Dynamics Modeling and Parameter Identification for a Coupled-Drive Dual-Arm Nursing Robot

Abstract

A dual-arm nursing robot can gently lift patients and transfer them between a bed and a wheelchair. With its lightweight design, high load-bearing capacity, and smooth surface, the coupled-drive joint is particularly well suited for these robots. However, the coupled nature of the joint disrupts the direct linear relationship between the input and output torques, posing challenges for dynamic modeling and practical applications. This study investigated the transmission mechanism of this joint and employed the Lagrangian method to construct a dynamic model of its internal dynamics. Building on this foundation, the Newton-Euler method was used to develop a dynamic model for the entire robotic arm. A continuously differentiable friction model was incorporated to reduce the vibrations caused by speed transitions to zero. An experimental method was designed to compensate for gravity, inertia, and modeling errors to identify the parameters of the friction model. This method establishes a mapping relationship between the friction force and motor current. In addition, a Fourier series-based excitation trajectory was developed to facilitate the identification of the dynamic model parameters of the robotic arm. Trajectory tracking experiments were conducted during the experimental validation phase, demonstrating the high accuracy of the dynamic model and the parameter identification method for the robotic arm. This study presents a dynamic modeling and parameter identification method for coupled-drive joint robotic arms, thereby establishing a foundation for motion control in humanoid nursing robots.

1 Introduction

According to China's Seventh National Population Census, by the end of 2020, individuals aged 60 years and older constituted 18.7% of the population, with those aged 65 years and older representing more than 13.5% [1]. Amid the ongoing demographic shift towards an older population, coupled with declining birth rates, addressing the needs of seniors with disabilities has become a focal point in both societal discussions and scientific research [2]. Daily caregiving for these seniors covers a spectrum of activities ranging from changing clothes and feeding to bathing, managing excretion, assisting mobility, and lifting. It is particularly challenging to move seniors from beds to wheelchairs, bathrooms, or baths, which pose the greatest physical demand for caregivers [3, 4].

To address the challenges of patient transfer, leading research institutions have developed an array of nursing robots, including those designed for lifting [5], sliding [6], integrated bed-chair transitions [7], and humanoid back-hugging [8]. However, although specialized, these robots present operational complexities and may fall short in providing the desired comfort and safety, thus narrowing their range of applications. Recently, dual-arm collaborative robots have emerged at the forefront, capturing attention because of their versatility, efficiency, and anthropomorphic design [9].

Recognizing this, researchers have seamlessly integrated dual-arm functionalities into nursing robots [10] and crafted models tailored for the elderly to aid in the transition between beds and wheelchairs. Designed with a focus on user comfort and safety, these robots are capable of handling heavy loads and achieving seamless human-machine collaboration and are also skillfully equipped to navigate complex, unstructured environments such as homes and hospitals [11].

Although dual-arm nursing robots have achieved notable research breakthroughs, they have not yet matured into products primed for widespread adoption [12]. A primary hindrance is the ill-fitting of traditional robotic arm designs for caregiving applications. Most nursing and rehabilitation robot joints use a single motor combined with a decelerator, making it difficult to generate sufficient torque to safely lift and support the human body [13]. Furthermore, the perpendicular orientation of the joint axes caused pronounced protrusions in the mechanical arm. This leads to an undue localized pressure upon human contact, thereby diminishing user comfort [14]. Therefore, it is crucial to move beyond conventional design approaches and delve into pioneering joint and structural designs of nursing robots.

In recent years, the field of research has seen growing interest in coupled drive joints [15]. By combining the output torques of multiple motors, this design significantly enhances the load-bearing capability of the joint, thus heralding a novel stride in joint technology [16]. Two primary configurations exist for this joint mechanism: rope–pulley and gear transmissions, with the former being predominant. Using coupled drive joint technology, Olarui launched a bipedal robot called Sherpa to enhance leg movement efficiency [17]. Hagn employed this approach to craft a Miro surgical robot, increasing the operational flexibility during surgery [18].

The dynamic establishment is crucial for robots to execute force control [19]. In conventional robot dynamics modeling, factors such as joint internal inertia and flexibility are often overlooked, leading to a straightforward correlation between the output torque of the motor and the joint response [20]. Although this simplification is acceptable for traditional collaborative robots with simplistic designs and negligible transmission inertia, it is not applicable for coupled-drive joints. Given their intrinsic coupling effects, ignoring their transmission inertia would be incorrect, necessitating a deeper exploration of their internal dynamic frameworks. Research on dynamic modeling techniques for the coupled drive joints of robots is still in its nascent stage [21].

Nevertheless, valuable insights can be gained from studies focusing on differential transmission structures in sectors such as automobiles, wind power, and aircraft engines. Xiang et al. focused on the differential speed control of wind turbine systems and devised a triaxial dynamic model for their transmission mechanism. Using the Lagrangian approach, he derived the corresponding dynamic equations [22]. Similarly, Che et al. constructed dynamic equations for a wind turbine differential transmission system based on a multi-body dynamic framework [23].

In the real-world operation of robotic arms, beyond the effects of motor-driven torque, multiple nonlinear disturbances are encountered, with frictional forces being particularly influential on the arm's motion dynamics [24, 25]. Friction arises from a combination of factors, such as relative velocity, acceleration, displacement, lubrication status, and contact conditions, and is present at every moving interface [26]. For simplicity, when devising a friction model, it is customary to localize the frictional force at the pivot point of the joint, representing it as a function primarily tied to the speed [27]. Traditional friction models, either discontinuous or segmented in continuity, present challenges in accurately describing friction as speed approaches zero [28]. This can result in undesirable vibrations and jolting during directional shifts.

This study introduces a novel coupled-drive nursing robot aimed at enhancing the safety and comfort of human-robot interactions. For this design, we conducted in-depth dynamic modeling of the coupled-drive joints and a robotic arm composed of these joints. To precisely address the joint friction and impacts during direction changes, we integrated a continuous and differentiable friction model into the dynamic equations. Based on this model, we linearize the dynamic equations and derive a minimal set of inertia parameters. Furthermore, we established a mapping between the frictional force and the motor current using a particle swarm optimization algorithm to identify the parameters of the continuous friction model. Additionally, we optimized the excitation trajectory and employed machine-learning techniques to identify the inertia parameters of the robotic arm. In the final phase of this study, we set up an experimental platform and conducted trajectory-tracking experiments, the results of which validated the high accuracy of our dynamic model and parameter identification.

2 Design and Kinematic Analysis of Coupled Drive Nursing Robots

2.1 Coupled Drive Nursing Robot Design

In this study, a novel coupled drive joint was designed to meet the requirements for large-area human-machine contact. The joint employs two motors as the inputs and achieves a coupled output through a differential transmission mechanism comprising three equal-diameter bevel gears, as shown in Figure 1.

Figure 1

Coupled-Drive Joint

This joint provides dual rotational flexibility for both pitch and rotation. Harnessing the combined output torque of the two motors significantly boosted their load-bearing capacity. Its sleek, cylindrical design ensures a smooth exterior devoid of protrusions for comfortable human interaction and maintains a compact form, mirroring the diameter of a human arm. Linking three of these joints yielded a robotic arm with six degrees of freedom. Additionally, with the integration of the designed waist, hip joints, and mobile platform, we established a comprehensive dual-arm transfer nursing robot. The complete design of the dual-arm nursing robot is shown in Figure 2.

Figure 2

Nursing robot using coupled-drive joints

2.2 Coupled Transmission Principle Analysis

In the study of coupled transmissions, we can skip the drive chain from the motor to the input bevel gear and delve directly into the differential mechanism. This setup comprises three uniformly sized bevel gears and a linkage rod. Bevel gears 1 and 2 serve as the input, while bevel gear 3, along with rod H, form the output. From a structural standpoint, this configuration aligns with the 2K-H type differential gear system, as shown in Figure 3.

Figure 3

Coupled drive principle

As shown in Figure 3, φ₁ and φ₂ denote the angles of the two input bevel gears, whereas q₁ and q₂ correspond to the output angles for rotation and pitch. The direction in which the input bevel gears rotate determines the output's degrees of freedom (DOF). When both gears rotate at identical speeds in the same direction, only the pitch degree of freedom is activated. Conversely, if they spin at the same rate, but in opposite directions, only the rotational degree becomes active. In any other scenario, the joint produces outputs for both the pitch and rotation, leading to a coupled motion.

For the kinematic analysis, we selected two output angles from the coupled drive joint as the generalized coordinates, denoted q₁ and q₂. Consequently:

$$\dot{q}_{{1}} =\omega_{{3}} ,\dot{q}_{{2}} = \omega_{H} ,$$

(1)

where ω₃ and ω_H represent the angular velocities of bevel gear 3 and the linkage rod, respectively.

Using the calculated transmission ratio for the rotary gear system, we performed a kinematic analysis of the coupled drive joint as follows:

$$\left\{ {\begin{array}{*{20}l} {i_{12}^{H} = \frac{{\omega_{{1}} - \omega_{H} }}{{\omega_{{2}} - \omega_{H} }} = - \frac{{z_{3} z_{2} }}{{z_{1} z_{3} }} = - \frac{{z_{2} }}{{z_{1} }},} \\ {i_{13}^{H} = \frac{{\omega_{{1}} - \omega_{H} }}{{\omega_{3} }} = \frac{{z_{3} }}{{z_{1} }},} \\ \end{array} } \right.$$

(2)

where $i_{12}^{H}$ and $i_{13}^{H}$ denote the transmission ratios between bevel gears 1 and 2 and bevel gears 1 and 3, respectively, when the reverse angular velocity $- \omega_{H}$ is applied to the linkage rod. z₁, z₂, and z₃ denote the tooth counts on the bevel gears; z₁ = z₂ = z₃. By merging Eqs. (1) and (2), we establish a link between the generalized coordinate velocity and the input speed of the bevel gear. Following the integration process, the kinematic decoupling relationship of the coupled drive joint was identified.

$$\begin{gathered} \left\{ {\begin{array}{*{20}l} {q_{1} = {{(\varphi_{1} - \varphi_{2} )} \mathord{\left/ {\vphantom {{(\varphi_{1} - \varphi_{2} )} 2}} \right. \kern-0pt} 2},} \\ {q_{2} = {{(\varphi_{1} + \varphi_{2} )} \mathord{\left/ {\vphantom {{(\varphi_{1} + \varphi_{2} )} 2}} \right. \kern-0pt} 2},} \\ \end{array} } \right. \hfill \\ \left\{ {\begin{array}{*{20}l} {\varphi_{1} = q_{1} + q_{2} ,} \\ {\varphi_{2} = - q_{1} + q_{2} .} \\ \end{array} } \right. \hfill \\ \end{gathered}$$

(3)

2.3 Kinematic Analysis of the Robotic Arm

While the robot joints exhibited coupled movements, the robotic arm architecture comprised six sequentially connected rotational DOF. We employed a refined D-H parameter method to outline the kinematic model of the robot. The foundational coordinate system was anchored at the base joint of the robotic arm, with the z₀ axis extending vertically, the x₀ axis perpendicular to z₀ and oriented inward towards the plane of the paper, and the y₀ axis direction set by the right-hand rule. The six subsequent coordinate systems, O_i-X_iY_iZ_i (i = 1–6), were designated for each rotating joint, as shown in Figure 4. Note that the left and right arms of the nursing robot mirror each other in terms of structure and spatial orientation. For clarity, our analysis focused primarily on the kinematic attributes of the right arm.

Figure 4

Coordinate system of the robotic arm

Figure 4 shows the locations of the six rotational axes of the robotic arm. Each coordinate system was seamlessly integrated with its respective fundamental components. Within this context, θ_i signifies the joint variable, d_i indicates the link offset, a_i-1 represents the link length, and α_i designates the link twist angle. Table 1 lists a detailed breakdown of the D–H parameters of the robotic arm.

Table 1 D-H parameters of the robotic arm

The following parameters were set in the design of the nursing robot used in this study: a₂ = 110 mm, d₂ = 242 mm, and d₄ = 28 mm. Upon establishing the joint angles θ_i (i = 1–6), we can employ the kinematic equations to derive the pose matrix for the end effector of the robotic arm.

3 Dynamic Modeling of the Nursing Robot

3.1 Dynamic Model of Coupled Drive Joint

After investigating the force transmission dynamics of the coupled drive joint, we embarked on dynamic modeling. Considering its distinct transmission features, we divided the joint transmission chain into two primary sections: fixed-axis transmission and differential transmission. The former spans from the motor to the pseudo-hyperbolic gear, whereas the latter bridges the input bevel gear to the output bevel gear and the linkage rod. Note that the output of the fixed-axis section is seamlessly integrated as an input to the differential section. Given the unwavering positions of the components within the fixed-axis section and input bevel gear, they can collectively be viewed as an integrated entity. Therefore, the joint aptly mirrors a differential mechanism bifurcated into two pathways: the left and right branches. A simplified transmission diagram of the coupled-drive joint is shown in Figure 5.

Figure 5

Simplified transmission diagram of the coupled drive joint

Using the Lagrangian approach [29], we executed dynamic modeling of the coupled drive joint. When integrated with the kinematic relationship of the joint (as described in Eq. (3)), the kinetic energy of the differential mechanism can be expressed as follows:

$$\begin{aligned} E = \frac{1}{2}J_{11} \dot{q}_{1}^{2} + J_{12} \dot{q}_{1} \dot{q}_{2} + \frac{1}{2}J_{22} \dot{q}_{2}^{2} , \hfill \\ \left\{ {\begin{array}{*{20}l} {J_{11} = J_{1} + J_{2} + J_{3} ,} \\ {J_{12} = J_{1} - J_{2} ,} \\ {J_{22} = J_{1} + J_{2} + J_{H} + m_{3} r_{H}^{2} ,} \\ \end{array} } \right. \hfill \\ \end{aligned}$$

(4)

where J₁ and J₂ correspond to the equivalent moments of inertia from Motor 1 to Bevel Gear 1 and Motor 2 to Bevel Gear 2, respectively. J₃ and J_H denote the moments of inertia for Bevel Gear 3 and the linkage rod, respectively. m₃ signifies the mass of Bevel Gear 3, and r_H represents the radius at which the center of mass of Bevel Gear 3 rotates around the axis of the linkage rod. The kinetic energy E of the system is differentiated with respect to the generalized coordinate velocities and generalized coordinates as follows:

$$\left\{ {\begin{array}{*{20}l} {\frac{\partial E}{{\partial \dot{q}_{1} }} = J_{11} \dot{q}_{1} + J_{12} \dot{q}_{2} ,} \\ {\frac{\partial E}{{\partial \dot{q}_{2} }} = J_{12} \dot{q}_{1} + J_{22} \dot{q}_{2} ,} \\ \begin{aligned} \frac{d}{{d_{t} }}(\frac{\partial E}{{\partial \dot{q}_{1} }}) =& J_{11} \ddot{q}_{1} + \frac{{\partial J_{11} }}{{\partial q_{1} }}\dot{q}_{1}^{2} + \frac{{\partial J_{11} }}{{\partial q_{2} }}\dot{q}_{1} \dot{q}_{2} \\ &+J_{12} \ddot{q}_{2} + \frac{{\partial J_{12} }}{{\partial q_{1} }}\dot{q}_{1} \dot{q}_{2} + \frac{{\partial J_{12} }}{{\partial q_{2} }}\dot{q}_{2}^{2} , \\ \end{aligned} \\ \begin{aligned} \frac{d}{{d_{t} }}(\frac{\partial E}{{\partial \dot{q}_{2} }}) =& J_{12} \ddot{q}_{1} + \frac{{\partial J_{12} }}{{\partial q_{1} }}\dot{q}_{1}^{2} + \frac{{\partial J_{12} }}{{\partial q_{2} }}\dot{q}_{1} \dot{q}_{2} \\ &+J_{22} \ddot{q}_{2} + \frac{{\partial J_{22} }}{{\partial q_{1} }}\dot{q}_{1} \dot{q}_{2} + \frac{{\partial J_{22} }}{{\partial q_{2} }}\dot{q}_{2}^{2} , \\ \end{aligned} \\ {\frac{\partial E}{{\partial q_{1} }} = \frac{1}{2}\frac{{\partial J_{11} }}{{\partial q_{1} }}\dot{q}_{1}^{2} + \frac{1}{2}\frac{{\partial J_{22} }}{{\partial q_{1} }}\dot{q}_{2}^{2} + \frac{{\partial J_{12} }}{{\partial q_{1} }}\dot{q}_{1} \dot{q}_{2} ,} \\ {\frac{\partial E}{{\partial q_{2} }} = \frac{1}{2}\frac{{\partial J_{11} }}{{\partial q_{2} }}\dot{q}_{1}^{2} + \frac{1}{2}\frac{{\partial J_{22} }}{{\partial q_{2} }}\dot{q}_{2}^{2} + \frac{{\partial J_{12} }}{{\partial q_{2} }}\dot{q}_{1} \dot{q}_{2} .} \\ \end{array} } \right.$$

(5)

The potential energy U of the system can be expressed as follows:

$$U = - m_{He} gr_{He} \cos q_{2} - m_{3} gr_{H} ,$$

(6)

where m_He​ represents the mass of all components involved in the rotational motion, r_He​ denotes the distance from the centroid to the zero-potential energy surface, and g is the gravitational acceleration. The partial derivative of the potential energy U with respect to the generalized coordinates is expressed as follows:

$$\frac{\partial U}{{\partial q_{1} }} = 0,\;\;\frac{\partial U}{{\partial q_{2} }} = m_{He} gr_{He} \sin q_{2} .$$

(7)

The power P of the system can be expressed as follows:

$$\begin{aligned} P =& Q_{1} \dot{q}_{1} + Q_{2} \dot{q}_{2} \\ =& T_{1} \omega_{1} + T_{2} \omega_{2} + T_{3} \omega_{3} + T_{H} \omega_{H} , \\ \end{aligned}$$

(8)

where Q₁ and Q₂ represent the generalized forces and T₁, T₂, T₃, and T₄ denote the input and output torques, respectively. By incorporating the kinematic relationships, the generalized force can be expressed as follows:

$$\left\{ {\begin{array}{*{20}l} {Q_{1} = T_{1} - T_{2} + T_{3} ,} \\ {Q_{2} = T_{1} + T_{2} + T_{H} ,} \\ \end{array} } \right.$$

(9)

Substituting Eqs. (5), (9), and (7) into the Lagrangian equation, the dynamics of the coupled drive joint can be expressed as follows:

$$\varvec{J\ddot{q} + Gq} = {\varvec{\tau}},$$

(10)

where J represents the inertia matrix, G denotes the gravity matrix, and τ signifies the generalized force matrix.

$$\left\{ \begin{aligned}& {\varvec{J}} = \left[ {\begin{array}{*{20}l} {J_{11} } & {J_{12} } \\ {J_{12} } & {J_{22} } \\ \end{array} } \right],\; \hfill \\ &\varvec{G} =\left[ {\begin{array}{*{20}l} 0 & 0 \\ 0 & {m_{He} gr_{He} \sin q_{2} } \\ \end{array} } \right], \hfill \\ \;\varvec{\ddot{q}} = \left[ {\begin{array}{*{20}l} {\ddot{q}_{1} } \\ &{\ddot{q}_{{2}} } \\ \end{array} } \right]\;,\;{\varvec{\tau}} = \left[ {\begin{array}{*{20}l} {Q_{1} } \\ {Q_{1} } \\ \end{array} } \right]. \hfill \\ \end{aligned} \right.$$

(11)

Parameters, including the moment of inertia, mass, and location of the center of mass for each component within the joint, can be precisely determined using 3D design software, as listed in Table 2.

Table 2 Joint component parameters

3.2 Dynamic Model of the Nursing Robotic Arm

Coupling was present within the joint; however, the robotic arm maintained a multi-link serial configuration. For the dynamic modeling of such arms, the predominant methods are the Newton-Euler and Lagrangian approaches. We opted for the Newton-Euler technique because it eliminates the need for partial derivatives of the joint angles in the calculations and offers a more straightforward programming implementation. To streamline the derivation, we set aside the influence of friction, which leads us to a relationship focused solely on the link factors between the joint output torque and joint angle.

$${\varvec{M}}({\varvec{q}})\varvec{\ddot{q}} + {\varvec{C}}({\varvec{q}},\dot{\varvec{q}})\dot{\varvec{q}} + {\varvec{G}}({\varvec{q}}) + {\varvec{\tau}}_{ext} = \varvec{\tau ,}$$

(12)

where ${\varvec{M}}({\varvec{q}})$ represents the inertia matrix; ${\varvec{C}}({\varvec{q}},\dot{\varvec{q}})$ denotes the Coriolis and centrifugal force matrices; ${\varvec{G}}({\varvec{q}})$ denotes the gravitational matrix; ${\varvec{\tau}}_{ext}$ denotes the external force matrix; and ${\varvec{\tau}}$ represents the joint torque matrix.

3.3 Friction Model

The traditional friction model becomes non-differentiable at zero, creating uncertainties when describing the frictional forces at zero velocity. Such nuances result in oscillations and directional shifts in the controllers grounded in these models. To mitigate this challenge, this study integrated the continuous friction model introduced by Makkar [30]. This model is consistently differentiable across the time domain, and it is notably versatile and adeptly addresses the anisotropic challenges in frictional forces stemming from varying sliding directions. The continuous differentiable friction model is expressed as follows:

$$\begin{aligned} \tau_{f} (\dot{q}) =& \gamma_{1} (\tanh (\gamma_{2} \dot{q}) - \tanh (\gamma_{3} \dot{q})) \\&+ \gamma_{4} \tanh (\gamma_{5} \dot{q}) + \gamma_{6} \dot{q}, \\ \end{aligned}$$

(13)

where γ_k (k = 1–6) represents the coefficient of the continuous differentiable friction model. By incorporating this friction model into the dynamic model of the robotic arm, the dynamic equation can be expressed as follows:

$${\varvec{M}}({\varvec{q}})\varvec{\ddot{q}} + {\varvec{C}}({\varvec{q}},\dot{\varvec{q}})\dot{\varvec{q}} + {\varvec{G}}({\varvec{q}}) + {\varvec{\tau}}_{{\varvec{f}}} + {\varvec{\tau}}_{{{\varvec{ext}}}} = {\varvec{\tau}},$$

(14)

where ${\varvec{\tau}}_{f}$ represents the friction torque.

3.4 Linearization of Dynamic Models

Linearizing the dynamic model of the robotic arm means expressing the torque within it as a product of the coefficient matrix and inertia parameters. Although conventional friction models are typically linearized alongside the dynamic model, the complexity of the continuous friction model led us to defer its linearization and identify the friction model parameters later. While ensuring the elements in τ remain unchanged, we have adjusted the parameter positions in the dynamic model as follows:

$${\varvec{\tau}} = {\varvec{Y}}({\varvec{q}},\dot{\varvec{q}},\varvec{\ddot{q}}){\varvec{U}},$$

(15)

where, U represents the inertia parameters of the robot, and Y represents the coefficient matrix. A single joint consists of 10 inertia parameters, resulting in 60 inertia parameters for a robotic arm with six degrees of freedom. However, owing to the structural constraints on the relative motion of the joints, the Y matrix is not always full rank [31]. This means that we cannot identify every element in U but can only recognize a set of minimal inertia parameters. Therefore, we derived a dynamic model based on this minimal set of inertia parameters, allowing the model to be further expressed as follows:

$${\varvec{\tau}} = \tilde{\varvec{Y}}({\varvec{q}},\dot{\varvec{q}},\varvec{\ddot{q}}){\varvec{U}}_{\min } ,$$

(16)

where $\tilde{\varvec{Y}}({\varvec{q}},\dot{\varvec{q}},\varvec{\ddot{q}})$ represents the column full-rank coefficient matrix obtained through ${\varvec{Y}}({\varvec{q}},\dot{\varvec{q}},\varvec{\ddot{q}})$ decomposition and ${\varvec{U}}_{\min }$ denotes the minimal set of inertia parameters. Using the method proposed by Gautier and Kawasaki [32], which derived the minimal inertia parameter set through categorical reorganization, we identified 36 parameters in the minimal set. The minimum inertia parameters are listed in Table 3.

Table 3 Minimum inertia parameter set

4 Parameter Identification

4.1 Parameter Identification of Friction Model

Typically, friction is gauged by assessing the torque output of the joints of a robot during steady movement. Without a torque sensor, the output can be inferred from the current readings of the motor, providing an indirect measure of friction. Dynamic variations in a robot's joints can result in inconsistencies in motor current data. To derive a more accurate measure of friction, it is essential to filter out factors such as inertia, Coriolis force, centrifugal force, gravity, and external forces using kinematic principles.

When the pitch movement axis was aligned perpendicular to the ground, the gravitational impact on the coupled drive joint remained steady. This specific orientation could be used to counteract the influence of gravity. During consistent motion without external loads, the joint acceleration is nullified, ensuring that the product of inertia and acceleration remains zero. This suggests that steady motion can effectively neutralize the effects of inertia. When the joint engages in a steady back-and-forth movement restricted to the pitch motion, the correlation between friction and torque produced by the motor is expressed as follows:

$$\left\{ {\begin{array}{*{20}l} {\tau_{f1} (\dot{q}) + \tau_{n1} (\dot{q}) = T_{1} (\dot{q}) - T_{2} (\dot{q}),} \\ {\tau_{f1} ( - \dot{q}) + \tau_{n1} ( - \dot{q}) = T_{1} ( - \dot{q}) - T_{2} ( - \dot{q}),} \\ \end{array} } \right.$$

(17)

where τ_n1 represents the modeling error of friction during pitch motion. When the joint engages in a steady back-and-forth movement restricted to rotational motion, the correlation between the friction and torque produced by the motor is expressed as follows:

$$\left\{ {\begin{array}{*{20}l} {\tau_{f2} (\dot{q}) + \tau_{n2} (\dot{q}) = T_{1} (\dot{q}) + T_{2} (\dot{q}),} \\ {\tau_{f2} ( - \dot{q}) + \tau_{n2} ( - \dot{q}) = T_{1} ( - \dot{q}) + T_{2} ( - \dot{q}),} \\ \end{array} } \right.$$

(18)

where τ_n2 represents the modeling error of friction during rotational motion. Potential modeling errors may arise from assembly variances, observational biases, or temperature shifts. Although these inaccuracies were not governed by the speed magnitude, they were influenced by its direction as follows:

$$\left\{ \begin{aligned}& \tau_{fi} ( - \dot{q}) = - \tau_{fi} (\dot{q}), \, \hfill \\ &\tau_{ni} ( - \dot{q}) = \tau_{ni} (\dot{q}). \hfill \\ \end{aligned} \right.$$

(19)

Therefore, by executing oscillatory movements and integrating their outcomes, the modeling discrepancies can be minimized. Once the gravity, inertia, and modeling discrepancies are considered and removed, the correlation between the friction and torque output of the motor is expressed as follows:

$$\left\{ {\begin{array}{*{20}c} {\tau_{f1} (\dot{q}) = {{(T_{1} (\dot{q}) - T_{2} (\dot{q}) - T_{1} ( - \dot{q}) + T_{2} ( - \dot{q}))} \mathord{\left/ {\vphantom {{(T_{1} (\dot{q}) - T_{2} (\dot{q}) - T_{1} ( - \dot{q}) + T_{2} ( - \dot{q}))} 2}} \right. \kern-0pt} 2},} \\ {\tau_{f2} (\dot{q}) = {{(T_{1} (\dot{q}) + T_{2} (\dot{q}) - T_{1} ( - \dot{q}) - T_{2} ( - \dot{q}))} \mathord{\left/ {\vphantom {{(T_{1} (\dot{q}) + T_{2} (\dot{q}) - T_{1} ( - \dot{q}) - T_{2} ( - \dot{q}))} 2}} \right. \kern-0pt} 2}.} \\ \end{array} } \right.$$

(20)

Given the complexities associated with fully mitigating the effects of friction during the movement of a robotic arm, we implemented a phased identification strategy. This method focused on individually assessing the frictional forces of each coupled drive joint. It is important to note that all the coupled drive joints followed identical experimental and identification protocols. For illustrative purposes, we elaborate on this using a coupled drive joint situated at the forearm as a representative example.

We configured the pitch output to move uniformly within the [− 0.5π, 0.5π] interval. The joint initiated its movement at a pace of 1°/s. With each reciprocating motion, the speed was increased by 0.5°/s, interspersed with a 5-second interval. This regimen was performed in over 60 consistent speed-tracking experiments. The detailed procedure is shown in Figure 6.

Figure 6

Pose for friction force collection experiment

Convert the current data to joint output torque values as follows:

$$T_{i} = n\eta K_{m} I_{mi} (i = 1, \, 2),$$

(21)

where n represents the total reduction ratio, with n = 3060; η stands for the total transmission efficiency, and η = 82%; K_m denotes the sensitivity coefficient, with K_m = 29.3 mN·m/A; and I_mi refers to the motor current.

Although traditional identification methods face difficulties in handling nonlinear challenges, this study utilized the Particle Swarm Optimization (PSO) algorithm for this task [33]. Note that the algorithm integrates a compression factor to optimize its global and local search functions, thereby boosting its overall performance. During the identification phase, we designated an iteration limit of 200 and defined a particle dimensionality of six. The objective function depended on the variance between the frictional force captured by the joint and its anticipated counterpart. The primary goal of optimization is to reduce this differential. The fitness curve obtained during the identification process is shown in Figure 7.

Figure 7

Evolutionary curve of fitness algorithm

The parameters for the continuous differentiable friction model of the yaw and pitching motions are listed in Table 4.

Table 4 Continuous differentiable friction model parameters for rotation and pitching motion

A comparison between the fitted friction model and the actual sampled data is shown in Figure 8.

Figure 8

Comparison of the fitted friction model and sampled points

We employed the goodness-of-fit metric to gauge the precision of identifying the friction model. Specifically, the R² value for the friction model of rotational motion was 0.931, whereas that for the pitch motion was 0.927. These data underscore the excellent performance of the particle swarm optimization algorithm for discerning a continuous friction model.

4.2 Inertia Parameter Identification

Before identifying the inertia parameters, it is essential to establish the excitation trajectory. This trajectory represents the path of the joint motion designed to stimulate the joint dynamic parameters maximally [34]. Within the realm of robot dynamic identification, this excitation trajectory is commonly articulated as a finite-term Fourier series as follows:

$$q_{i} (t) = q_{i0} + \sum\limits_{l = 1}^{N} {\left(\frac{{a_{l}^{i} }}{{\omega_{f} l}}\sin (\omega_{f} lt) - \frac{{b_{l}^{i} }}{{\omega_{f} l}}\cos (\omega_{f} lt)\right)} ,$$

(22)

where q_i represents the angle of the i-th joint, N represents the Fourier series, and N = 3 is chosen. ω denotes the fundamental frequency of the trajectory, and in this case, ω_f is set as 0.2π. $q_{i0} ,a_{l}^{i} ,b_{l}^{i}$ represents the parameters to be optimized in the excitation trajectory.

During system identification, the selection of the excitation trajectory parameters cannot be arbitrary. This is owing to the constraints that the robot faces, such as joint angles, angular velocities, and angular accelerations, during operation. To ensure the stability and avoid oscillations, the initial angle, angular velocity, and angular acceleration of the excitation trajectory must approach zero. Moreover, the trajectory should be confined to the operational workspace of the robot. Therefore, when setting the parameters for this trajectory, the essential boundary constraints are:

$$\Lambda_{con} = \left\{ {\begin{array}{*{20}l} {\left| {q_{i} (t)} \right| \le q_{i\max } ,} \\ {\left| {\dot{q}_{i} (t)} \right| \le \dot{q}_{i\max } ,} \\ {\left| {\ddot{q}_{i} (t)} \right| \le \ddot{q}_{i\max } ,} \\ {q_{i} (t_{0} ) = q_{i} (t_{f} ) = 0,} \\ {\dot{q}_{i} (t_{0} ) = \dot{q}_{i} (t_{f} ) = 0,} \\ {\ddot{q}_{i} (t_{0} ) = \ddot{q}_{i} (t_{f} ) = 0,} \\ {W(q(t)) \subset W_{s} ,} \\ \end{array} } \right.$$

(23)

where $\Lambda_{con}$ denotes the constraint conditions. $q_{i\max }$,$\ddot{q}_{i\max }$, and $\ddot{q}_{i\max }$ signifies the maximum angle, angular velocity, and angular acceleration for the ith joint, respectively. $q_{i\max } ,\dot{q}_{i\max }$ and $\ddot{q}_{i\max }$ illustrate the angle, angular velocity, and angular acceleration at the beginning of each cycle of the excitation trajectory, respectively. $q_{i} (t_{f} ), \, \dot{q}_{i} (t_{f} ), \, \ddot{q}_{i} (t_{f} )$ conveys the same parameters but at the conclusion of each cycle. $W( \cdot ), \, W_{s}$ represents both the workspace function and the overall workspace of the robot.

Within these constraints, we address both linear and nonlinear inequalities. Consequently, determining the coefficients for the optimal excitation trajectory requires a nonlinear algorithm. To optimize the excitation trajectory, this study employed the minimum condition number as the primary criterion. As outlined in this research, the six DOF nursing robot arm had joint motion boundary conditions, as listed in Table 5.

Table 5 Motion boundary of the nursing robot

To address the challenge of determining the minimum value for this complex nonlinear function, we used the fmicon function in MATLAB. The relevant parameters derived from the computations are listed in Table 6.

Table 6 Excitation trajectory parameters

The incentive trajectory obtained from the optimized parameters is shown in Figure 9.

Figure 9

The incentive trajectory of the nursing robot

Once the incentive trajectory was determined, the nursing robot was instructed to execute it and collect data on the joint position, velocity, and motor current. Each incentive trajectory lasted 10 s. It was operated over ten continuous cycles to reduce the effects of measurement noise. The methodology for identifying the inertia parameters of the robotic arm is shown in Figure 10.

Figure 10

Inertial parameter identification process

The motor-current data exhibited pronounced variation, warranting preliminary filtering. Given its origin in the incentive trajectory, the data should reflect the frequency attributes of the trajectory. To address this, we employed a low-pass Butterworth filter featuring a 0–15 Hz bandwidth and 4 dB ripple for data refinement. The results of this filtering process are shown in Figure 11.

Figure 11

Current data filtering

After filtering, the motor current yields the output torque of the motor when multiplied by its sensitivity coefficient. However, in the identification process, the primary focus was on the output torque of the joint. Therefore, conversion of the motor current data to the joint output torque is imperative. This transition necessitates the incorporation of the kinematics, dynamics, and frictional forces specific to the coupled drive joint. A comprehensive depiction of this conversion is shown in Figure 12.

Figure 12

The conversion process of output torque

Due to disturbances and various external influences, it is necessary to process the directly collected data for noise reduction. Considering the cyclical nature of the incentive trajectory, we implement mean filtering of the signal within each cycle as follows:

$$\left\{ {\begin{array}{*{20}l} {\overline{q}_{1} (k) = \frac{1}{{N_{j} }}\sum\limits_{i = 1}^{{N_{j} }} {q_{1i} (k),} } & {\overline{q}_{2} (k) = \frac{1}{{N_{j} }}\sum\limits_{i = 1}^{{N_{j} }} {q_{2i} (k),} } \\ {\overline{\dot{q}}_{1} (k) = \frac{1}{{N_{j} }}\sum\limits_{i = 1}^{{N_{j} }} {\dot{q}_{1i} (k),} } & {\overline{\dot{q}}_{2} (k) = \frac{1}{{N_{j} }}\sum\limits_{i = 1}^{{N_{j} }} {\dot{q}_{2i} (k),} } \\ {\overline{\tau }_{f1} (k) = \frac{1}{{N_{j} }}\sum\limits_{i = 1}^{{N_{j} }} {\tau_{f1i} (k),} } & {\overline{\tau }_{f2} (k) = \frac{1}{{N_{j} }}\sum\limits_{i = 1}^{{N_{j} }} {\tau_{f2i} (k),} } \\ {\overline{T}_{mc1} (k) = \frac{1}{{N_{j} }}\sum\limits_{i = 1}^{{N_{j} }} {T_{mc1i} (k),} } & {\overline{T}_{mc2} (k) = \frac{1}{{N_{j} }}\sum\limits_{i = 1}^{{N_{j} }} {T_{mc2i} (k),} } \\ \end{array} } \right.$$

(24)

where K represents the number of sampling times within each cycle, N_j is the number of cycles in the incentive trajectory, $q_{1i} (k), \, q_{2i} (k)$ represents the yaw and pitch joint angles, $\tau_{f1i} (k), \, \tau_{f2i} (k)$ signifies the yaw and pitch frictional forces, and $T_{mc1} (k), \, T_{mc2} (k)$ corresponds to the yaw and pitch joint output torques. Therefore, the robot's output torque can be expressed as follows:

$$\left\{ \begin{aligned} &\varvec{\tau = }\left[ {\tau_{1} ,\tau_{2} ,\tau_{3} ,\tau_{4} ,\tau_{5} ,\tau_{6} } \right]^{{\text{T}}} , \hfill \\ &\tau_{i} = \overline{T}_{mc1} - \overline{\tau }_{fi} , \, i = 1\sim 6. \hfill \\ \end{aligned} \right.$$

(25)

Utilizing Eq. (16), we form a dataset where the joint torque observation matrix τ serves as the target and the coefficient matrix $\tilde{\varvec{Y}}({\varvec{q}},\dot{\varvec{q}},\varvec{\ddot{q}})$ acts as the features. Of this dataset, 70% was designated for training, and the remaining 30% for testing. Once standardized, the curated dataset was input into a ridge regression model for training. The dynamic inertial parameters of the identified robotic arm are listed in Table 7.

Table 7 Identified inertia parameters

5 Experiment

In transfer care, caregivers are primarily responsible for moving patients from their beds to wheelchairs and addressing essential needs such as restroom use, feeding, and mobility. To simulate this, we created the experimental scenario shown in Figure 13. To address these multifaceted challenges, our dual-arm care robot was engineered to control several motors in unison, necessitating exceptional real-time responsiveness, functionality, and stability from its controller. To achieve this, we integrated the robot's control system with the EtherCAT fieldbus by selecting the Elmo MOLTWI-10/100EEOT driver and an Apache industrial PC with an i7-8700 processor as the central control unit. Moreover, this control unit offers remote operability, enabling distant management of care robots.

Figure 13

Experimental scenario

To validate the accuracy of the identification results, the robotic arm was directed to follow a verification trajectory that differed from the original stimulation trajectory. We then assessed the alignment between the computed joint torques and the actual measured values. The verification trajectory presented in this study was formulated by adjusting the coefficients of the stimulation trajectory, as shown in Figure 14.

Figure 14

Verify trajectory

We captured motor current and speed data from the robotic arm. Drawing on the coupled drive joint dynamics model, friction model, and the discerned parameters for both the joint dynamics and friction models, we determined the joint torque and treated it as the measured value. Throughout this procedure, we employed the noise-reduction filtering algorithm outlined in Section 4.2 to refine the motor current and speed data. Next, by leveraging the recognized inertia parameters of the robotic arm and its linearized dynamic model, we computed the theoretical joint torque. Finally, we juxtaposed the measured torque with its theoretical counterpart, as shown in Figure 15.

Figure 15

Verification of Dynamic Parameter Identification

To evaluate the accuracy of the identification results, the root mean square errors (RMSE) of the actual and theoretical torques were used as a quantitative metric. The torque root mean square error can be expressed as follows:

$$\varepsilon_{RMS,i} = \sqrt {\frac{1}{{k_{n} }}\sum\limits_{k = 1}^{{k_{n} }} {(\hat{H}_{i} (k) - H_{i} (k))^{2} } } ,i = 1\sim 6,$$

(26)

where $\varepsilon_{RMS,i}$ represents the root mean square error of each joint, k_n denotes the number of sampling points, $\hat{H}_{i} (k)$ represents the predicted joint torque, $H_{i} (k)$ represents the measured joint torque, and i represents the joint index. The RMSE between the measured and theoretical torques for each joint are listed in Table 8.

Table8 The RMSE between the measured and theoretical torques for each joint

As listed in Table 8, it is evident that the root mean square error between the measured and theoretical torques is notably small when juxtaposed with the joint drive torque. This underscores the precision of the dynamic parameter identification shown in Figure 15, thereby attesting to the efficacy of the proposed dynamic modeling and identification techniques. Furthermore, fluctuations in the measured torque surrounding the theoretical values are shown in Figure 15, indicating discrepancies between the dynamic identification outcomes and the anticipated results. These discrepancies likely arose from pronounced variations in the motor current and the intricate design of the robotic arm. The dynamic model, which is based on a streamlined mechanical representation, neglects elements such as lubrication and the impact of gear meshing. In addition, imperfections during the machining and assembly phases introduce gaps between the transmission components, further influencing the precision of identification.

6 Discussion

Dynamic modeling and parameter identification are essential for achieving dynamic, compliant interactions between robots and humans. This study focuses on an innovative coupled-drive joint and explores its dynamic modeling, robotic arm dynamic modeling, friction model, friction model parameter identification, and inertia parameter identification methods. The accuracy of these methods was validated through experiments.

However, there are noticeable discrepancies between the experimentally collected and theoretical data. These discrepancies stem from unavoidable simplifications made during the mathematical modeling process, such as assuming uniform materials for components, ignoring machining errors, and neglecting the influence of small-inertia parts. Although identifying model parameters through actual data partially compensates for these modeling errors, it does not fundamentally resolve the mismatch between the actual working conditions and theoretical derivations.

Given that humans possess a certain degree of environmental adaptability, ensuring that the gap between actual conditions and theoretical derivations remains within a specific range allows robots to be effectively applied in human-robot interaction tasks. Therefore, despite errors, the methods presented in this study remain feasible. In the future, we plan to transcend the current research framework and construct dynamic models of complex human-robot systems entirely using data-driven methods.

7 Conclusions

This study explored methods for the dynamic modeling and parameter identification of a coupled drive joint robotic arm. The accuracy of these methods was validated through trajectory-tracking experiments using a nursing robot. The key findings are as follows:

(1) The dynamic model of the coupled-drive nursing robot comprises two parts: the joint dynamics model and the robotic arm dynamics model. The joint dynamics model describes the relationship between the motor torque and joint torque, whereas the robotic arm dynamics model describes the relationship between the joint torque and robot motion.

(2) By designing experiments to compensate for the effects of gravity, inertia, and modeling errors, friction can be represented using directly collected current data. The particle swarm optimization algorithm demonstrated high accuracy in identifying the friction model parameters. In this study, the coefficient of determination (R²) for joint turnover friction was 0.931, and for joint pitch friction was 0.927.

(3) The Fourier excitation trajectory, optimized according to the robot's motion constraints, effectively identifies the inertial parameters of the dynamic model. In this study, the root mean square error (RMSE) for the inertial parameter identification of joint 1 (base joint) is 14.372 N·m², and for joint 6 (end joint) is 3.152 N·m².