 - (m_3 f_3 - m_2 \frac{f_2}{l_2} l_3) \cos \theta_3 \atop -(m_3(l_3 - e_3) + m_2 \frac{e_2}{l_2} l_3) \cos \theta_3 - (m_3 f_3 - m_2 \frac{f_2}{l_2} l_3) \sin \theta_3 \right] \dot{\theta}_3 \tag{9} \\
&\quad + \left[ -(m_{tot}(l_4 - o_1) - m_2 \frac{e_2}{l_2} l_4 - m_3 l_4 - m_4(l_4 - e_4)) \sin \theta_4 + \atop (m_{tot} o_2 + m_2 \frac{f_2}{l_2} l_4 - m_4 f_4) \cos \theta_4 \atop (m_{tot}(l_4 - o_1) - m_2 \frac{e_2}{l_2} l_4 - m_3 l_4 - m_4(l_4 - e_4)) \cos \theta_4 + \atop (m_{tot} o_2 + m_2 \frac{f_2}{l_2} l_4 - m_4 f_4) \sin \theta_4 \right] \dot{\theta}_4. \nonumber
\end{align}
$$

We desire that $L_S = 0$ for any $\dot{\theta}_i$, which results in force balance.
The force balance conditions are therefore fulfilled by setting the
coefficients of $\dot{\theta}_1$, $\dot{\theta}_3$, and $\dot{\theta}_4$ to 0:

$$
\begin{align}
m_1 e_1 + m_2 \left(1 - \frac{e_2}{l_2}\right) l_1 &= 0 \notag \\
m_1 f_1 - m_2 \frac{f_2}{l_2} l_1 &= 0 \notag \\
m_3 (l_3 - e_3) + m_2 \frac{e_2}{l_2} l_3 &= 0 \tag{10} \\
m_3 f_3 - m_2 \frac{f_2}{l_2} l_3 &= 0 \notag
\end{align}
$$

$$
\begin{align*}
m_{tot}(l_4 - o_1) - m_2 \frac{e_2}{l_2} l_4 - m_3 l_4 - m_4(l_4 - e_4) &= 0 \\
m_{tot}o_2 + m_2 \frac{f_2}{l_2} l_4 - m_4 f_4 &= 0.
\end{align*}
$$

As the system has 6 equations and 18 variables, it is under-
determined and either 1) 12 parameters can be selected with the

remaining 6 solved, 2) the constrained minimum of some objective
function can be found. By setting constraints for the position of
the center of mass ($o_1$ and $o_2$), different kinematic configurations
can be explored, which are analyzed in detail in [17]. Both are
relevant for concepts that will be presented in Section 3.1.

**2.2 Force Balanced Five Bar Linkage.** A five-bar linkage (5BL) is a 2-DOF mechanism consisting of five links connected with 5 revolute joints. The 5BL can be considered as a closed loop made up of 2 open chains. The end effector is attached to one of the nonactuated links.

General solutions for the force balance of 2,3, and 4 chain planar
parallel manipulators consisting of 3 serial revolute joints (RRR)
were presented by van der Wijk [17]. From that, a five bar link-
age was derived as a joint redundant case of a 2-RRR parallel
manipulator.

The kinematics of the five bar linkage is shown in Figure 2. The joints A₁ and A₂ are connected to the fixed bar, although they can be two separate points on the base. The two chains meet at joint C. The force balance equations for a five bar linkage are shown in (11).

$$
\begin{align*}
m_{21}e_{21} + m_{22}l_{21} \frac{l_{22} - e_{22}}{l_{22}} &= 0 & m_{21}f_{21} + m_{22}l_{21} \frac{f_{22}}{l_{22}} &= 0 \\
m_{11}e_{11} + m_{12}l_{11} + m_{22}l_{11} \frac{e_{22}}{l_{22}} &= 0 & m_{11}f_{11} + m_{22}l_{11} \frac{f_{22}}{l_{22}} &= 0
\end{align*}
$$

$$
m_{12}e_{12} + m_{22}l_{12} \frac{e_{22}}{l_{22}} = 0 \qquad m_{12}f_{12} + m_{22}l_{12} \frac{f_{22}}{l_{22}} = 0 \tag{11}
$$

3 Method

**3.1 Concepts.** To facilitate the analysis and comparison of high-level concepts within these modeling constraints, the following modeling assumptions (A) were made.

A.1 The CoM of each link is modeled as inline, that is, it lies along the length of the link so that $f_{ij} = 0$. This simplifies the kinematic design and positioning of balancing counter masses. It also results in the CoM being along the fixed bar of the mechanism ($f_S = 0$ in (10) and (11)).

A.2 The link geometries are symmetric and all links in the ma-
nipulator have the same profile (i.e., same density, cross-
sectional inertia). Additional masses on the links (e.g. ac-
tuators, joints) are also inline to the link geometry, and
assumed to be point masses in inertia calculations.

A.3 The counter masses are assumed to be point masses with
only a mass contribution to the link inertia.

Based on the scope, design constraints, and modeling assump-
tions, 3 concepts are generated from inherently force balanced four
and five bar linkage theory. Each joint is indicated with a corre-
sponding generalized coordinate or position vector (for a closed-
loop constraint joint).

3.1.1 Five-Bar Linkage (5BL2) with 2 actuated joints. This concept is an application of the 5BL presented in Section 2.2 and shown in Figure 2. The joints A₁ and A₂ with generalized co-ordinates θ₁₁ and θ₂₁ are attached to a fixed bar with length lₐ and actuated with motors. The chain is closed at position pₑ with length l₁₂ along link 12 and l₂₂ along link 22.

The end effector is attached to link 22 with length l<sub>22</sub> + l<sub>e</sub> in the
frame of θ<sub>22</sub> and extends past the closed-loop joint. The maximum
extension of the manipulator is l<sub>21</sub> + l<sub>22</sub> + l<sub>e</sub>. Counter masses are
attached behind the links to reposition the CoM at e<sub>ij</sub> and adjust
the mass to m<sub>ij</sub> required to fulfill the force balance conditions (11).
The concept is shown in Figure 3.
### 3.1.2 Four-Bar Linkage with 2 actuated DOF (4BL2).
This design can be interpreted in two different ways, which have equivalent balancing conditions and kinematic properties:

(1) A four bar linkage with 2 actuated degrees of freedom ($\theta_1$ and $\theta_4$), with center of mass at A ($e_S = l_4$) that is a static point mounted to the base (see Figure 1).

(2) A five bar linkage with $A_1$ and $A_2$ at the same point, and $\theta_{11}$ and $\theta_{21}$ actuated (see Figure 2). The center of mass of the manipulator is positioned along the fixed link, between $A_1$ and $A_2$, through the force balance conditions (11).

The second interpretation is used, as it allows for easier modeling of the inverse kinematics (Section 3.4). The high-level parameters for this concept are shown in Figure 4. The kinematic properties such as the actuated and passive joints, maximum extension, lengths are the same as 5BL2. In fact, as $l_d \to 0$, the kinematic behavior of 5BL2 approaches that of 4BL2.

An important feature of this design is that the entire manipulator can potentially be rotated about the center of mass through the simultaneous motion of $\theta_{11}$ and $\theta_{21}$. Although this may not be entirely feasible in practice due to joint limit and interference constraints in the out-of-plane axis (y-axis in Figure 4), it allows for a wider workspace and greater maneuverability.

Furthermore, $\theta_{11}$ and $\theta_{21}$ can be selected to result in two different sets of orientations for the end effector ($\theta_{22}$). However, switching from one set orientation to another is difficult as the manipulator passes through singularities in between.

### 3.1.3 Synthesized Four Bar Linkage (S4BL).
This concept was inspired by Wu and Gosselin [32] who presented several concepts for the synthesis of dynamically balanced 1-DOF four bar linkages into higher DOF planar and spatial manipulators. A suitable inherently balanced 4BL configuration for robot manipulation is aligning the CoM $e_S$ with link 4, actuating $\theta_1$ providing a 1-DOF motion at the output joint $\theta_3$, with the end effector attached to link 3. For synthesis, a crossed-over kinematic configuration is chosen for link 2, which is more compact and allows simpler force balance where counter masses are only needed on links 1 and 3.

The synthesis requires the outermost 4BL to be force balanced first, after which it is modeled as a single static load. This 4BL is then mounted as the end effector of another four bar linkage, which provides the second DOF of motion. By actuating $\theta_1$ in each four bar linkage, 2-DOF motion is obtained. The concept is shown in Figure 5.

## 3.2 Modeling.
The force balance equations for the 4BL (10) and 5BL (11) are based on the aggregated mass distribution of the link. However, the link kinematic parameters for these designs are dependent on parameters such as the length, cross-section, counter mass positions and additional loads (from components such as joints, end effector etc.).

The following features of the manipulator links are defined in the coordinate frame for each joint as are used for the CoM of the link ($e_{ij}$):

* A uniform link base geometry in terms of cross sectional shape with area A and material density $\rho$. This is extended to form the profile of the link and connects all the components from the furthest ends of the link, such as counter masses and any applied object loads (e.g. joints and end effector payload).

* The position $e_{ij,c}$ and the mass $m_{ij,c}$ of the counter masses in the joint frame.

* Additional loads $n$, with index $k$ have CoM at position $e_{ij,k}$ with mass $m_{ij,k}$ and centroid y-inertia of $J_{ij,k}$. They are attached inline to the link at the position vector $e_{ij,k}$.

The link geometry extends from $e_{ij,1} = \min[0, e_{ij,c}, p_{ij,1}, \dots, p_{ij,n}]$ to $e_{ij,2} = \max[e_{ij,c}, l, p_{ij,1}, \dots, p_{ij,n}]$.

Note that this ensures $e_{ij,2} > e_{ij,1}$. The mass $m_{ij,l}$ and center of mass vector $e_{ij,l}$ for the link profile are calculated as the following:

$$m_{ij,l} = A\rho(e_{ij,2} - e_{ij,1}) \quad (12)$$

$$e_{ij,l} = \frac{e_{ij,2} - e_{ij,1}}{2}. \quad (13)$$

We use these kinematic parameters to aggregate the link profile, additional components, and counter-masses to a CoM position $e_{ij}$ and mass $m_{ij}$ for the whole link:

$$m_{ij} = m_{ij,l} + \sum_{k=1}^{n} m_{ij,k} + m_{ij,c}e_{ij,c} \quad (14)$$

$$e_{ij} = \frac{1}{m_{ij}} \left( m_{ij,l}e_{ij,l} + \sum_{k=1}^{n} m_{ij,k}e_{ij,k} + m_{ij,c}e_{ij,c} \right) \quad (15)$$

Although calculation of inertias is not required for force balance, we require them for the dynamics model of the manipulator. Using the Parallel Axis theorem, the inertias of each link about its joint coordinate position (the origin of $\theta_{ij}$) as the following:

$$I_{ij,y} = J_{ij,l} + m_{ij,l}e_{ij,l}^2 + \sum_{k=1}^{n} (J_{ij,k} + m_{ij,k}e_{ij,k}^2) + m_{ij,c}e_{ij,c}^2 \quad (16)$$

where $J_{ij,l}$ and $J_{ij,k}$ are y moment of inertias the of link base geometry and additional components about their centroids.

## 3.3 Optimization.
The mass $m_{ij,c}$ and position $e_{ij,c}$ of the counter masses need to be selected to force balance the initial unbalanced design of the manipulator, consisting of its base geometry and attached components (joints, actuators, payload). As the force balance equations (10) (11) are underdetermined, we have many possible solutions depending on the variables that we decide to fix.

This allows us to optimize for a given goal within constraints, depending on our design requirements. We select total mass minimization as our objective function and use nonlinear optimization to find the set of counter mass and positions that force balance the mechanism. The optimization is set up as the following constrained nonlinear multivariable problem:

$$\begin{array}{ll} \min_{\mathbf{x}} & f(\mathbf{x}), \\ \text{subject to} & b(\mathbf{x}) = \mathbf{0}, \\ & \mathbf{x}_{\min} \le \mathbf{x} \le \mathbf{x}_{\max}, \end{array} \quad (17)$$

where:

* The optimization vector is $\mathbf{x} = [\mathbf{m}_c \ \mathbf{e}_c]^T$, where $\mathbf{m}_c$ is the set of counter mass variables and $\mathbf{e}_c$ is the set of counter mass position vector variables.

* The objective function is $f(\mathbf{x}) = m_{tot}(\mathbf{x}) = \sum m_{ij}(\mathbf{x})$. We find the link masses counter masses by first selecting $\mathbf{x}$, then calculate the aggregated masses of all the links (14).

* The nonlinear constraints $b(\mathbf{x})$ are the force balance equations, (10) for the 4BL and (11) for the 5BL.

* $\mathbf{x}_{\min}$ is the lower bound and $\mathbf{x}_{\max}$ is the upper bound. For $\mathbf{m}_c$ the lower bound is 0 (as counter masses cannot be negative) and we vary the upper bound appropriately. We vary the bounds on $\mathbf{e}_c$ to investigate its relationship to the total mass. The further the boundaries, the lower the total mass as the counter masses can be positioned arbitrarily behind the loads.
Fixed variables for each optimization are: *l* (link lengths), *l*<sub>e</sub>, and additional component masses **m**<sub>k</sub> and positions **p**<sub>k</sub>. These are initialized first to determine the geometries and unbalanced kinematic parameters of the mechanism. They impact the total mass and workspace of the unbalanced and balanced manipulator variants.

### 3.3.1 Optimization of 4BL2 and 5BL2.
For the 4BL2 and 5BL2 manipulators, the optimization procedure is as follows:

(1) Initialize the fixed variables for link lengths, link profile and additional components.

(2) Select the boundaries $x_{min}$ and $x_{max}$.

(3) Carry out (17) to find the optimal counter mass $m_c$ and $e_c$ that minimizes the total mass of the mechanism.

(4) Calculate $m$ (the vector of aggregated link masses) and $e$ (the aggregated link CoM positions) for the balanced mechanism with the optimized $m_c$ and $e_c$.

### 3.3.2 Optimization of S4BL.
For the S4BL concept, 2 stages of optimization are required:

(1) Initialize the fixed variables for link lengths, link profile and additional components.

(2) Select the boundaries $x_{min}$ and $x_{max}$.

(3) Carry out (17) to find the optimal counter mass $m_c$ and $e_c$ that minimizes the total mass of the 4BL1 (indexed using $i=1$), which contains the end-effector.

(4) Calculate $m_1$ (the vector of aggregated link masses) and $e_1$ (the aggregated link CoM positions) of 4BL1, as well as the total mass and CoM along Link 14.

(5) Carry out (17) to find the optimal counter mass $m_c$ and $e_c$ that minimizes the total mass of the second 4BL2 (indexed using $i=2$). 4BL1 is attached a single additional component with mass added at distance $p_{13,1} = l_{13} + l_m$, and CoM $e_{13,1} = l_{23} + l_m + l_{14} - e_{S,1}$, where $e_{S,1}$ is the CoM of 4BL1.

(6) Calculate $m_2$ (the vector of aggregated link masses) and $e_2$ (the aggregated link CoM positions), as well as the total mass and CoM of 4BL2.

For comparison, the unbalanced designs (pre-optimization) for each concept are also generated by finding $m$ and $e$ for $x=0$. The link profiles are shorter and do not extend behind the joint as no counter masses are added.

## 3.4 Inverse Kinematics.
Using numerical solvers for the constrained kinematics of both mechanisms, an initial configuration for the joint angles $\theta$ that fulfils the closed loop constraints can be found. From that, achieving a task requires generating a trajectory for the end-effector, and finding its inverse kinematics in the joint angle space to apply on the actuated joints.

These trajectories are generated in the velocity space, where the inverse differential kinematics are defined in closed form. Given a set of end-effector cartesian velocities $\dot{p}_e = v_e$, a method of calculating the joint velocities $\dot{\theta}$ is used. The 4BL2 and 5BL2 manipulator uses the following algorithm [39]:

(1) Let the transform for each serial chain composed of variables $\theta_i = [\theta_{i1} \ \theta_{i2}]^\mathrm{T}$ to the closed loop position $p_i$ be $T_i \in SE(2)$. Note: the velocity of the closed-loop position $v_c$ based on the desired end-effector velocity is required if $l_e > 0$, using the relation $v_e = v_c + J_{ce}\theta$, where $J_{ce}$ is the Jacobian from the closed loop position to the end-effector.

(2) The closed loop constraint equation is defined $T_1(\theta_1) = T_2(\theta_2)$ in position space. Its equivalent in velocity space is $J_1(\theta) \dot{\theta}_1 = J_2(\theta) \dot{\theta}_2$, formulated as $[J_1 - J_2] [\dot{\theta}_1 \ \dot{\theta}_2] = 0$, where $J_i \in \mathbb{R}^{2 \times 2}$ is the Jacobian of chain $\theta_i$ with respect to the cartesian velocities $[\dot{x} \ \dot{z}]^\mathrm{T}$.

(3) Partition $\theta$ into actuated coordinates $\theta_a = [\theta_{11} \ \theta_{21}]^\mathrm{T}$ and passive $\theta_p = [\theta_{12} \ \theta_{22}]^\mathrm{T}$.

(4) Rearrange $[J_1 - J_2]$ according to $\theta_a$ and $\theta_p$, where $[H_a \ H_p] [\dot{\theta}_a \ \dot{\theta}_p] = 0$.

(5) Find the passive joint velocities which are $\dot{\theta}_p = H_p^{-1} H_a \dot{\theta}_a = [e_1 \ e_2] \dot{\theta}_a$.

(6) Find actuated joint velocities according to the inverse kinematics of one of the serial chains $v_e = J_1 [\begin{matrix} 1 & 0 \\ e_1 \end{matrix}] \dot{\theta}_a$.

(7) Calculate actuated joint velocities according to (5), reorganise the components of $\dot{\theta}$ into the original order.

For the S4BL, we require a different approach as the two kinematic chains for 4BL1 and 4BL2 do not meet at the same point. We use a constrained inverse kinematics solution, incorporating the closed loop constraints of 4BL1 and 4BL2:

$$ 
\begin{aligned}
& J_{\theta_{21} \to \theta_{23}}^{p_2} \dot{\theta} = 0 \\
& (J_{\theta_{21} \to \theta_{13}}^{p_1} - J_{\theta_{21} \to \theta_{23}}^{p_1}) \dot{\theta} = 0
\end{aligned} 
\qquad (18) 
$$

where $J_{a \to b}^{p_i}$ is the Jacobian of point $p_i$ through the chain from $a$ to $b$ at the current configuration $\theta$.

The inverse kinematic solution for the joint velocities $\dot{\theta}$ is found through a constrained least squares error minimization:

$$ \min_{\dot{\theta}} \| J_{\theta_{11} \to \theta_{23}}^{p_e} \dot{\theta} - v_e \| \qquad (19) $$

$$ \text{such that} \quad \left[ (J_{\theta_{21} \to \theta_{13}}^{p_1} - J_{\theta_{21} \to \theta_{23}}^{p_1}) \dot{\theta} \right] = \mathbf{0} \qquad (20) $$

# 4 Results

## 4.1 Simulation environment.
The method in Section 3 was implemented to generate different variants of the design concept, with different kinematic parameters, and to analyze the manipulators for mass, workspace and dynamic behavior such as reaction forces/moments at the base. In addition to this, the balanced vs. unbalanced manipulators were benchmarked to validate force balancing through CoM displacement and find the improvement in manipulation performance from force balancing.

Manipulator design modeling and balance optimization was conducted in MATLAB as defined in section 3.2. We modeled additional masses on the links, such as the payload and revolute joints. The maximum extension of all manipulator designs was set at 0.5. During optimization, only counter masses, their positions and required link dimensions were varied. These assumptions simplified concept generation and allowed for a fair comparison between concepts.

The nonlinear optimization of balancing counter masses (Section 3.3) was implemented using MATLAB's `fmincon`. The boundaries (minimum and maximum values) of $e_c$ were considered at the following ratios of the original link lengths $l$, $1.5l$, and $2l$, as they provide a snapshot of the qualitative relationship between counter mass positions and the increase in total mass within feasible boundaries of a practical mechanical design. The optimized parameters are $m_c$, and $e_c$, which was aggregated to $m$ (14) and $e$ (15) for each link.

The first stage involved generating a manipulator, conducting the force-balance optimization of the counter masses and link size, and modeling using the MATLAB Robotics Library. The steps were the following:
(1) Initialization of link profile, and additional mass objects for each connector (joint) and payload.

(2) Creation of a manipulator type class that contains the link profile and additional masses for all the links.

(3) Optimization of counter masses while achieving force balance (17) using `fmincon` to create a balanced manipulator.

(4) Building balanced and unbalanced manipulator dynamic models by converting the parameters of the aggregated links into a MATLAB robotics object.

(5) Workspace Analysis: an initial configuration which fulfills the closed loop constraints was found using the MATLAB Robotics generalized Inverse Kinematics solver, and the workspace is determined through constant linear velocity trajectories at different angles in a radial sweep.

(6) Trajectory simulation: The joint velocities for the manipulator executing each trajectory were calculated using the algorithms shown in section 3.4. The CoM displacement was verified for balancing behavior between the balanced and unbalanced manipulator models.

The comparisons analyzed the impact of geometric parameters on workspace and mass, analyzing the relationship between mass and workspace for different link lengths. We did not make any additional assumptions on the workspace, such as joint limits, as it is mechanically feasible to design manipulators in 3D to facilitate the full range of motion. The maximum extent of the workspace was determined as the point at which a kinematic singularity occurs (measured as an absolute angular velocity above 5 rad/s).

Note that MATLAB Robotics only simulates serial mechanisms with closed loop position constraints, and hence this is only suitable for kinematic analysis. Without a constrained Lagrangian or Differential Algebraic Solver, the correct reaction forces/moments and joint torques of closed loop parallel mechanisms cannot be determined. For this reason, the selected designs from each concept were simulated in MSC ADAMS, a multibody dynamics software, to obtain accurate estimates of the reactions and joint torques.

## 4.2 Kinematic Parameters vs. Mass and Workspace.
Variations in link lengths and within a maximum extension of 0.5m were considered, which affected the workspace through span and singularities, along with different counter mass position boundaries that influenced the mass and the feasibility of the design.

Since the optimization objective was to minimize mass, the optimization generally placed the counter masses as far as possible toward the boundaries of $e_c$ to balance the link profile and any additional loads. However, the counter mass values changed as there was a balancing contribution given by the link profile being extended, which also affected the total mass. This phenomenon is explained by the force-moment relationship $\tau = Fr$, as the moment of a mass can be increased by increasing its distance from the balance position. As such, the main value of optimization is to solve the nonlinear force balance equations while minimising increase in mass from variations in parameters such as counter mass placement, link profile etc. at a more detailed level.

For the 5BL2 and 4BL2 designs, we maintained a relationship of $l_{11} = l_{21}$ and $l_{12} = l_{22}$ for the link lengths. The symmetry of the manipulator resulted in a symmetric workspace, which reduced the set of singularities. To simplify the workspace and inverse differential kinematics, $l_e = 0$ was also set. This reduces any additional inertial contribution from an offset of the end effector from the closed loop joint.

The profile of the links was uniform, with a rectangular shape that had a cross-sectional height of 2 cm, a width of 1 cm, and a thickness of 3 mm. The material chosen was aluminum with a density of 2710 kg.m⁻².

The joints were standardized at an ABS plastic density for a cylindrical bearing with a radius of 2 cm and a depth of 3 cm, each having a mass of 37.7 g and a moment of inertia around the y-axis of $7.5398 \times 10^{-6} \text{kg.m}^2$.

The payload mass at the end effector was 300 g, and it was assumed that it had a y-moment of inertia of $1 \times 10^{-4} \text{kg.m}^2$. The motors were added to the fixed bars at the positions of the actuated joints, with each motor having a mass of 82 g, corresponding to that of a Dynamixel XM430-W350T.

The fixed bar mass was not included in the mass calculation of the 4BL2 and 5BL2 designs, as it was not considered in the force balance equations. However, since the S4BL design required the balancing of the fixed bar (in the first 4BL1 mechanism), it was included as the mass of the fixed bars 14 and 24. An additional distance of 0.3 m was also introduced between the base and the first joint of the mechanism (mount distance).

### 4.2.1 5BL2 Design.
The impact of kinematic parameters on the mass and workspace of the 5BL2 design is described in Tables 1 and 2.

The following phenomena can be observed from our analysis:

(1) The workspace improves as the fixed link length approaches 0, i.e. the 5BL2 approaches the 4BL2 design.

(2) By having some space between $\theta_{11}$ and $\theta_{21}$, the manipulator can be narrower on the axis in the plane (y); however, $l_d$ should be minimized to maximize the workspace. The smallest value for $l_d$ in the 5BL2 design is therefore where the 2 motors actuating the joints are stacked on top of each other, which was determined to be 8 cm.

(3) The optimization solution is always at the minimum value boundary of $e_c$ for 3 out of 4 links, and counter masses can be omitted on link 11 or 21 while still achieving force balance.

(4) If $l_{i1} > l_{i2}$, there is a high chance of passing through a singularity due to the geometry of the manipulator. However, as there is no measured improvement in workspace, it is advisable to set $l_{i1} \le l_{i2}$ to avoid these issues.

(5) The total mass for the balanced designs does not vary significantly between different length parameters, as such we can optimize purely for workspace area which is a suitable objective for robot manipulation.

For these reasons, the link lengths of $l_{11} = l_{21} = 0.25m$, $l_{12} = l_{12} = 0.25m$, and $l_d = 0.08m$ were selected as the best set of geometric parameters for the 5BL2 concept.

The workspace is shown in Figure 6 with $e_c = -l$. The mass of the unbalanced manipulator is 0.759kg, and force balancing increases the total mass by 2.8 times.

The dynamic parameters for the selected link lengths of the 5BL2 design are summarized in Table 3, for $e_c = -l$. The motor mass was added to links 11 and 21, and the end-effector payload mass was added to link 22. The optimization did not require counter masses on link 12, which simplified the counter mass arrangement and reduced the chance of self-interference.

### 4.2.2 4BL2 Design.
As mentioned in Section 3.1.2, this is considered as the 5BL with $l_d = 0$. The results for different link length and counter mass extension combinations are shown in Table 4.

The workspace was maximized when all lengths were the same, as the entire mechanism can rotate about the fixed joints and also contract without reaching singularities. For this reason, link lengths of 0.25m was selected as the best geometry for this concept. The workspace is shown in Figure 7. Phenomena 4 and 5 of the 5BL analysis in Section 4.2.1 was also observed for this manipulator concept.

The balanced design parameters for the chosen link lengths of the 4BL2 design are summarized in Table 5, for $e_c = -l$. The motor mass has been added to Links 11 and 21, and the end-effector payload mass was added to L22. Similar to 5BL2, counter masses could be removed on link 12 while still maintaining force balance.
### 4.2.3 S4BL Kinematic Design.
Some thumb rules were applied when analyzing the S4BL design parameters. For example, since 4BL(1) was mounted on 4BL(2) and force balanced, it made sense for its link lengths to be smaller, similar to a serial manipulator. This reduced the counter mass required, which lowered the total mass, and allowed for greater maneuverability in the workspace. To simplify the kinematics, $l_{i1} = l_{i3}$ and $l_{i2} = l_{i4}$ were set as additional design constraints. According to Gosselin et al. [40], this is a prerequisite for dynamic balance and therefore also improves reaction moments under force balance.

The impact of geometric parameters within a range of ratios for $l_{i1}:l_{i3}$ was analyzed. Due to the increased number of joints, there was a higher probability of singularities, which often results in a non-convex workspace. For example, as $l_{i1}$ approaches $l_{i2}$, the probability of the manipulator collapsing into a singular configuration equivalent to a serial link is very high. In practice, controlling a closed-loop parallel mechanism with two collapsed links is infeasible. If $l_{i1}$ is too small, there is a large impact in the angle and velocity of $\theta_{i3}$ for small variations in $\theta_{i1}$, which can also result in the manipulator approaching singularities. Through trial and error, the optimal range for link ratios was found to be $\frac{2}{3}l_{i2} \le l_{i1} \le \frac{4}{5}l_{i2}$.

Four different combinations of link length are presented for different values of $e_c$. The mount distance $l_m$ (see Figure 5) is set to 0.15m, which fixes $l_{22} = l_{24} = 0.15$ m and $l_e = 0.2$ m. 4BL2 is resized with 2 different combinations of parameters. This was done keeping in mind practical implementation aspects of providing enough space for the extended links with the counter masses to maneuver during manipulation.

The results for the mass and the workspace with different geometric parameters are summarized in Table 6. The selected concept, with an optimal workspace, is $l_{11} = 0.12$ m, $l_{12} = 0.15$ m, $l_{21} = 0.2$ m, $l_{22} = 0.2$ m. The workspace is shown in Figure 8.

The mass and inertial parameters for the selected design are summarized in Table 7. The motor mass was added to links 14 and 24, and the end-effector payload mass was added to link 13. The balanced design only required counter masses on links 11, 13 (4BL1) and 21, 23 (4BL2), which simplified the design and reduced the chance of interference with the base.

Unlike the previous two designs, the decrease in mass from the change in counter mass position $e_c$ was not as noticeable. Furthermore, due to the increased chance of singular configuration due to the kinematic structure of the mechanism, the workspace was greatly reduced. The selected set of parameters were $l_{11} = l_{13} = 0.12$ m, $l_{12} = l_{14} = 0.15$ m, $l_{21} = l_{23} = 0.16$ and $l_{22} = l_{24} = 0.2$ m.

### 4.3 Trajectory Simulation Results.
The selected concepts, for counter mass extension $e_c = -l$ were simulated for 3 different end-effector trajectories, to validate and compare the respective balancing performances. For each trajectory, the y-axis reaction moment $\tau_R$, reaction forces $F_x$ and $F_z$ on the base mount at position $([x \ z]^T = [0 \ 0]^T$, and actuated joint torques $\tau_{11}$ and $\tau_{21}$. The balanced manipulators exhibited a constant reaction force $F_z = -m_{tot}g$ and $10^{-9}m$ order of magnitude displacement of the base CoM. The tracking error for the end effector was negligible as we did not model any random disturbances and assumed a perfect inverse differential kinematic controller. The trajectories and their respective results for each design are now outlined.

#### 4.3.1 Line Trajectory.
A smooth quintic polynomial interpolating two waypoints from the start position to a relative offset $[x \ z]^T = [0.15 \ 0.1]^T$ m in 1s, hold for 0.5s, and the reverse motion back to the start within 1s. The end-effector velocity profile is shown in Figure 9. The results for the concepts are in Figures 10-12. Table 8 summarizes the key metrics across the 3 designs.

#### 4.3.2 Circle Trajectory.
The end-effector traced a circle of radius 0.1 m in 5 s at a constant speed. The starting point along the perimeter of the circle and the initial configuration was set so

that the trajectory remained within the concept workspace. The end-effector velocity profile is shown in Figure 13. The results of the concepts are shown in Figures 14-16. Table 9 summarizes the key metrics for all designs.

#### 4.3.3 Load Trajectory.
a quintic polynomial interpolating 3 waypoints: start to relative position $[x \ z]^T = [0.1 \ -0.2]^T$ m in 1s, hold for 1.5 while a mass of 0.3kg is applied to the end effector and motion to end waypoint $[x \ z]^T = [0.2 \ -0]^T$ m (relative to start). The end-effector velocity profile is shown in Figure 17. The results of the concepts are shown in Figures 18-20. Table 10 summarizes the key metrics for all designs.

### 4.4 Observations.
From the trajectory simulations, the following validations of force balancing can be observed:

* Force balanced manipulators have lower reaction moments than the unbalanced variants, as the static force is kept constant and balanced by the base support rather than compensated by the joint torques, and the dynamic forces are removed.

* The joint torques only need to induce rotational acceleration of the links, which despite an increase in inertia are lower than the gravity compensation torques for the acceleration profiles in our trajectories. At some high accelerations, we can expect the joint torques to be higher for the balanced manipulator than the unbalanced manipulator; however, in general for a robotics application it is unlikely such high accelerations would be required. As a result, energy consumption is also reduced for a balanced manipulator.

* Although the counter masses balance a fixed end-effector payload, further addition of a mass load (for the load trajectories) still results in lower reaction moments and joint torques for the balanced manipulator design in comparison to the unbalanced, as the mass of the links is already balanced for manipulator motions.

* The balanced 5BL2 and 4BL2 designs have lower reaction moments and joint torques than the S4BL design in all simulated cases. This is due to the fact that the motors of both designs are attached to the base and do not require balancing, whereas in S4BL the motor for 4BL(1) moves along with the whole linkage and requires balancing when attached to 4BL(2). Furthermore, the number of links in 5BL2 and 4BL2 designs is lower, so the initial mass is also lower, and a lesser mass addition is needed for balancing. However, through more realistic modeling of links for the S4BL design, such as non-uniform link profiles, this concept could be further improved.

* The S4BL has far higher maximum torque and energy consumption than the other concepts when unbalanced. This can be explained through the extension of the end effector, as it is much further from the base and has a longer kinematic chain. On average, the distance between the end effector and the furthermost actuated joint (11) position is 0.65m compared to 0.25m for the other concepts. As the unbalanced design also has higher mass (1.354kg vs. 0.759kg), the gravity compensation of maintaining the end effector configuration requires much higher joint torques, and during motion higher inertia of the links results in additional energy consumption.

* Depending on the requirements for minimizing reactions vs. maximizing workspace, the 5BL2 or 4BL2 design can be chosen. The 4BL2 design has a larger workspace, but higher reaction moments and joint torques. Therefore, we can conclude that increasing $l_d$ results in a trade-off between reactions and the workspace.
# 5 Conclusion

This research paper explored several concepts of 2-DOF planar force balanced manipulators using a high-level design methodology and simulation. A quantitative analysis and comparison of the concepts was presented, showing the various trade-offs in mass and workspace from different combinations of geometric parameters and counter mass constraints for an original unbalanced design. The dynamic behavior of both balanced and unbalanced designs was analyzed for different trajectories using multibody dynamics simulation, measuring relevant metrics such as reaction forces/moments and joint torques. This analysis validated the improvements in terms of reduction of reaction moments and joint torques through force balancing, and also allowed for a direct comparison of the considered concepts with each other.

It was shown that placing the counter mass as far behind as possible decreases the mass addition for a balanced design, however, that has to be carefully considered in practice for space limitations in the particular application. Furthermore, designs with motors attached to the fixed mount, such as the Five Bar Linkage with 2 actuated DOF (5BL2) and its Four Bar Linkage counterpart (4BL2), perform better in terms of reactions, total mass, joint torques, and workspace, than a manipulator built out of synthesizing two 1-DOF four bar linkages together (S4BL).

Between the 5BL2 and 4BL2 designs, there are tradeoffs that have to be taken into consideration when choosing a concept for detailed mechanical design and implementation to a robotic application. For an increase in reaction moments and joint torques, a larger workspace can be achieved with the 4BL2 design. These designs were analyzed at a high level, assuming feasibility in implementing the planar design in 3D without self-interference. The analysis provides a valuable understanding of the different aspects of force balancing and how they impact manipulator performance. The design methodology also serves as a guide for selecting a balanced manipulator concept for application to a robotics application.

# Acknowledgment

## Funding Data

The research leading to these results has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement no 101034319 and from the European Union — NextGenerationEU.

Part of this work was carried out in the framework of the project “DynAeRobot—Development and validation of a new dynamically balanced aerial manipulator” (project no. BIRD213590), funded under the BIRD 2021 program promoted by the University of Padova.

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<figure><img src="image_2.png" /><figcaption>Fig. 1 A four bar linkage.</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Fig. 2 Five bar linkage as a closed-loop of two RRR chains. The actuated joint for each chain is $A_i$ and passive second joint is $B_i$. The generalized coordinate for the joint degree of freedom is $\theta_{ij}$, link length $l_{ij}$ and CoM vector is $[e_{ij}, f_{ij}]^\top$ of each link in the joint reference frame as shown. The indexing system uses $i$ to represent the chain number where $\{A, B\}: \{1,2\}$, and $j$ as the joint number in the chain.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 3 The 5BL2 manipulator concept. Counter masses (black circles) are mounted to adjust the the link mass and CoM position (checkered circles). The inline vector of the link and mass relative to the joint is $e_{ij}(m_{ij})$. The closed loop joint position is $p_c$ and end-effector is at $p_e$.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 4 The 4BL2 concept. Counter masses (black circles) are mounted to adjust the the link mass and CoM position (checkered circles). The inline vector of the link and mass relative to the joint is $e_{ij}(m_{ij})$. The closed loop joint position is $p_c$ and end-effector is at $p_e$.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 5 The S4BL manipulator concept. Counter masses (black circles) are mounted to adjust the the link mass and CoM position (checkered circles). The inline vector of the link and mass relative to the joint is $e_{ij}(m_{ij})$. The closed loop joint position for 4BL1 is $p_1$, for 4BL2 is $p_2$, and end-effector is $p_e$.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 6 The workspace for the selected set of parameters of 5BL2. A resolution of 10 degrees for the radial sweep, with a constant end-effector velocity of 0.1 m/s was used. The fixed link is black, end-effector is green. Links are lines and joints are circles, with red indicating actuated and blue indicating passive.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Fig. 7 The workspace for the selected set of parameters of 4BL2. A resolution of 10 degrees for the radial sweep, with a constant end-effector velocity of 0.1 m/s was used. The fixed link is black, end-effector is green. Links are lines and joints are circles, with red indicating actuated and blue indicating passive.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 8 The workspace for the selected set of parameters of S4BL. A resolution of 10 degrees for the radial sweep, with a constant end-effector velocity of 0.1 m/s was used. The fixed links are black, and end-effector is green. Links are lines and joints are circles, with red indicating actuated and blue indicating passive.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 9 End-effector velocities for the line trajectory.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 10 Results for 5BL2 with line trajectory. The solid lines are balanced and dotted lines are unbalanced.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 11 Results for 4BL2 with line trajectory. The solid lines are balanced and dotted lines are unbalanced.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 12 Results for S4BL with line trajectory. The solid lines are balanced and dotted lines are unbalanced.</figcaption></figure>
<figure><img src="image_14.png" /><figcaption>Fig. 13 End-effector velocities for the circle trajectory.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>Fig. 14 Results for 5BL2 with circle trajectory. The solid lines are balanced and dotted lines are unbalanced.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Fig. 16 Results for S4BL with circle trajectory. The solid lines are balanced and dotted lines are unbalanced.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>Fig. 17 End-effector velocities for the load trajectory.</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>Fig. 15 Results for 4BL2 with circle trajectory. The solid lines are balanced and dotted lines are unbalanced.</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>Fig. 18 Results for 5BL2 with load trajectory. The solid lines are balanced and dotted lines are unbalanced.</figcaption></figure>
<figure><img src="image_20.png" /><figcaption>Fig. 19 Results for 4BL2 with load trajectory. The solid lines are balanced and dotted lines are unbalanced.</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>Fig. 20 Results for S4BL with load trajectory. The solid lines are balanced and dotted lines are unbalanced.</figcaption></figure>
<table><caption>Table 1 Mass for different parameters of the 5BL2 design concept. The chosen parameters for concept comparison are in bold.</caption><thead><tr><th>l<sub>11</sub>, l<sub>21</sub><br>(m)</th><th>l<sub>12</sub>, l<sub>22</sub><br>(m)</th><th>Mass (kg),<br>e<sub>c</sub> = -l</th><th>Mass (kg),<br>e<sub>c</sub> = -1.5l</th><th>Mass (kg),<br>e<sub>c</sub> = -2l</th></tr></thead><tbody><tr><td>0.35</td><td>0.15</td><td>2.053</td><td>1.631</td><td>1.467</td></tr><tr><td>0.32</td><td>0.18</td><td>2.072</td><td>1.641</td><td>1.473</td></tr><tr><td>0.3</td><td>0.2</td><td>2.185</td><td>1.651</td><td>1.477</td></tr><tr><td>0.28</td><td>0.22</td><td>2.099</td><td>1.657</td><td>1.481</td></tr><tr><td><strong>0.25</strong></td><td><strong>0.25</strong></td><td><strong>2.119</strong></td><td>1.664</td><td>1.487</td></tr><tr><td>0.22</td><td>0.28</td><td>2.138</td><td>1.674</td><td>1.492</td></tr><tr><td>0.2</td><td>0.3</td><td>2.151</td><td>1.681</td><td>1.495</td></tr></tbody></table>

<table><caption>Table 2 Workspace (WS) for different link and fixed bar lengths of the 5BL2 design. The chosen parameters for concept comparison in bold.</caption><thead><tr><th>l<sub>11</sub>, l<sub>21</sub><br>(m)</th><th>l<sub>12</sub>, l<sub>22</sub><br>(m)</th><th>l<sub>d</sub> (m)</th><th>WS Area<br>(m<sup>2</sup>)</th><th>WS Span<br>(m)</th></tr></thead><tbody><tr><td>0.3</td><td>0.2</td><td>0.08</td><td>0.358</td><td>0.905</td></tr><tr><td>0.28</td><td>0.22</td><td>0.08</td><td>0.403</td><td>0.905</td></tr><tr><td><strong>0.25</strong></td><td><strong>0.25</strong></td><td><strong>0.08</strong></td><td><strong>0.446</strong></td><td><strong>0.905</strong></td></tr><tr><td>0.22</td><td>0.28</td><td>0.08</td><td>0.402</td><td>0.905</td></tr><tr><td>0.2</td><td>0.3</td><td>0.08</td><td>0.357</td><td>0.904</td></tr><tr><td>0.3</td><td>0.2</td><td>0.12</td><td>0.339</td><td>0.869</td></tr><tr><td>0.28</td><td>0.22</td><td>0.12</td><td>0.383</td><td>0.869</td></tr><tr><td>0.25</td><td>0.25</td><td>0.12</td><td>0.429</td><td>0.869</td></tr><tr><td>0.22</td><td>0.28</td><td>0.12</td><td>0.385</td><td>0.869</td></tr><tr><td>0.2</td><td>0.3</td><td>0.12</td><td>0.338</td><td>0.869</td></tr></tbody></table>
<table><caption>Table 3 Masses, CoM positions and Inertias (about the joint position $\theta_{ij}$)) for all the links of the final 5BL2 design.</caption><thead><tr><th rowspan="2">Link</th><th colspan="3">Unbalanced</th><th colspan="6">Balanced</th></tr><tr><th>ij</th><th>m<sub>ij</sub> (kg)</th><th>e<sub>ij</sub> (m)</th><th>I<sub>ij,yy</sub> (kg.m<sup>2</sup>)</th><th>m<sub>c,ij</sub>(kg)</th><th>e<sub>c,ij</sub>(m)</th><th>m<sub>ij</sub>(kg)</th><th>e<sub>ij</sub>(m)</th><th>I<sub>ij,yy</sub> (kg.m<sup>2</sup>)</th></tr></thead><tbody><tr><td>11</td><td>0.137</td><td>0.050</td><td>1.16 &times; 10<sup>-3</sup></td><td>0.065</td><td>-0.25</td><td>0.257</td><td>-0.063</td><td>7.81 &times; 10<sup>-3</sup></td></tr><tr><td>12</td><td>0.093</td><td>0.074</td><td>1.16 &times; 10<sup>-3</sup></td><td>0</td><td>0</td><td>0.093</td><td>0.074</td><td>1.16 &times; 10<sup>-3</sup></td></tr><tr><td>21</td><td>0.137</td><td>0.041</td><td>1.16 &times; 10<sup>-4</sup></td><td>0.802</td><td>-0.25</td><td>0.994</td><td>-0.202</td><td>5.39 &times; 10<sup>-2</sup></td></tr><tr><td>22</td><td>0.393</td><td>0.209</td><td>1.99 &times; 10<sup>-2</sup></td><td>0.327</td><td>-0.25</td><td>0.327</td><td>-0.009</td><td>4.29 &times; 10<sup>-2</sup></td></tr></tbody></table>

<table><caption>Table 4 The mass and workspace (WS) of the 4BL2 for different kinematic parameters. The most feasible design, selected for dynamic analysis, are in bold.</caption><thead><tr><th>l<sub>11</sub>, l<sub>21</sub> (m)</th><th>l<sub>12</sub>, l<sub>22</sub> (m)</th><th>Mass (kg),<br>e<sub>c</sub> = -l</th><th>Mass (kg),<br>e<sub>c</sub> = -1.5l</th><th>Mass (kg),<br>e<sub>c</sub> = -2l</th><th>WS<br>(m<sup>2</sup>)</th><th>Area</th><th>WS Span (m)</th></tr></thead><tbody><tr><td>0.3</td><td>0.2</td><td>2.086</td><td>1.647</td><td>1.477</td><td>0.428</td><td></td><td>0.944</td></tr><tr><td>0.28</td><td>0.22</td><td>2.099</td><td>1.657</td><td>1.481</td><td>0.471</td><td></td><td>0.944</td></tr><tr><td><b>0.25</b></td><td><b>0.25</b></td><td><b>2.118</b></td><td><b>1.664</b></td><td><b>1.487</b></td><td><b>0.511</b></td><td></td><td><b>0.944</b></td></tr><tr><td>0.22</td><td>0.28</td><td>2.138</td><td>1.674</td><td>1.495</td><td>0.470</td><td></td><td>0.944</td></tr><tr><td>0.2</td><td>0.3</td><td>2.217</td><td>1.681</td><td>1.496</td><td>0.427</td><td></td><td>0.944</td></tr></tbody></table>

<table><caption>Table 5 Masses, CoM positions and Inertias (about the joint position $\theta_{ij}$) for all the links of the final 4BL2 design.</caption><thead><tr><th rowspan="2">Link</th><th colspan="3">Unbalanced</th><th colspan="6">Balanced</th></tr><tr><th>ij</th><th>m<sub>ij</sub> (kg)</th><th>e<sub>ij</sub> (m)</th><th>I<sub>ij,yy</sub> (kg.m<sup>2</sup>)</th><th>m<sub>c,ij</sub>(kg)</th><th>e<sub>c,ij</sub>(m)</th><th>m<sub>ij</sub>(kg)</th><th>e<sub>ij</sub>(m)</th><th>I<sub>ij,yy</sub> (kg.m<sup>2</sup>)</th></tr></thead><tbody><tr><td>11</td><td>0.137</td><td>0.05</td><td>1.16 &times; 10<sup>-3</sup></td><td>0.065</td><td>-0.25</td><td>0.0257</td><td>-0.063</td><td>7.80 &times; 10<sup>-3</sup></td></tr><tr><td>12</td><td>0.093</td><td>0.074</td><td>1.16 &times; 10<sup>-3</sup></td><td>0</td><td>0</td><td>0.093</td><td>0.074</td><td>1.16 &times; 10<sup>-3</sup></td></tr><tr><td>21</td><td>0.0137</td><td>0.05</td><td>1.16 &times; 10<sup>-3</sup></td><td>0.802</td><td>-0.25</td><td>0.994</td><td>-0.202</td><td>5.39 &times; 10<sup>-2</sup></td></tr><tr><td>22</td><td>0.093</td><td>0.209</td><td>1.99 &times; 10<sup>-2</sup></td><td>0.327</td><td>-0.25</td><td>0.775</td><td>-0.009</td><td>4.29 &times; 10<sup>-2</sup></td></tr></tbody></table>

<table><caption>Table 6 Mass and workspace for different S4BL parameters. The selected concept is in bold.</caption><thead><tr><th>l<sub>11</sub>, l<sub>13</sub><br>(m)</th><th>l<sub>12</sub>, l<sub>14</sub><br>(m)</th><th>l<sub>21</sub>, l<sub>23</sub><br>(m)</th><th>l<sub>22</sub>, l<sub>24</sub><br>(m)</th><th>Unbalanced<br>Mass (kg)</th><th>Mass (kg),<br>e<sub>c</sub> = -l</th><th>Mass (kg),<br>e<sub>c</sub> = -1.5l</th><th>Mass (kg),<br>e<sub>c</sub> = -2l</th><th>WS<br>(m<sup>2</sup>)</th><th>Area</th><th>WS Span<br>(m)</th></tr></thead><tbody><tr><td>0.1</td><td>0.15</td><td>0.16</td><td>0.2</td><td>1.435</td><td>2.422</td><td>2.119</td><td>2.014</td><td>0.227</td><td></td><td>0.744</td></tr><tr><td><b>0.12</b></td><td><b>0.15</b></td><td><b>0.16</b></td><td><b>0.2</b></td><td>1.354</td><td><b>2.33</b></td><td>2.438</td><td>1.887</td><td><b>0.227</b></td><td></td><td><b>0.744</b></td></tr><tr><td>0.1</td><td>0.15</td><td>0.2</td><td>0.3</td><td>1.497</td><td>2.325</td><td>2.056</td><td>1.975</td><td>0.209</td><td></td><td>0.756</td></tr><tr><td>0.12</td><td>0.15</td><td>0.2</td><td>0.3</td><td>1.416</td><td>2.251</td><td>2.008</td><td>1.858</td><td>0.205</td><td></td><td>0.756</td></tr></tbody></table>

<table><caption>Table 7 Masses, CoM positions and Inertias (about the joint position $\theta_{ij}$) for all the links of the final S4BL design.</caption><thead><tr><th rowspan="2">Link</th><th colspan="3">Unbalanced</th><th colspan="6">Balanced</th></tr><tr><th>ij</th><th>m<sub>ij</sub> (kg)</th><th>e<sub>ij</sub> (m)</th><th>I<sub>ij,yy</sub> (kg.m<sup>2</sup>)</th><th>m<sub>c,ij</sub>(kg)</th><th>e<sub>c,ij</sub>(m)</th><th>m<sub>ij</sub>(kg)</th><th>e<sub>ij</sub>(m)</th><th>I<sub>ij,yy</sub> (kg.m<sup>2</sup>)</th></tr></thead><tbody><tr><td>11</td><td>0.0640</td><td>0.025</td><td>1.34 &times; 10<sup>-4</sup></td><td>0.054</td><td>-0.12</td><td>0.145</td><td>-0.045</td><td>1.20 &times; 10<sup>-3</sup></td></tr><tr><td>12</td><td>0.0706</td><td>0.035</td><td>2.55 &times; 10<sup>-4</sup></td><td>0</td><td>0</td><td>0.071</td><td>0.035</td><td>2.55 &times; 10<sup>-4</sup></td></tr><tr><td>13</td><td>0.408</td><td>0.120</td><td>1.28 &times; 10<sup>-2</sup></td><td>0.215</td><td>-0.12</td><td>0.649</td><td>0.123</td><td>1.65 &times; 10<sup>-2</sup></td></tr><tr><td>14</td><td>0.153</td><td>0.097</td><td>2.12 &times; 10<sup>-3</sup></td><td>0</td><td>0</td><td>0.153</td><td>0.097</td><td>2.12 &times; 10<sup>-3</sup></td></tr><tr><td>21</td><td>0.073</td><td>0.039</td><td>1.34 &times; 10<sup>-4</sup></td><td>0.061</td><td>-0.0158</td><td>0.169</td><td>-0.057</td><td>2.49 &times; 10<sup>-3</sup></td></tr><tr><td>22</td><td>0.0816</td><td>0.054</td><td>2.55 &times; 10<sup>-4</sup></td><td>0.001</td><td>0.089</td><td>0.082</td><td>0.054</td><td>5.91 &times; 10<sup>-4</sup></td></tr><tr><td>23</td><td>0.106</td><td>0.0997</td><td>9.27 &times; 10<sup>-4</sup></td><td>0.7576</td><td>-0.1597</td><td>0.898</td><td>-0.126</td><td>2.13 &times; 10<sup>-2</sup></td></tr><tr><td>24</td><td>0.164</td><td>0.127</td><td>2.12 &times; 10<sup>-3</sup></td><td>0</td><td>0</td><td>0.164</td><td>0.127</td><td>3.87 &times; 10<sup>-3</sup></td></tr></tbody></table>

<table><caption>Table 8 Summarized metrics (maximum absolute values) for the line trajectory. Bal. indicates balanced (Y) and un-balanced (N) designs. $\tau_R$ is the reaction moment, and $\tau_{11}$ and $\tau_{21}$ are the joint torques, measured in kg.m$^2$. Energy is measured in J.</caption><thead><tr><th>Design</th><th>Bal.</th><th>τ<sub>R</sub></th><th>τ<sub>11</sub></th><th>τ<sub>21</sub></th><th>Energy</th></tr></thead><tbody><tr><td rowspan="2">5BL2</td><td>Y</td><td>0.3923</td><td>0.1197</td><td>0.3939</td><td>0.0536</td></tr><tr><td>N</td><td>1.2034</td><td>0.3495</td><td>0.9080</td><td>0.1690</td></tr><tr><td rowspan="2">4BL2</td><td>Y</td><td>0.5196</td><td>0.1433</td><td>0.5495</td><td>0.0701</td></tr><tr><td>N</td><td>1.2034</td><td>0.3495</td><td>0.9080</td><td>0.1909</td></tr><tr><td rowspan="2">S4BL</td><td>Y</td><td>0.5950</td><td>0.3720</td><td>0.6713</td><td>0.3371</td></tr><tr><td>N</td><td>1.6656</td><td>2.9953</td><td>2.0618</td><td>4.2508</td></tr></tbody></table>

<table><caption>Table 9 Summarized metrics (maximum absolute values) for the circle trajectory. Bal. categorizes balanced (Y) and unbalanced (N) designs. $\tau_R$ is the reaction moment, and $\tau_{11}$ and $\tau_{21}$ are the joint torques, measured in kg.m$^2$. Energy is measured in J.</caption><thead><tr><th>Concept</th><th>Bal.</th><th>τ<sub>R</sub></th><th>τ<sub>11</sub></th><th>τ<sub>21</sub></th><th>Energy</th></tr></thead><tbody><tr><td rowspan="2">5BL2</td><td>Y</td><td>0.1382</td><td>0.0705</td><td>0.1108</td><td>0.0762</td></tr><tr><td>N</td><td>1.8802</td><td>0.9483</td><td>0.9240</td><td>1.3207</td></tr><tr><td rowspan="2">4BL2</td><td>Y</td><td>0.2687</td><td>0.1123</td><td>0.2006</td><td>0.1192</td></tr><tr><td>N</td><td>1.402</td><td>0.8971</td><td>0.8748</td><td>1.2915</td></tr><tr><td rowspan="2">S4BL</td><td>Y</td><td>0.1528</td><td>0.2145</td><td>0.0742</td><td>0.0341</td></tr><tr><td>N</td><td>1.7764</td><td>3.0201</td><td>2.3233</td><td>5.3779</td></tr></tbody></table>
<table><caption>Table 10 Summarized metrics (maximum absolute values) for the load trajectory for all the simulated concepts. Bal. categorizes balanced (Y) and unbalanced designs. $τ_R$ is the reaction moment, and $τ_{11}$ and $τ_{21}$ are the joint torques, measured in kg.m². Energy is measured in J.</caption><thead><tr><th>Concept</th><th>Bal.</th><th>τ<sub>R</sub></th><th>τ<sub>11</sub></th><th>τ<sub>21</sub></th><th>Energy</th></tr></thead><tbody><tr><td rowspan="2">5BL2</td><td>Y</td><td>1.6027</td><td>0.8451</td><td>0.6979</td><td>0.9078</td></tr><tr><td>N</td><td>3.1456</td><td>1.7681</td><td>1.3979</td><td>1.9722</td></tr><tr><td rowspan="2">4BL2</td><td>Y</td><td>1.3915</td><td>0.9912</td><td>0.6529</td><td>0.2332</td></tr><tr><td>N</td><td>2.3525</td><td>1.8965</td><td>1.1713</td><td>1.0342</td></tr><tr><td rowspan="2">S4BL</td><td>Y</td><td>1.458</td><td>1.2980</td><td>1.5558</td><td>0.2321</td></tr><tr><td>N</td><td>2.9784</td><td>2.9893</td><td>3.8245</td><td>0.9402</td></tr></tbody></table>
## List of Figures

<table><tr><td>1</td><td>A four bar linkage.</td><td>10</td></tr><tr><td>2</td><td>Five bar linkage as a closed-loop of two RRR chains. The actuated joint for each chain is A<sub>i</sub> and passive second joint is B<sub>i</sub>. The generalized coordinate for the joint degree of freedom is θ<sub>ij</sub>, link length l<sub>ij</sub> and CoM vector is [e<sub>ij</sub>, f<sub>ij</sub>]<sup>T</sup> of each link in the joint reference frame as shown. The indexing system uses i to represent the chain number where {A, B}: {1, 2}, and j as the joint number in the chain.</td><td>10</td></tr><tr><td>3</td><td>The 5BL2 manipulator concept. Counter masses (black circles) are mounted to adjust the link mass and CoM position (checkered circles). The inline vector of the link and mass relative to the joint is e<sub>ij</sub>(m<sub>ij</sub>). The closed loop joint position is p<sub>c</sub> and end-effector is at p<sub>e</sub>.</td><td>10</td></tr><tr><td>4</td><td>The 4BL2 concept. Counter masses (black circles) are mounted to adjust the link mass and CoM position (checkered circles). The inline vector of the link and mass relative to the joint is e<sub>ij</sub>(m<sub>ij</sub>). The closed loop joint position is p<sub>c</sub> and end-effector is at p<sub>e</sub>.</td><td>10</td></tr><tr><td>5</td><td>The S4BL manipulator concept. Counter masses (black circles) are mounted to adjust the link mass and CoM position (checkered circles). The inline vector of the link and mass relative to the joint is e<sub>ij</sub>(m<sub>ij</sub>). The closed loop joint position for 4BL1 is p<sub>1</sub>, for 4BL2 is p<sub>2</sub>, and end-effector is p<sub>e</sub>.</td><td>10</td></tr><tr><td>6</td><td>The workspace for the selected set of parameters of 5BL2. A resolution of 10 degrees for the radial sweep, with a constant end-effector velocity of 0.1 m/s was used. The fixed link is black, end-effector is green. Links are lines and joints are circles, with red indicating actuated and blue indicating passive.</td><td>10</td></tr><tr><td>7</td><td>The workspace for the selected set of parameters of 4BL2. A resolution of 10 degrees for the radial sweep, with a constant end-effector velocity of 0.1 m/s was used. The fixed link is black, end-effector is green. Links are lines and joints are circles, with red indicating actuated and blue indicating passive.</td><td>11</td></tr><tr><td>8</td><td>The workspace for the selected set of parameters of S4BL. A resolution of 10 degrees for the radial sweep, with a constant end-effector velocity of 0.1 m/s was used. The fixed links are black, and end-effector is green. Links are lines and joints are circles, with red indicating actuated and blue indicating passive.</td><td>11</td></tr><tr><td>9</td><td>End-effector velocities for the line trajectory.</td><td>11</td></tr><tr><td>10</td><td>Results for 5BL2 with line trajectory. The solid lines are balanced and dotted lines are unbalanced.</td><td>11</td></tr><tr><td>11</td><td>Results for 4BL2 with line trajectory. The solid lines are balanced and dotted lines are unbalanced.</td><td>11</td></tr><tr><td>12</td><td>Results for S4BL with line trajectory. The solid lines are balanced and dotted lines are unbalanced.</td><td>11</td></tr><tr><td>13</td><td>End-effector velocities for the circle trajectory.</td><td>12</td></tr><tr><td>14</td><td>Results for 5BL2 with circle trajectory. The solid lines are balanced and dotted lines are unbalanced.</td><td>12</td></tr><tr><td>15</td><td>Results for 4BL2 with circle trajectory. The solid lines are balanced and dotted lines are unbalanced.</td><td>12</td></tr><tr><td>16</td><td>Results for S4BL with circle trajectory. The solid lines are balanced and dotted lines are unbalanced.</td><td>12</td></tr><tr><td>17</td><td>End-effector velocities for the load trajectory.</td><td>12</td></tr><tr><td>18</td><td>Results for 5BL2 with load trajectory. The solid lines are balanced and dotted lines are unbalanced.</td><td>12</td></tr><tr><td>19</td><td>Results for 4BL2 with load trajectory. The solid lines are balanced and dotted lines are unbalanced.</td><td>13</td></tr><tr><td>20</td><td>Results for S4BL with load trajectory. The solid lines are balanced and dotted lines are unbalanced.</td><td>13</td></tr></table>

## List of Tables

<table><tr><td>1</td><td>Mass for different parameters of the 5BL2 design concept. The chosen parameters for concept comparison are in bold.</td><td>14</td></tr><tr><td>2</td><td>Workspace (WS) for different link and fixed bar lengths of the 5BL2 design. The chosen parameters for concept comparison in bold.</td><td>14</td></tr><tr><td>3</td><td>Masses, CoM positions and Inertias (about the joint position θ<sub>ij</sub>) for all the links of the final 5BL2 design.</td><td>15</td></tr><tr><td>4</td><td>The mass and workspace (WS) of the 4BL2 for different kinematic parameters. The most feasible design, selected for dynamic analysis, are in bold.</td><td>15</td></tr><tr><td>5</td><td>Masses, CoM positions and Inertias (about the joint position θ<sub>ij</sub>) for all the links of the final 4BL2 design.</td><td>15</td></tr><tr><td>6</td><td>Mass and workspace for different S4BL parameters. The selected concept is in bold.</td><td>15</td></tr><tr><td>7</td><td>Masses, CoM positions and Inertias (about the joint position θ<sub>ij</sub>) for all the links of the final S4BL design.</td><td>15</td></tr><tr><td>8</td><td>Summarized metrics (maximum absolute values) for the line trajectory. Bal. indicates balanced (Y) and unbalanced (N) designs. τ<sub>R</sub> is the reaction moment, and τ<sub>11</sub> and τ<sub>21</sub> are the joint torques, measured in kg.m<sup>2</sup>. Energy is measured in J.</td><td>15</td></tr><tr><td>9</td><td>Summarized metrics (maximum absolute values) for the circle trajectory. Bal. categorizes balanced (Y) and unbalanced (N) designs. τ<sub>R</sub> is the reaction moment, and τ<sub>11</sub> and τ<sub>21</sub> are the joint torques, measured in kg.m<sup>2</sup>. Energy is measured in J.</td><td>15</td></tr><tr><td>10</td><td>Summarized metrics (maximum absolute values) for the load trajectory for all the simulated concepts. Bal. categorizes balanced (Y) and unbalanced designs. τ<sub>R</sub> is the reaction moment, and τ<sub>11</sub> and τ<sub>21</sub> are the joint torques, measured in kg.m<sup>2</sup>. Energy is measured in J.</td><td>16</td></tr></table>