Stiffness Enhancement by Means of Situational Coupling of Two Collaborative Robots

Abstract

While collaborative robots are designed to enable flexible and safe human–robot interactions, their comparatively low structural stiffness poses a challenge for high-precision machining and heavy-assembly tasks. Addressing this limitation is essential for enhancing their performance and improving their overall efficiency in manufacturing processes. This paper proposes an approach for enhancing the stiffness by means of situational coupling of two collaborative robots. Therefore, an analysis is conducted to determine the kinematic limitations of coupled collaborative robots. The stiffness of coupled collaborative robots is then modeled using the finite element method. Furthermore, experimental stiffness measurements of a single collaborative robot are conducted to establish a quantitative reference, which is both to validate the model and to quantify the stiffness enhancement achieved through coupling. On the basis of the combined experimental and numerical results, it is demonstrated that the approach of coupling has the potential to enhance stiffness by up to 37.19 times in comparison with a solitary collaborative robot.

1. Introduction

Flexible process automation is playing an increasingly decisive role in the economic competitiveness and sustainability of modern production systems [1]. Due to global market volatility, rising production costs and demographic changes, the use of robots is becoming progressively more important to ensure efficiency gains and production reliability. This is reflected in the continuously rising annual installation rate of robots. This amounted to over 500,000 units each year from 2021 to 2023, and further acceleration of this growth is forecast for the coming years [2]. Collaborative robots (cobots), in particular, are gaining importance due to their unique advantages, the most significant of which is their capability for direct and safe human–robot interaction—a key distinction from industrial robots [3]. This is also reflected in their steadily increasing market share, which reached almost 10% of total robot sales in 2023 [2].

Despite these advantages, the comparably low structural stiffness k of cobots remains one of the major challenges for their application in high-precision machining processes and assembly work with high process forces F_P. Low structural stiffness k can lead to deformations that negatively impact machining speed, process stability and energy efficiency. The open kinematic chain of a serial six-axis cobot causes the structure to bend, similar to a machine tool frame in C-design, which leads to a deterioration in the pose accuracy T at the tool center point (TCP) and, thus, to errors in the execution of the cobot’s task, see Figure 1a.

Higher structural stiffness k not only improves the pose accuracy T, payload capacity and process stability but also extends the range of industrial tasks that cobots can reliably perform, and thus, facilitates a more flexible and safe production. Numerous research projects, therefore, focus on increasing structural stiffness k, where a distinction can be made between design and compensatory measures. Design measures focus on increasing structural stiffness k by modifying the robot structure. Implementation of these measures is generally only possible retrospectively or iteratively, if at all, by means of extensive conversion work. According to Brecher et al. [4], the differentiation of robots in terms of their kinematics can be categorized into three distinct classifications: open, closed and partially closed chains. The serial arrangement of arm members and the resulting long, cantilevered arm structure result in open kinematic chains that exhibit low structural stiffness k. Integration of a parallelogram mechanism between axes two and three alleviates the load on the drive of axis two, thereby enhancing the structural stiffness k and positional accuracy T of the robot [5]. However, the closed kinematic sub-chain imposes constraints on flexibility and working space. Completely closed kinematic chains are also characterized by high structural stiffness k with lower flexibility and mobility. In this context, Tanev, Yeshmukhametov et al., and Fang et al. investigated hybrid serial–parallel kinematic chains [6,7,8]. Wang et al. [9] present an elastostatic-stiffness modeling approach for hybrid robotic structures, in which the parallel and serial modules can be analyzed separately. In contrast, numerous robot manufacturers prioritize increasing the robot arm’s cross-section to enhance its structural stiffness, as evidenced by the findings of Wu et al. [10]. However, an increase in the arm’s cross-section results in an elevated dead weight, concurrently compromising the robot’s dynamic performance. Chen et al. [11] present a modeling approach that simulates robot stiffness considering joint flexibility through a rigid–flexible coupling. Time-varying stiffness models are developed and applied for trajectory planning. An alternative approach was presented by Lai et al. [12], where two serial 3-axis industrial robots were permanently connected. The main structure is equipped with a spindle at the end effector. The support structure is attached to the end effector of the main structure. During the machining process, the support structure is anchored to a fixed point within in the working space. In this configuration, the system employs parallel kinematics with serial arms. In this coupled operating state, a parallel connection of the two Cartesian stiffnesses k_kart of the serial arms has been postulated in regard to the structural stiffness k. Conversely, compensatory measures are intended to enhance positional accuracy T without the need for structural modifications. The prevailing focus of research in this field is the development of globally applicable compensation measures. These measures are based on models that describe the causes of error in the robot’s kinematic model [13,14,15]. Depending on the complexity of the models under consideration, Roth et al. [16] introduced a distinction between three calibration categories:

• Level-1 calibration:

  →

  Compensation for joint axis angle errors,

• Level-2 calibration:

  →

  Compensation for geometric errors (regarding angle errors and arm lengths),

• Level-3 calibration:

  →

  Compensation for non-geometric errors (e.g., elastic and thermal deformations, friction, gear backlash, gear ratio errors).

As demonstrated in numerous studies, the implementation of a level-2 calibration results in a 90% enhancement in positional accuracy T [17]. However, it proves inadequate for processes characterized by high process forces F_P and accuracy requirements due to the low structural stiffness k. While the gradual increase in model complexity enhances model quality, it concurrently complicates parameter identification. Furthermore, the complexity of the model is constrained by the necessity of a sufficiently brief calculation time.

In light of the aforementioned context, this paper proposes an innovative approach to improving the stiffness k by means of situational coupling of two six-axis cobots. Due to the optimized force flow in the coupled operating state, a significant increase in stiffness k is expected, similar to a machine tool in O-design, seen in Figure 1b. Coupling of industrial robots is proposed by Goebels et al. [18], Neusser et al. [19], Mühlbeier et al. [20] and Ye et al. [21]. Notably, Ye et al. [21] demonstrated experimentally that stiffness-oriented cooperative placement and motion planning for physically coupled industrial robots is feasible and beneficial in manufacturing contexts. However, these studies are limited to conventional industrial robots. The systematic coupling of cobots has not yet been explored, and the characteristic features of cobots fundamentally impose different requirements for achieving safe and fully synchronized cobot coupling [22]. Most importantly, the lightweight and compliant design, driven by safety requirements, is a key characteristic of cobots and should be preserved. Therefore, it is essential to specifically enhance the structural stiffness k through the coupling of two cobots in order to ensure precise and stable collaboration without compromising their fundamental safety advantages.

2. Concept for the Situational Coupling

From the results of the aforementioned studies concerning the coupling of conventional industrial robots, it is assumed that situational coupling in the overlapping workspace of two cobots offers the opportunity of direct integration of tasks with high process force F_P and accuracy requirements. Therefore, it is hypothesized that such a configuration will enhance flexibility in cobot-based process chains, thereby facilitating automation and increasing productivity, even with small batch sizes. The concept investigated in this paper is based on two cobots working independently in a decoupled mode while remaining within mutual reach. Since such configurations are already present in many industrial cobot-based process chains, the proposed approach can be readily integrated into existing industrial environments. When a process task requires a stiffness exceeding the achievable stiffness of a single cobot, mechanical coupling is activated to form a serial–parallel system. The switching criterion is defined by a task-dependent stiffness requirement k_req, which is compared to the estimated single-cobot stiffness k_single in the current configuration. Coupling is initiated when k_req > k_single and released once k_req ≤ k_single. This strategy ensures that mechanical coupling is applied only when necessary, providing high stiffness k during force-critical phases, while maintaining the advantages of collaborative operation during all other phases. The stiffness requirement k_req is derived from the task description and online process monitoring. When the condition k_req > k_single is detected, the system automatically triggers the coupling mechanism. After completing its current task, the secondary cobot moves into position and establishes the mechanical connection, forming a coupled system. Once the force-critical phase is completed and the stiffness required k_req falls below the single cobot stiffness k_single, the system autonomously decouples and returns to compliant single-cobot operation. This way, the approach ensures that both cobots maintain their inherent high degree of flexibility and dynamics in the decoupled operating state, see Figure 2.

This requires the use of a flexible coupling interface, which enables a fully automated coupling and decoupling process of both cobots and fulfills the conflicting requirements of lightweight construction, high stiffness, high precision and low costs. Furthermore, the native control system of both cobots has to be adapted. It is essential to ensure that both cobots follow a synchronized trajectory.

The proposed situational coupling strategy is particularly well-suited for robotic tasks that are characterized by alternating phases of high compliance and high stiffness demand. Typical examples include contact-rich operations such as surface finishing, grinding, or precision assembly. In these tasks, high stiffness k is required only during specific process phases, while flexible behavior remains advantageous, e.g., during approach and positioning. However, tasks that demand high positioning accuracy at high dynamic speeds may be limited by the dynamic coupling effects introduced by the second robot and the coupling interface.

3. Kinematic Model for the Coupled Operating State

In order to both coordinate the cobots in the coupled operating state and to generate possible joint angle configurations for the stiffness model, a mathematical representation of the coupled kinematics is required. The cooperative cobot system is considered a combination of two serial cobots, I, I ϵ {A|B}. In addition to the two Cartesian base coordinate systems K_I, I ϵ {A|B}, a working coordinate system K_x is introduced. The constant homogeneous transformation matrix ${}_{\mathrm{I}}{}^{\mathrm{X}}\mathsf{T}$ represents the rotation and displacement of K_x relative to K_I. The homogeneous transformation matrix between the base coordinate system and the end effector flange ${}_{\mathrm{I}}{}^{\mathrm{EE}\text{\_}\mathrm{I}}\mathsf{T}$ can be determined according to Formula (1).

$${}_{\mathrm{I}}{}^{\mathrm{EE}\text{\_}\mathrm{I}}\mathsf{T}={}_{\mathrm{I}}{}^{\mathrm{X}}\mathsf{T} · {}_{\mathrm{X}}{}^{\mathrm{AP}}\mathsf{T} · {}_{\mathrm{AP}}{}^{\mathrm{EE}\text{\_}\mathrm{I}}\mathsf{T}=\left(\begin{array}{cc}{}^{\mathrm{I}}{\mathsf{R}}_{{\mathrm{K}}_{\mathrm{I}}\to \mathrm{EE}\text{\_}\mathrm{I}}& {}^{\mathrm{I}}{\mathsf{x}}_{{\mathrm{K}}_{\mathrm{I}}\to \mathrm{EE}\text{\_}\mathrm{I}}\\ 0& 1\end{array}\right),$$

(1)

In Formula (1), ${}_{\mathrm{X}}{}^{\mathrm{AP}}\mathsf{T}$ represents the homogeneous transformation matrix between the working coordinate system and the working point, while ${}_{\mathrm{AP}}{}^{\mathrm{EE}\text{\_}\mathrm{I}}\mathsf{T}$ represents the homogeneous transformation matrix between the working point and the end effector flange of the cobot. Figure 3 illustrates the definitions of the homogeneous transformation matrices.

The so-defined coupled kinematics was implemented in the software MATLAB R2023b of the company The Mathworks Inc., Natick, MA, USA. Figure 4 exemplifies the resulting coupled working space given a base distance d_h = 1500 mm.

For modeling purposes, the forward and backward kinematics of the cobot of type UR10e from the company Universal Robots A/S, Odense, Denmark, were implemented. For the sake of simplicity, it will be referred to as UR10e from here on out. Since the end effectors of the cobots should be connected to each other, preventing any relative motion, a point constraint was defined accordingly. The distance between the base coordinate systems of the cobots was set to d_h = 1500 mm, as this provides the optimal ratio of workspace to installation area. The maximum working space between the cobots corresponds to the maximum reach of the cobots, which is l_UR10e = 1300 mm. Evidently, due to inherent structural limitations, including the diameter of the cobot’s base d_b = 190 mm, a base distance d_h = 1300 mm is infeasible. To ensure sufficient space, this results in a base distance of d_h = 1500 mm. Consequently, the resulting working space volume is determined to be V_A,coup = 1.06 m³. Comparing the usable working space volume of a single UR10e, V_A,UR10e ≈ 3.50 m³, with that of the coupled cobots reveals that the latter is only about 30.23%. In order to avoid noticeable loss of flexibility through permanent coupling, the implementation of fully automated coupling mechanisms is of crucial importance. This enables either a high degree of flexibility or high stiffness k to be achieved, depending on the requirements of the application.

4. Stiffness Tests of the Cobot

In order to enable an increase in system stiffness k throughout the coupling of cobots, it is essential to investigate the stiffness k of a single UR10e. Therefore, stiffness tests were carried out. According to the manufacturer’s datasheet, the UR10e’s maximum payload capacity is m_P,UR10e = 12.50 kg. Its reach measures to l_R,UR10e = 1300 mm and its net weight to m_E,UR10e = 33.50 kg [23].

4.1. Experimental Procedure

To determine the stiffness k of the UR10e, a radial force F_rad is applied to its end effector, and the corresponding displacement u of the end effector is measured. To carry out the tests, the end effector of the UR10e is moved to a specific position. This position along the base coordinate system’s x-axis corresponds both to the UR10e’s nominal operational configuration and to the midpoint of the simulated working space. The positions in the y- and z-directions remained invariant, due to the given environmental conditions. The stiffness measurement was performed in this deliberately selected reference configuration, which is kinematically extended and located in a mid-workspace region known to be among the most compliant and critical for serial manipulators. Since the objective of this work is to demonstrate the achievable stiffness gain through coupling, this pose represents a conservative and representative test case. While the absolute stiffness k of a single robot varies over the workspace, the relative stiffness gain introduced by mechanical coupling is primarily determined by the serial–parallel topology and is, therefore, largely independent of the specific robot pose. Consequently, the measured stiffness increase can be interpreted as a representative indicator of the coupling effect. During the application of the radial force F_rad at the designated operating point, the UR10e is in switched-on mode without performing any movement, so that the experiments could be conducted under quasistatic conditions. The radial force F_rad is applied to the UR10e’s end effector in the respective opposite direction from which it has been moved to its target position. A type 9257B multi-component dynamometer from the company Kistler Group, Winterthur, Switzerland, is used to measure the radial force F_rad. The radial force F_rad is calculated using the LabVIEW interface from the company National Instruments, Austin, TX, USA, and is applied by a fine-thread screwing device, which is connected to the multi-component dynamometer by a screw connection. The displacement u of the end effector is measured using a laser triangulator of the type ILD2200-10 from the company MICRO-EPSILON Optronic GmbH, Dresden, Germany. Both radial force F_rad and displacement u are recorded at a sampling rate of f_s = 1 Hz. The stiffness k was identified under quasistatic loading conditions, where the robot was allowed to settle after each force increment. Therefore, a sampling rate of f_s = 1 Hz is sufficient, as no dynamic effects were evaluated and all measurements correspond to static equilibrium points. The entire measurement procedure was repeated three times. Figure 5 illustrates the experimental setup.

4.2. Results and Evaluation

Figure 6 shows the mean values of the measured displacement u in the x- and y-directions, subject to radial force F_rad imposed on the end effector, along with the respective standard deviation. In order to determine the in-plane and out-of-plane stiffness k of the UR10e, radial forces F_rad were applied in both the x- and y-directions, the amplitude of which ranged up to two orders of magnitude. Utilizing a weighted least square algorithm, the average stiffness of k_x = 95.94 N/mm in the x-direction and k_y = 57.42 N/mm in the y-direction was obtained on the basis of the three repetitions. Thus, the UR10e exhibits a structural stiffness k that is about 67% higher in the x-direction than in the y-direction.

The maximum displacement in the x-direction, u_x,max ≈ 3.09 mm, occurred at the corresponding maximum radial force of F_rad,x,max ≈ 220 N. In accordance with a priori expectations, the UR10e demonstrated a capacity of a maximum radial force of only F_rad,y,max ≈ 82 N in the y-direction, resulting in a displacement of u_y,max ≈ 7.01 mm. This is consistent with the observations made during the experiments, where at a threshold of radial force F_rad,y ≈ 75 N and displacement u_y ≈ 1.42 mm, any additional rotation of the fine-thread screwing device would result in only a slight increase in the measured radial force F_rad,y, while a substantial increase in the corresponding displacement u_y occurred. In the x-direction, the same amplitude of radial force F_rad resulted in a displacement of merely u_x = 0.73 mm. To characterize these spatial differences in the UR10e’s behavior, the pointwise stiffness k_pw, as defined in Formula (2), was determined.

$${\mathrm{k}}_{\mathrm{pw},\mathrm{i}}:={\displaystyle \frac{{\Delta \mathrm{F}}_{\mathrm{rad}}}{\Delta \mathrm{u}}}={\displaystyle \frac{{\mathrm{F}}_{\mathrm{rad},\mathrm{i}+1}-{\mathrm{F}}_{\mathrm{rad},\mathrm{i}}}{{\mathrm{u}}_{\mathrm{i}+1}-{\mathrm{u}}_{\mathrm{i}}}}$$

(2)

Figure 7, accordingly, illustrates the mean values of pointwise stiffness k_pw in the x- and y-directions, subject to radial force F_rad imposed on the end effector, along with the respective standard deviation.

The observed anisotropy in the stiffness k may be attributed to the UR10e’s kinematic structure, in particular the general arrangement and alignment of its joints and the resulting transmission of force and momentum, as well as the pose under consideration within the conducted tests.

4.3. Stiffness Evaluation of Cobots in Coupled Operating State

To investigate the effect of coupling two cobots on compound stiffness k_C, a simulation-based analysis was conducted. For this purpose, a multibody simulation model was implemented in the finite element software Ansys 2025 R2 provided by the company Ansys Inc., Canonsburg, PA, USA. In order to obtain general insights into the overall effect, certain assumptions have been made, e.g., the simplification to a quasistatic problem due to slow force application, negligible influence of material properties as all arms are assumed to be stiff and discrete localization of all compound compliances s_C in the six or twelve cobot joints, respectively [24]. The joints were configured as purely rotational, with one degree of freedom each, and independent torsional stiffnesses c_T,i. These torsional stiffnesses c_T,i were utilized as parameters for phenomenologically adjusting the behavior of a single cobot. Therefore, experimental data presented in Section 4.2 have been reproduced by numerically varying each joint’s torsional stiffness c_T. In the coupled operating state, the coupling object between the two cobots was modeled as stiff, preventing any relative motion between the end effectors and reflecting a typical state of pre-tension in these components, seen in Figure 3. Boundary conditions at the bases were set according to the fixed clamping of both cobots to the ground. The model has been implemented such that the second cobot is mirrored to the first one with respect to the y’z-plane, resulting in a relative angle of α = 107°, seen in Figure 8.

Loading was configured according to single-cobot experiments as stated in Section 4.2 and was applied to the common node of the end effectors. The simulation could prove the exemplary expediency of the concept; as in the coupled operating state, increased compound stiffnesses of k_C,x ≈ 212 N/mm and k_C,y ≈ 2.95 kN/mm were observed. An additional simulation, besides the stiff-beam model, was conducted, in which the coupling object shown in Figure 3 is modeled as a compliant beam. Figure 9 illustrates the differences in compound stiffness k_cx and k_cy between the compliant beam, the stiff beam, and the experimental measurements. The actual coupling mechanism has not yet been developed, but it is expected to exhibit an effective stiffness that lies between the two considered extremes, namely, compliant and stiff simulated behavior, due to mechanical preloading and the constrained and interlocked kinematic structure. It should be noted that an additional side effect of the angled arrangement is that both cobots compensate for the more compliant spatial direction of the other. This results in an overall stiffer structure, as the results illustrate.

5. Conclusions

In this paper, the concept of situational coupling of two cobots was presented. For this purpose, the resulting working space of the coupled cobots was determined by a kinematic model. Furthermore, pose-related stiffness in the planar main-axis system of a single cobot of type UR10e was investigated experimentally. Therefore, radial forces F_rad,i in the x- and y-directions were applied to the UR10e’s end effectors, and the resulting displacement u_i was measured. The results of the analysis yield a stiffness ratio of η ≔ k_x/k_y ≈ 1.67 with respect to the UR10e’s base coordinate system.

Next, a multibody simulation model of the UR10e was implemented and parameterized digitally, utilizing experimentally obtained measurement data. The adapted parameter values then were used to numerically investigate the coupling of two UR10e. The compound stiffnesses k_C,i exhibited an enhancement in comparison to the single operating state, with an increase of approximately 200% in the x-direction and approximately 5940% in the y-direction. In light of these findings, an angular arrangement of the two cobots was found to be advantageous in terms of compensating for the respective more compliant direction of movement. These findings substantiate the conceptual viability of situationally coupled cobots and indicate the necessity for further research to investigate the scope of applications in which the structural advantages of collaborative industrial robots can be leveraged. Future research, therefore, may aim to extend this investigation throughout the entire working space, incorporating stiffness characteristics of the cobots’ arm segments. Furthermore, the determination of an optimal angle α in the coupled operating state with regard to compound stiffness k_C and optimal pre-tension of the end effectors represent potential subjects for further investigation.