Nonlinear Gear-Spring Design for Gravity Balancing of Robotic Manipulators With Variable Payloads: Methods and Comparison Open Access

Modern manufacturing puts robotic manipulators at the central of many industrial operations where precision and efficiency are critical, such as welding, material handling, packaging and labeling, painting, assembly, and machining [1]. However, industrial robots are often integrated with powerful actuators to fulfill required tasks, which consume significant energy during operation [2]. One of the most significant factors that contribute to the energy consumption of industrial robots is the weight of the links and payload [3,4]. For low-speed operation, the energy consumed for handling the robot's weight is dominant among other factors [5–7].

The gravity balancing of robotic manipulators refers to a technique used to counteract the effects of gravity on the moving parts of a robot, especially its joints and links [8,9]. The ultimate goal of gravity balancing is to achieve a state in which a robot can move with minimal resistance from gravitational forces, allowing for more precise and energy-efficient operation [10–12]. Gravity balancing can be regarded as a green technology, particularly valuable in applications where robots interact with the environment or are required to handle heavy payloads, such as assembly tasks [13,14] and human-weight support [15–17]. The principle of constant potential energy is commonly employed to determine the appropriate parameters for gravity balancing designs in robotic systems [18–20]. Since the potential energy of a robot varies with its configuration, energy storage elements, such as counterweights and springs, are integrated into the robot's links or joints to compensate for these variations by storing and releasing energy as needed [6,21].

Counterweight methods involve attaching supplemental counterweights to the moving parts of a robot to balance the gravitational forces acting on those parts [22–24]. Counterweight methods can provide a simple and effective solution for gravity balancing. However, notable drawbacks of these methods include negative impacts on the system dynamics due to the increased mass and inertia, inefficiency with variable payloads as the counterweights are typically fixed to the links or joints, and their large space requirements [5,6,10].

Spring methods utilize mechanical springs to counteract gravitational forces by storing and releasing potential energy through auxiliary mechanisms [25–28]. Compared to counterweight methods, spring methods offer several advantages, including reduced weight, adaptability to variable payloads, improved space efficiency, and greater versatility in multi-axis systems with complex motion profiles [10,29]. Owing to these advantages, spring methods are widely adopted in the gravity balancing design of robotic manipulators [30–33].

The most fundamental problem in gravity balancing is counteracting the unbalanced force generated during the rotation of a single-degree-of-freedom (1-DoF) mass [6,34]. Many researchers have solved this problem by using various mechanisms for spring attachment [35–40]. For instance, Hung and Kuo [35] used a Cardan gear mechanism, Zhang et al. [36] used a bidirectional spiral pulley negative-stiffness mechanism, Arakelian and Zhang [37] used an inverted slider-crank mechanism, Fedorov and Birglen [38] designed a differential noncircular pulley mechanism, and Shieh and Chou [39] used a Scotch yoke mechanism with a pair of gears.

Typical serial manipulators are equipped with multiple DoFs to provide high flexibility and versatility in performing complex motions. Gravity balancing for such manipulators is a complex and challenging task, as each joint is influenced by the gravitational forces induced by multiple interconnected links [41–43]. For ease of design, a multi-DoF auxiliary system is integrated into the manipulator to decouple the gravitational torques acting at individual joints. Typical auxiliary systems employed for gravity balancing include parallelogram linkages [44–49], pulley-belt systems [42,43], and bevel-gear systems [41,50]. With the aid of these systems, gravity balancing for each joint can be accomplished independently. For example, with parallelogram linkages, Koser [44] integrated a cam-follower mechanism and a spring into each joint, and Zhou et al. [48] used a Cardan gear mechanism and a spring. With pulley-belt systems, Lee and Seo [42] implemented a multiwinding mechanism and spring to each joint, and Peng and Bu [31] employed a double-gear-slider mechanism and spring. By using bevel-gear systems, Kim et al. [41] incorporated a slider-crank mechanism and spring into each joint for gravity balancing, and Min et al. [50] employed a pulley-wire mechanism and spring. Moreover, spring-based designs can also be directly integrated into a robot for gravity balancing, such as the geared five-bar mechanism with spring [30] and spring suspension designs [51,52].

Robotic manipulators are widely utilized across various industries to perform a broad spectrum of tasks, often involving the manipulation of objects with differing sizes, shapes, and weights. To accommodate these diverse operational requirements, manipulators must be capable of handling a range of payloads with efficiency and adaptability [53,54]. Thus, gravity balancing for robotic manipulators with variable payloads is crucial, yet challenging, as changes in payload compromise the balance condition. To accommodate variable payloads, it is necessary to implement an adjustment procedure for at least one parameter within the gravity balancing design, allowing compensation for the change in potential energy resulting from payload variation [55–57]. The most common approach to making a gravity balancing design compatible with variable payloads is to adjust the preload of a spring. For instance, Takesue et al. [58] introduced an eccentric-follower mechanism with two orthogonal springs, while Yang and Lan [56] proposed a double-slider mechanism incorporating both an extension and a compression spring. Although these methods allow straightforward preload adjustment, typically implemented using a screw mechanism, their primary drawback lies in the considerable amount of energy required to achieve such adjustments. An alternative approach to achieving variable-payload capability involves adjusting the number of active coils in a spring using a spindle-nut mechanism, as proposed by Van Dorsser et al. [59]. Note that friction, complex design, and a limited range of payloads are considered the disadvantages of this method. The variable-payload capability can also be achieved by altering the position of the spring ends using, for example, a cam-rack-gear mechanism made by Chu and Kuo [60], a lock-slider mechanism by Van Dorsser et al. [57], and a rack-pinion mechanism by Chew et al. [61]. Note that these mechanisms were designed using zero-free-length springs, which were practically implemented through pulley-cable systems. Consequently, the overall gravity balancing design becomes relatively complex and requires a considerable amount of space for spring installation. More importantly, the gravity balancing methods discussed earlier are limited to single-DoF systems with variable rotating masses.

This article presents a nonlinear gear-spring design for gravity balancing of robotic manipulators with variable payloads. Three design methods for serial manipulators are introduced and compared, all based on the compact gear-spring mechanism (CGSM) previously proposed by the author in Ref. [62]: direct installation, integration with parallelogram linkages, and integration with a pulley-belt system. Compared to traditional gravity balancing methods for serial manipulators, the proposed approaches offer several key advantages: (i) they enable compact designs with high performance, (ii) they accommodate variable payloads, and (iii) they incorporate gear friction losses into the performance analysis. In this study, the effectiveness of the proposed methods is evaluated through numerical simulations and experimental validation using a robot arm.

The remainder of this article is organized as follows. Sections 2 and 3 present the nonlinear gear-spring design and gear friction analysis, respectively. Then, the performance assessment of the robotic manipulators is shown in Sec. 4. Next, Sec. 5 describes an experimental study. Sections 6 and 7 present the discussion and conclusion, respectively.

This section uses a 3-DoF planar serial manipulator as a case study. Then, the nonlinear gear-spring design of the manipulator is illustrated through three methods: direct installation, parallelogram, and pulley-belt.

Figure 1 shows the nonlinear gear-spring design of the 3-DoF serial manipulator based on the direct installation method. The manipulator consists of three links (links 1, 2, and 3) articulated to the ground (link 0). Link i (i = 1, 2, 3) is connected to link i–1 by a revolute joint O_i. On each link i, a CGSM is implemented so that the center of gear i1 coincides with the revolute joint O_i, and the centers of gears i2 and i3 (A_i, B_i) are placed on the longitudinal axis of link i. Gear i1 is fixed to link i–1, as the base of CGSM i, and gears i2 and i3 can rotate about joints A_i and B_i relative to link i. Two cranks i1 and i2 are fixed to gears i2 and i3 so that the longitudinal axes of cranks i1 and i2 pass through the centers of gears i2 and i3, respectively. Spring i is connected between cranks i1 and i2. The spring ends (P_i, Q_i) are positioned on the longitudinal axes of the cranks by adopting sliders i1 and i2, respectively. The movement of the sliders enables the change in the positions of the spring ends.

In Eqs. (1)–(3), T_si, T_si1, and T_si2 stand for the spring torques exerted on link i, gear i2, and gear i3, respectively; A_i, B_i, P_i, and Q_i are the position vectors of points A_i, B_i, P_i, and Q_i, respectively; F_si1 and F_si2 are the elastic forces caused by spring i; and η_i1 and η_i2 the gear transmission ratios of CGSM i.

Note that Eq. (7) can be satisfied even when the configuration of the manipulator changes (or the joint angles θ_i vary with respect to time).

In Eqs. (8)–(10), L_m represents the Lagrangian function; W_i and U_i represent the kinetic and potential energies of link i (i = 1, 2, 3), respectively; W_ij and U_ij are the kinetic and potential energies of gear ij (j = 1, 2, 3), respectively; W_e and U_e represent the kinetic and potential energies of the payload, respectively; and $θ˙$ the angular velocity vector of the manipulator.

In Eqs. (11)–(14), m_ci, m_ij, and m_e stand for the masses of link i at the center C_i, gear ij (i, j = 1, 2, 3), and the payload, respectively; a_i and s_i are the length of link i and distance from the center of mass C_i to joint O_i, respectively; and g is the gravitational acceleration (g = 9.81 m/s²). Assume that links 1–3 are thin rods with symmetrical geometry, leading to s_i = a_i/2.

From Eqs. (11)–(15), one can observe that when the payload m_e changes, the spring stiffnesses k_i (i = 1, 2, 3) or the crank lengths r_i4 can be adjusted to maintain the balancing condition. The former can be achieved by adjusting the number of active spring coils, as presented in Ref. [59], which is quite complicated. The latter is the approach used in this study by using two sliders (P_i, Q_i) on CGSM i to adjust the crank lengths r_i4, as shown in Fig. 2. Theoretically, the variable-payload adjustment is energy free when performed at the initial configuration (θ_i = 0), as the spring torques T_si are zero.

Figure 3 shows the nonlinear gear-spring design of the 3-DoF serial manipulator based on the parallelogram method. Similar to the direct installation presented in Sec. 2.1, the parallelogram method employs three CGSMs attached to the manipulator. The main difference is that it exploits two parallelogram linkages (O₁D₁D₂O₂, O₂D₂D₃O₃) to maintain the orientations of the reference links (O₂D₂, O₃D₃), respectively. Gears 21 and 31 are fixed to links O₂D₂ and O₃D₃, respectively, ensuring that the orientations of gears 11, 21, and 31 remain the same during manipulator operation.

In Eqs. (16)–(19), k_i, θ_i, and r_i4 (i = 1, 2, 3) represent the stiffness coefficient of spring i, the joint angle of link i, and the crank length of CGSM i, respectively; m_ci and s_i (i = 1, 2,…,7) are the mass of link i and its center of location from the pivot point.

Equations (17)–(19) show that the quantities Ψ_i depend on the payload m_e. When the payload varies, Eq. (20) can still be held by adjusting the crank lengths r_i4. In other words, the parallelogram method is capable of accommodating variable payloads.

Figure 4 shows the nonlinear gear-spring design of the 3-DoF serial manipulator based on the pulley-belt method. In this design, a pulley-belt system (four pulleys and two belts) is adopted to maintain the orientations of gears 21 and 31. Two identical pulleys are attached to joints O_i (i = 1, 2), and two others are mounted at joints O_i (i = 2, 3). Belt 1 is connected from pulleys 1 to 2, while belt 2 is from pulleys 3 to 4.

In Eqs. (22)–(24), m_pu (u = 1, 2, 3, 4) represents the mass of pulley u.

Similar to the parallelogram method, the pulley-belt method can also accommodate variable payloads by adjusting the crank lengths r_i4.

This section analyzes the effect of gear friction on the nonlinear gear-spring design of the serial manipulator. According to Diez-Ibarbia's studies [64,65], gear friction appears at the interface of two gears, and it causes errors in power and force transmissions. Let us consider the gear friction analysis between gears 11 and 12, as shown in Fig. 5. In this figure, point G stands for the instantaneous contact point between the two gears, which moves along the contact path as the gears rotate. Key points along the contact path include the contact beginning point H₁, the beginning of single contact point H₂, pitch H₃, the ending of single contact point H₄, and the contact ending point H₅.

In Eqs. (26)–(28), μ_c1 and H_V1 stand for Niemann's friction coefficient [66] and gear loss factor [67] along the contact path of the gear system, respectively; F_Nmax represents maximum normal force; w_g represents gear width; V_C is the total speed of the contact point along the linear vector; ρ_c is the reduced radius of curvature at the pitch point; η_oil is the dynamic oil viscosity; R_a is the arithmetic mean roughness of the teeth profile; X_L is the lubricant factor; z₁ is the number of teeth on the pinion gear; β_b is the base helix angle; ε_α is the transverse contact ratio (1 ≤ ε_α ≤ 2); and ε₁ and ε₂ are the addendum contact ratios of the mating gears. More details of the gear friction analysis can be found in Refs. [64,65].

Note that when gear friction is considered, the spring torques in Eqs. (32)–(34) are used to determine the actuator torques T_mi (Eq. (8)) rather than those in Eqs. (4)–(6). The spring torques in Eqs. (32)–(34) are not taken for the derivation of the balancing condition (Eq. (15)).

This section assesses the effectiveness of the proposed design methods for gravity balancing with consideration of payload variation.

In Eqs. (35)–(40), T_mi(k_i ≠ 0) and T_mi(k_i = 0) represent the actuator torques at joint O_i in the balanced state (with spring) and unbalanced state (without spring), respectively; P_mi(k_i ≠ 0) and P_mi(k_i = 0) are the powers; E_mi(k_i ≠ 0) and E_mi(k_i = 0) are the energies; P_hi is the power loss due to heat; k_bi and k_ti are the back-electromotive tension constant and torque constant, respectively; E_Ii and E_Ri are the energy consumptions due to motor inductance and resistance [4]; t₀ and t₁ are the start and end time, respectively.

In Eqs. (41)–(43), GTRR λ_t, GPRR λ_p, and GERR λ_e range from 0% to 100% (lowest to highest performance). Without considering frictions and dynamic torques, perfect gravity balancing can achieve the highest performance (λ_t = λ_p = λ_e = 100%), while imperfect gravity balancing produces lower values (λ_t, λ_p, λ_e < 100%). Moreover, gravity balancing is ineffective with λ_t = λ_p = λ_e = 0%.

Let us consider two manipulators designed by using the direct installation and pulley-belt methods, respectively. As presented in Sec. 2, the direct installation maintains the serial architecture, while using a pulley-belt system results in a quasi-serial architecture. For comparison, assume that these manipulators have the same link lengths, masses, gear pitch radii, and gear masses. The parameters of the manipulators are listed in Table 1.

The serial manipulator cannot achieve perfect gravity balancing over the overall workspace, as the actuator torques are subjected to the initial setting angles. In contrast, the quasi-serial manipulator can make the actuator torques equal zero over the workspace, as shown in Fig. 6. When the initial setting angles “θ_0i = 0” are taken for the serial manipulator, the performance of the two manipulators over the overall workspace is detailed in Table 2. It is observed that, for gravity balancing, the maximum and mean values of the actuator torques are increased in the serial manipulator. For example, at joint O₁, the maximum and mean actuator torques increase from 7.98 to 13 N-m and 2.39 to 6.88 N-m. These results imply that the direct installation method harms the balancing performance. In the quasi-serial manipulator, the maximum and mean actuator torques at all the joints are reduced to zero.

In this study, the serial and quasi-serial manipulators are requested to track along a trajectory, as shown in Fig. 7. It is assumed that the orientation of link 3 remains unchanged during the operation of both manipulators. The motion profiles are detailed in Fig. 8. The initial configuration (θ₁ = 30 deg, θ₂ = 120 deg, θ₃ = –105 deg) is chosen for the serial manipulator. The gear parameters of the CGSMs are listed in Table 3.

Figure 9 shows the actuator torques of the manipulators during trajectory tracking. For the serial manipulator, the maximum unbalanced and balanced torques at joint O₁ are 6.31 and 0.46 N-m, respectively. These results imply that the LTRR at this joint is about 92.7%. The maximum torques at joints O₂ and O₃ are reduced from 3.73 to 0.58 N-m and 1.22 to 0.38 N-m with gravity balancing, resulting in LTRRs of 84.5% and 68.9%, respectively. The balanced torques of the quasi-serial manipulator are smaller than those of the serial one. For example, at joint O₁, the maximum balanced torque of the quasi-serial manipulator is 0.22 N-m, indicating an LTRR of 96.5%. The LTRRs at joints O₂ and O₃ can achieve up to 95.2% and 92.6%, respectively.

Figures 10 and 11 show the power and energy consumption of the manipulators during trajectory tracking, respectively. For the serial manipulator, the maximum power at joint O₁ is decreased from 3.16 to 0.23 W with gravity balancing, showing an LPRR of 92.7%. The LPRRs at joints O₂ and O₃ are 81.8% and 79.3%, respectively. The powers of the quasi-serial manipulator are also reduced, showing LPRRs of 97.5%, 91.9%, and 100% at joints O₁, O₂, and O₃, respectively. The energies of the serial and quasi-serial manipulators are also reduced with gravity balancing. The LERRs of the serial manipulator are 95.4% at joint O₁, 91% at joint O₂, and 78.8% at joint O₃; those of the quasi-serial one are 97.7% at joints O₁, 94.1% at joint O₂, and 100% at joint O₃. The performance of the manipulators is detailed in Table 4.

Figure 12 shows the required crank lengths of the serial manipulator for different spring stiffnesses k_i = (500, 5000) N/m and payloads m_e = (0, 2) kg. For a given spring stiffness, the increase of the payload requires a longer crank length. For instance, for k₁ = 500 N/m, the crank length r₁₄ = 0.09 m when m_e = 1 kg and r₁₄ = 0.11 m when m_e = 2 kg.

Figure 13 shows the balancing performance of the manipulators with variable payloads. Here, the manipulators are requested to track along the same trajectory, as shown in Fig. 7. For the serial manipulator, the GTRR, GPRR, and GERR are generally reduced when the payload increases. For instance, the GTRR is 92.3% when m_e = 0 kg and is decreased to 89% when m_e = 2 kg. The GPRR and GERR are also reduced from 80.6% to 76.4% and from 93.3% to 91.3% when the payload varies from 0 to 2 kg. These results are due to the payload increase that raises the unbalanced loads at the joints, lowering the balancing performance. Meanwhile, the GTRR, GPRR, and GERR of the quasi-serial one have relatively the same values within a range of payloads from 0 to 2 kg. In comparison, the GTRR, GPRR, and GERR of the quasi-serial manipulator are higher than those of the serial one for any applied payload. For example, with m_e = 2 kg, the GTRR, GPRR, and GERR of the quasi-serial manipulator are 96%, 96.4%, and 98.3%; those of the serial one are 89%, 76.4%, and 91.3%, respectively.

The pulley-belt method was adopted to build a prototype of a robot arm. The 3D model and prototype of the robot are shown in Figs. 14 and 15, respectively. It possesses 4 DoFs of motion in which the first joint provides a rotation about the yaw axis, and the remaining three joints enable rotations about the pitch axes. As rotation about the yaw axis is not affected by gravity, it was neglected in the gravity balancing design. The parameters of the robot are listed in Table 1. For each CGSM, two linear springs (MISUMI AWY 20–70) were installed. The robot was designed to provide a load capability of 1 kg and a reach of 540 mm. Four Dynamixel servo motors were integrated into the joints (XM540-W270-R models for the first and second joints and XM430-W350-R models for the third and fourth joints) to actuate the robot. The motors were arranged in series and were connected to a computer through a communication interface (U2D2) and a U2D2 Power Hub. dynamixel wizard 2.0 was used to control these motors.

To evaluate the performance of the robot, the joint angles were controlled to move from “θ_i = 0” deg to “θ_i = 90” deg and then turn back; each joint was tested sequentially. This operation was conducted for two cases: without spring and with spring. For example, the motion profiles of the actuator at joint O₁ are illustrated in Fig. 16. The actuator torques of the robot were calculated by multiplying the current through the windings with the torque constant, as shown in Fig. 17. When the payload is zero, the unbalanced and balanced torques at joint O₁ are 2.69 and 0.49 N-m, respectively. These results imply an LTRR of 81.8% achieved at this joint. The LTRRs at joints O₂ and O₃ are 72.5% and 74.1%, respectively. When a 1-kg payload is attached, the actuator torque at joint O₁ reduces from 5.61 to 1.46 N-m with gravity balancing, presenting an LTRR of 74%. The gravity balancing design also decreases the actuator torques at joints O₂ and O₃ from 4.13 to 1.01 N-m and from 2.56 to 0.8 N-m, respectively. These results show LTRRs of 75.5% at joint O₂ and 68.8% at joint O₃.

Figures 18 and 19 illustrate the power and accumulated energy of the robot with different payloads, respectively. The gravity balancing design generally reduces power and energy consumption across a range of payloads from 0 to 1 kg. With m_e = 0 kg, the maximum power and energy at joint O₁ reduce from 1.73 to 0.6 W and from 13.23 to 1.65 J, respectively. Thus, the LPRR and LERR achieve up to 65.3% and 87.5%, respectively. High performance is also obtained at joints O₂ and O₃, where the LPRRs are 62.8% and 61.5%, and the LERRs are 85.5% and 84.3%, respectively. More results of the torque, power, and energy consumption are presented in Table 5.

Compared to traditional approaches [55–57], the significance of the present work lies in the introduction of a nonlinear gear-spring design implemented through three gravity balancing methods—direct installation, integration with parallelogram linkages, and integration with a pulley-belt system—tailored for multi-DoF serial manipulators handling variable payloads. By accounting for payload variation in the design process, the flexibility and adaptability of the nonlinear gear-spring system are significantly enhanced, thereby making the manipulator suitable for a broad range of applications. The direct installation method offers only approximate gravity balancing, as it enforces zero actuator torque at a specific target position rather than throughout the entire workspace. The actuator torques are varied when the manipulator's end-effector moves far from the desired position. In contrast, the parallelogram and pulley-belt methods achieve perfect gravity balancing, maintaining zero actuator torques over the entire workspace. However, the direct installation method offers a significantly simpler design and a larger workspace, as it does not require auxiliary structures (such as parallelograms and pulley-belt systems) to preserve the orientation of the reference links. A comparison between the proposed methods is detailed in Table 6. From an application standpoint, the direct installation method is well suited for commercially available robots operating along predefined trajectories, due to its simple design, ease of implementation, and acceptable performance. Meanwhile, the parallelogram and pulley-belt methods are more appropriate for custom-built robots, particularly in scenarios where the workspace is not predefined during the design phase.

This article presented a nonlinear gear-spring design for gravity balancing of robotic manipulators with variable payloads. The first method was realized by installing the CGSMs directly onto the manipulator, while the two others used parallelogram linkages and a pulley-belt system for the spring installation, respectively. The effect of gear friction was also considered in the performance analysis. This work provided numerical examples to assess the effectiveness of the proposed balancing methods and then compare their performance. It was found that the parallelogram and pulley-belt methods were more effective than the direct installation method. Moreover, a prototype of a robot arm based on the pulley-belt method, which holds a load capability of 1 kg and a reach of 540 mm, was also built. Experimental results showed that the torque, power, and energy consumption of the robot were reduced by an average of 72.8%, 57.5%, and 89.3%, respectively.

The APC for this publication was supported by VinUniversity.

There are no conflicts of interest.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.