This paper proposes a spring efficiency assessment of a statically spring-balanced planar serial manipulator. The admissible spring configurations for the static balancing of planar serial manipulators without auxiliary links have been determined in the past. Gravity is balanced by the spring configuration systematically; however, the spring configuration also contains counter-effects between springs. Conceptually, with fewer counter-effects between springs, there is less burden on the spring system, which means that the springs are used more efficiently, and accordingly, the system would be safer, and its service life would be longer. In this study, the spring energy is represented in a quadratic form. The coefficients in a quadratic form represent the change in elastic energy with the relative position between links, which is named “elastic pseudo-stiffness”. Compared to the quadratic form of gravity energy, those elastic pseudo-stiffnesses for static balancing are regarded as positive contributions of a spring, while those that contain counter-effects are seen as negative ones. Spring efficiency is defined as the ratio of the elastic pseudo-stiffnesses, which has positive contributions for balancing to total elastic pseudo-stiffnesses. To use springs efficiently, the counter-effects, which are functions of spring parameters, need to be decreased, including spring stiffness and the attachment location of springs on links. A method to use spring efficiently by adjusting spring parameters is developed. Furthermore, it is found that, for a spring attached between adjacent links, the spring efficiency is 100 %, and the spring efficiency decreases while the number of joints over which the spring spans increases. In a spring manipulator system, the efficiency is negatively correlated to the payload. As an example, an efficiency assessment on a 3 degrees of freedom (DOF) manipulator is shown at the end.

In recent decades, new static balancing technologies and system designs have been developed for multiple applications, such as a wearable exoskeleton for rehabilitation and training (Arakelian and Ghazaryan, 2008; Lin et al., 2013; Tschiersky et al., 2019; Kuo et al., 2021) and robotic manipulators (Arakelian, 2016; Kim and Cho, 2017). These manipulators provide a better control of the performance and more efficiency because the actuating force that is sustaining the weight of their system is partially or fully balanced.

One of the spring balancing methods uses a serially connected, four-bar,
parallelogram mechanism; this mechanism uses auxiliary links to connect
other links vertically to the ground, forming a pseudo-base, and each
parallelogram is independently balanced by a single spring (Nathan, 1985; Rahman et al., 1995; Lin et al., 2010). Another kind of spring balancing method uses auxiliary links to form a pantograph mechanism that is located in the center of mass. Springs are then attached to the center of mass to keep the total energy constant (Agrawal and Fattah, 2004a, b; Fattah and Agrawal, 2006; Najafi and Sepehri, 2011). Besides the methods with auxiliary links, Jamshidifar et al. (2021) proposed an approach by using gravity compensator consisting of one spring and multiple pulleys, which mitigates the issue of workspace interference caused by auxiliary linkages. In addition to the perfect balance, several methods with the partial balancing of a robot are proposed. For instance, adding gear spring modules as gravity compensators on the three revolute–prismatic–spherical (3 RPS) parallel robot (Nguyen et al., 2020) can balance the gravity partially. However, applications with auxiliary links or additional devices have several disadvantages, such as motion interference caused by the auxiliary links. Also, the additional mass of additional devices may increase the load of springs. To mitigate these problems, balancing methods that do not use auxiliary links but directly attach the springs to the manipulator have been proposed (Lin et al., 2011; Deepak and Ananthasuresh, 2012; Lee and Chen, 2014; Juang and Chen, 2022) and used in this paper. The gravity balancing method without auxiliary links uses springs to balance the gravity of manipulators systematically. Ideally, the elasticity of all of the springs is used to cancel out the gravity, but the method, as the manipulator's degrees of freedom (DOF) are more than 2, requires multiple springs to be installed and therefore contains counter-effects between springs. In other words, only part of the elasticity is used to counter the gravity of manipulators; the other parts are redundant and cancel each other out. For example, for a 2 DOF
manipulator in Fig. 1, referring to Lee and Chen (2014), two springs

A spring-balanced 2 DOF manipulator.

A (

Diagram of a zero-free length (ZFL) spring installed between links

Conceptually, the fewer the counter-effects between springs, the more the burden of the spring system is decreased. This means that the springs are more efficiently used; hence, the system would be safer, and its service life would be extended. The gravity balancing method by springs without auxiliary links is developed (Lin et al., 2011; Deepak and Ananthasuresh, 2012; Lee and Chen, 2014; Juang and Chen, 2022), but there are no research papers discussing the issue of how to use springs efficiently to balance the gravity of a manipulator. This study aims to develop a method to use springs efficiently, and thereby, the assessment of the spring efficiency is presented here.

The paper is structured as follows. Section 2 discusses the gravitational and elastic energies of a spring manipulator system and the static balancing conditions. Section 3 presents the concept of spring efficiency and the criteria for the efficient use of a spring. Furthermore, Sect. 4 discusses the efficiency assessment of a spring manipulator system, and the efficient spring configurations are proposed. In addition, the characteristics of spring parameters and the effect of payload on the system spring efficiency are suggested. Section 5 discusses the efficiency assessment by presenting an illustrative example of a spring-balanced, 3 DOF planar manipulator. The adjustment of spring parameters and payload are also displayed. Section 6 serves as a conclusion of all the previous sections.

Schematic of the installation of the springs and the attachment parameters of a 3 DOF manipulator with a spring configuration

For a serial planar manipulator with revolute joints only, we assume that
the center of mass (COM) of each link is located on the axis between link
pivots. In Fig. 2, the symbols used to represent a link's dimensions in the
manipulator system are presented. While

For link

Figure 3 illustrates a zero-free length (ZFL) spring

The system's spring efficiency of the example 3 DOF manipulator in the workspace.

The potential energies of the example 3 DOF spring manipulator system in the workspace.

As a ZFL spring is attached between links

By substituting Eq. (8) into Eq. (9),

The trend of the system spring efficiency index of a 3 DOF manipulator and variable payload.

According to Eqs. (11a)–(11d), the elastic pseudo-stiffnesses provided by a spring

A spring attached between two adjacent links, where only one joint is
spanned over, is named a “mono-articulated spring”. Otherwise, a spring
attached between a pair of links that are not adjacent, where multiple
joints are spanned over, is named a “multi-articulated spring”. The number
of joints a spring spans over determines the number of elastic pseudo-stiffnesses provided by a spring. For a mono-articulated spring

In this paper, only the extension spring is used, i.e., the spring stiffness

The term,

The static balancing of a spring manipulator system is achieved when the total energy of the system is a constant in different postures of the manipulator. Comparing Eq. (5) with Eq. (10), the summation of the terms, which vary with the posture of the manipulator, should be zero when static balancing is achieved. Therefore, the static balancing of a spring manipulator can be simplified as the summation of the pseudo-stiffnesses between the links of the spring manipulator system as being zero.

Since the gravitational pseudo-stiffnesses only exist between the ground and
other links, the condition of gravitational energy balancing is that the
summation of pseudo-stiffnesses between the ground and other links is zero,
which can be expressed as follows:

It is shown that, besides the elastic pseudo-stiffnesses used to balance the gravitational pseudo-stiffnesses, i.e., Eq. (16a), there are counter-effects between springs, i.e., Eq. (16b). The counter-effects between springs consist of the remaining elastic pseudo-stiffnesses which need to be balanced (elastic balanced part), and the corresponding elastic pseudo-stiffnesses contributed by other springs is used to cancel out the elastic balanced part (elastic balancing part). The elastic pseudo-stiffnesses of a spring can therefore be separated into the “balancing part” and “balanced part”. To identify whether an elastic pseudo-stiffness is the balancing or the balanced part, more explanations are given in the following section.

The elastic pseudo-stiffness contributed by a spring can be classified into
two groups, i.e., the balancing part and the balanced part. Accordingly, the
spring efficiency is conceptually defined as ratio of the balancing part to
total elastic pseudo-stiffnesses. The efficiency of a spring

To achieve the gravitational balancing condition in Eq. (16a), the sign of the elastic pseudo-stiffness must be opposite to the sign of the corresponding gravitational pseudo-stiffness. Under this circumstance, such elastic pseudo-stiffness is regarded as the balancing part; otherwise, it is regarded as the balanced part.

In a previous study (Juang and Chen, 2022), the admissible attachment angles of springs for statically balanced planar articulated manipulators have been proposed. For a ground-connected spring

A criterion for using a ground-connected spring efficiently is proposed as C1, which is defined below.

C1 will ensure that a ground-connected spring is efficiently used to balance the gravity, so the ground-connected spring should be attached with

For such a ground-connected spring to satisfy C1, the elastic pseudo-stiffnesses

To ensure that a non-ground connected spring

A criterion for using non-ground-connected spring efficiently is proposed as C2 below.

C2 will ensure that a non-ground-connected spring is efficiently used to balance the elasticity, so the non-ground-connected spring should be attached with

In a spring manipulator system, if there are balanced elastic pseudo-stiffnesses that do not correspond to any balancing elastic pseudo-stiffness, then the additional spring needs to be installed. Not until all the balanced elastic pseudo-stiffnesses are offset can the elastic energy balancing condition in Eq. (16b) can be achieved.

For a multi-articulated spring

For a ground-connected spring

For a non-ground-connected spring

For a non-ground-connected spring

Note that, for a mono-articulated spring

In this section, how the spring attachment parameters affect the spring
efficiency is examined. Also, it is shown that, as the spring is installed
under the same condition, the lower the number of joints the spring spans
over, the better the spring efficiency. However, to achieve the static
balancing of the spring manipulator system, the spring attachment parameter
and the number of joints the spring spans over are not arbitrary. In the
following section, the constraints of the spring attachment parameter and
the assessment of system spring efficiency are discussed. Note that,
referring to Juang and Chen (2022), the non-ground-connected springs
can also be attached with

The efficiency of a spring manipulator system can be conceptually defined, as
follows, to show the performance of efficiency in the system.

The efficient spring configurations for the 1–3 DOF manipulators.

In the system, the spring attachment parameters are constrained by the
balancing equations as in, i.e., Eqs. (16a) and (16b), so the spring parameters are not arbitrarily used. According to Eqs. (16a) and (16b), to attain static balancing, the magnitude of the balancing elastic pseudo-stiffnesses should be equal to the corresponding magnitude of the balanced gravitational

Adjustment of the spring attachments for efficient spring configurations.

The spring parameters can be determined by the balancing Eqs. (16a) and (16b). We take an efficient spring configuration for a 3 DOF manipulator

On the other hand, according to Eqs. (27b) and (27d), the constraints indicate that the spring stiffness

The dimension, mass, and workspace for the links of an example 3 DOF planar manipulator.

Spring parameters for the example 3 DOF manipulator.

Similarly, for a 2 DOF manipulator with the spring configuration

R1 is used for a better

R2 is used for a better

R3 is used for a better

For a 1 DOF manipulator, only one mono-articulated spring

System efficiency index after adjustment of spring parameters.

Note that, according to the constraints of spring parameters in Eqs. (27b), (27d), and (27e), the spring stiffnesses are changed with adjustment of spring attachment points.

In the previous section, the dimensions and mass of the links are given; therefore,

For a given manipulator, when a payload

As the payload changed, the attachment of springs is adjusted accordingly.
Assuming that the spring stiffnesses are fixed, only the attachment
distances of the springs

The ABB IRB 140 robot arm (ABB Inc., Auburn Hills, MI 48326, USA), an industrial manipulator with six rotational axes, was used in the real world (Suárez and Heredia, 2013; Almaged, 2017). The example referring to the dimension, mass, and workspace of the links of the ABB IRB 140 industrial manipulator is proposed. The operation of the manipulator is assumed to work at a constant speed. Only a planar motion is considered, and an additional link is added to form a 3 DOF planar manipulator. The dimension, mass, and workspace are listed in Table 3.

After substituting the dimension and mass of the links into Eq. (7),

In the example, the spring configuration

The schematic of the manipulator and spring attachment parameters are presented in Fig. 4 (note that, in Fig. 4, the springs attached to a manipulator are ZFL springs that can be formed by adopting a cable pulley arrangement; Ou and Chen, 2017).

Substituting the spring parameters in Table 4 and

The system spring efficiency is a variable which is determined by the
posture of the links. Therefore, the “system spring efficiency index” is
proposed, which is defined as the mean of

To show the quality of static balancing, the gravitational and elastic energies in the workspace are presented in Fig. 6.

According to R2, the spring parameters can be adjusted to achieve a better

The system efficiency index variance over

When a payload

Note that the unit of the payload

This paper proposes a method to assess the spring efficiency and other methods for the efficient use of springs. The definition of pseudo-stiffness is provided, which can be regarded as the effect of the relative position between two links on potential energy. Static balancing condition can be simplified as the summation of pseudo-stiffnesses being zero. The elastic pseudo-stiffnesses of springs are classified into two categories, namely the balancing part and the balanced part. In this paper, spring efficiency is defined as the ratio of balancing part to total elastic pseudo-stiffnesses. Conceptually, for the efficient use of a spring, it requires an increase in the balancing part. Following this concept, the criteria for the efficient use of springs are proposed, where the pseudo-stiffnesses contributed by ground-connected springs should be used to compensate the gravitational pseudo-stiffnesses, and the pseudo-stiffnesses contributed by non-ground-connected springs should be used to compensate the elastic pseudo-stiffness retained by ground-connected springs. Also, a method to use a spring efficiently by adjusting the spring attachment points on the links and spring stiffness is developed. Furthermore, it is found that, as the number of joints that the spring spans over increases, so the spring efficiency decreases. By extending the result of the efficient use of a spring to the spring manipulator system, it is shown that spring configurations with the minimum number of joints that a spring spans over can be regarded as efficient spring configurations. Furthermore, considering that a payload is added at the COM of the end link, it is found that the system spring efficiency is negatively correlated to the amount of payload. Finally, a 3 DOF manipulator system spring efficiency assessment is shown as an illustrative example.

The code in this research is available upon request by contact with the corresponding author.

No data sets were used in this article.

The paper was written with the contributions of all authors. CWJ and CSJ developed the methodology and also completed the verification. CWJ wrote the paper. DZC guided the research and reviewed the paper. All authors have worked proportionally and given approval to the present research.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors gratefully acknowledge the support of the Ministry of Science and Technology (MOST).

This research has been supported by the Ministry of Science and Technology, Taiwan (grant no. 109-2221-E-002-002-MY3).

This paper was edited by Wuxiang Zhang and reviewed by Basilio Lenzo and two anonymous referees.