A Motion Propagation Force Analysis of Multi-DoF Systems Using the Partial Lagrangian Method

Abstract

A partial Lagrangian method is proposed as an inverse dynamics analysis method for multi-link systems. This method, combined with automatic differentiation, allows for the derivation of equations of motion and analytical extraction of motion-induced torque components. We introduce the concept of motion propagation force to describe joint torque components generated by the motion of other joints. This concept aligns with existing notions such as interaction torque, while providing a novel analytical perspective. The effectiveness of the proposed method is confirmed through simulations using a three-DoF arm model, where motion propagation torques are visualized and validated. This method is useful for motion analysis and impedance control in complex robotic systems.

1. Introduction

Motion propagation occurs in humans and multi-linked robots. By utilizing motion propagation, humans and robots can realize complex motions. We define the motion propagation force as the effect of joint motion transmitted to each link on the other joints. In this paper, we introduce the concept of motion propagation force to describe analytically derived components of joint torques that originate from the motion of other joints. While this is not a standard term, it is conceptually related to interaction torque and internal force transmission in multi-DoF systems and is used for clarity within our proposed analytical framework.

Motion propagation is a fundamental phenomenon observed in both human movement and robotic systems. It refers to the transmission of motion effects from one joint or link to another and is closely tied to the structural dynamics and control strategies of multi-link systems. Previous studies have shown that understanding force and motion transmission is essential in the analysis of parallel mechanisms [1] and that it plays a crucial role in impedance control for robot–environment interactions [2]. These insights confirm that motion propagation analysis is relevant not only for mechanical modeling but also for practical control design in robotics.

Here, visualization of motion propagation means extracting a specific component of motion as a force. However, visualizing the motion propagation of multi-degree-of-freedom systems is difficult because it requires solving complex equations of motion and extracting elements related to motion propagation from the analytical solution. In this study, the partial Lagrangian method [3], convenient for analyzing multi-linked systems, and an automatic differentiation method [4,5] for solving ordinary differential equations are used to derive motion propagation torque.

In this study, we focus on open-chain serial robotic manipulators, for which the proposed method is directly applicable. However, we also discuss the potential to extend the method to closed-chain systems, such as parallel robots, by incorporating constraint forces and internal interactions. The generalization of the method is considered in the Discussion section.

The Newton–Euler and Lagrangian methods are known as calculation methods in inverse dynamics analysis and are used for various applications [6,7,8,9]. The Newton–Euler method can handle multiple degrees of freedom [10,11,12]; however, it is often used for numerical analysis and is not suitable for producing an analytical solution. Although the Lagrangian method can produce an analytical solution from the calculation, the computational complexity increases exponentially with the number of links, making analysis with multiple links difficult. In addition, when the model is changed, the Lagrangian function must be recalculated, which is computationally expensive and makes it difficult to change the model. To address this drawback, Kusaka et al. proposed the partial Lagrangian method, which is characterized by dividing the calculation for each joint. As a result, this method can derive an analytical solution and can handle multi-link systems. In the Lagrangian method, each joint torque is obtained collectively from the energy of the entire system. In contrast, joint torques in the partial Lagrangian method are defined as partial torques for each link motion element based on the divide-and-conquer method [13]. By further decomposing the partial torques, the motion propagation force can be obtained. Since the partial Lagrangian method can solve multi-degree-of-freedom models analytically and efficiently, it can also be used in research to automatically generate equations of motion for manipulator robots with various configurations and degrees of freedom [14].

The Runge–Kutta method [15] is widely used for solving ordinary differential equations. In addition, it is an approximate method for solving complex mathematical equations and is useful for numerical analysis [16]. However, it does not retain an analytical solution in the calculation process. Therefore, it is not applicable in this study, in which the motion propagation force is analyzed by analytically solving equations using the partial Lagrangian method and then decomposing the equations. As a method for analytically solving ordinary differential equations, there is automatic differentiation, as implemented in tools such as PyTorch (2.0.0) [17]. Automatic differentiation is a technique used for computing gradients, often applied in machine learning [18]. It enables efficient and accurate calculation of derivatives for defined functions. It avoids the approximation errors of numerical differentiation and the complexity of symbolic differentiation by automatically deriving derivatives through the combination of basic operations (addition, subtraction, multiplication, division, and elementary functions) within a program. Automatic differentiation represents the computation process of a function as a computational graph, where each node corresponds to a basic operation and edges propagate the values of variables. By applying the chain rule to this computational graph, it becomes possible to efficiently and analytically compute derivatives even for complex functions. Therefore, we believe that automatic differentiation is effective in calculating motion propagation torque, which is solved analytically, and a part of the equation is extracted.

The purpose of this study is to determine the propagation of motion forces in open-link manipulators using the partial Lagrange method and automatic differentiation. In order to achieve motion analysis and impedance control using complex models, we first conducted simple experiments with a robotic arm to verify whether the targeted components of interest could be extracted. First, the motion propagation force is described, followed by an overview of the partial Lagrangian method and a method for deriving the motion propagation force. Next, automatic differentiation is explained, motion simulation of a three-link manipulator is performed, and the values and waveforms of the obtained motion propagation torque are discussed.

2. Materials and Methods

2.1. Motion Propagation Force

Transmission of force from proximal to distal and vice versa is important in the analysis of motion. For example, in baseball pitching, it has been shown that improving the efficiency of force transfer from the shoulder joint to the elbow joint and the hand joint during shoulder motion leads to improved pitching performance and that reducing the burden on the shoulder joint during hand motion leads to the prevention of injury [19]. Therefore, the acquisition of the kinetic chain is also important for improving pitching performance, and studies have been conducted on the mechanical analysis of the upper limb kinetic chain in the pitching motion [20]. However, an efficient method for determining the effects of motion at one joint on other joints has not yet been established. Therefore, we propose a method for efficiently extracting an arbitrary term from the forces applied to a joint by using the motion propagation force.

Although the concept of torque transfer among joints is related to established terms such as interaction torque and inverse dynamics, our focus differs in that we define “motion propagation force” as a specific term representing the torque generated at a joint due to the motion of another joint, isolated through analytical decomposition. This is not limited to inertia-driven effects but includes all analytically traceable components in joint space derived from the partial Lagrangian formulation.

The motion propagation force ${\tau }_{ij}^p$ is a force that is transferred to other joints owing to the transmission of joint motion to each link. The force can be defined as the torque given to joint i by the motion of joint j, as shown in Equations (1) and (2).

$$\begin{array}{c}\hfill {\tau }_{ij}^p\left(q_k^{\left(m\right)}\right)={\displaystyle \frac{\partial {\tau }_{ij}}{\partial q_k^{\left(m\right)}}}{q}_k^{\left(m\right)},\end{array}$$

(1)

$$\begin{array}{c}\hfill q_i^{\left(2\right)}=\ddot{q_i},\end{array}$$

(2)

where $q_i^{\left(m\right)}$ is the $m(m=0,1,2)$-order time derivative of $q_i$. As can be seen from Equation (1), the motion propagation force ${\tau }_{ij}^p\left(q_k^{\left(m\right)}\right)$ means that the torque is extracted by focusing on the acceleration when $m=2$, velocity when $m=1$, and the position when $m=0$. Therefore, by determining the motion propagation force, the motion propagation of interest can be identified. Figure 1 shows the conceptual diagram. The effect of any joint j on joint i can be extracted even between links.

To understand the generation of motion, interaction torque [21] has been proposed as an index by focusing on the kinematic chain. This index is based on understanding the generation of motion by decomposing the torque generated at the joints into muscle, gravitational, and interaction torques. However, the motion propagation torque is an index that arbitrarily extracts the motion of all links; thus, it is treated differently. Therefore, this indicator can be used to analyze the motion of multi-link systems and can also be used for impedance adjustment and controller design.

2.2. Derivation of Partial Torque Using the Partial Lagrangian Method

The Newton–Euler method is often used as a numerical algorithm, as in the recursive Newton–Euler algorithm [22]. Since the equation structure does not remain in such numerical algorithms, arbitrary terms cannot be extracted. The Lagrangian method can be solved analytically, but the number of terms in the equation is so large that it is difficult to extract the terms of interest from among them. The partial Lagrangian method decomposes even complex models and performs calculations, making it easier to extract the torque terms of interest because the equations and terms to focus on are compact. This advantage becomes more important as the complexity of the model increases. Therefore, in cases such as multi-linked systems, the partial Lagrangian method is the best choice because it can clearly identify the necessary terms.

The Lagrangian $\mathcal{L}$ in the Lagrangian equation of motion is defined based on the kinetic energy K and potential energy P as follows.

$$\mathcal{L}=K-P.$$

(3)

In the Lagrangian method, the equations of motion are obtained by the following solution:

$${\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{d}{d\dot{q_k}}}\mathcal{L}\right)-{\displaystyle \frac{d}{d{q}_k}}\mathcal{L}={\tau }_k,$$

(4)

where $q_k$ and ${\tau }_k$ are the k-th generalized coordinates and forces, respectively.

For simplicity of description, the differential operator $D_k$ is defined as follows:

$$D_k={\displaystyle \frac{d}{dt}}{\displaystyle \frac{\partial }{\partial \dot{q_k}}}-{\displaystyle \frac{\partial }{\partial q_k}}.$$

(5)

Using the energy distribution method, Equation (3) can be subsequently transformed for a system with n links as follows:

$$\mathcal{L}=\sum _{i=1}^n{K}_i-\sum _{i=1}^n{P}_i=\sum _{i=1}^n(K_i-P_i)=\sum _{i=1}^n{\mathcal{L}}_i.$$

(6)

The partial torque ${\tau }_{ki}$ of ${\tau }_k$ obtained by $D_k$ acting on the partial Lagrangian ${\mathcal{L}}_i$ of link i can be expressed from expressions (5) and (6) as follows:

$$D_k\mathcal{L}=D_k\sum _{i=1}^n{\mathcal{L}}_i=\sum _{i=1}^n\left(D_k{\mathcal{L}}_i\right)=\sum _{i=1}^n{\tau }_{ki}.$$

(7)

Now, consider the components of the partial Lagrangian ${\mathcal{L}}_i$. When $k\le i$, the partial Lagrangian ${\mathcal{L}}_i$ contains $q_k$, but when $k>i$, ${\mathcal{L}}_i$ does not contain $q_k$. Therefore, the partial torque $D_k{\mathcal{L}}_i$ can be expressed as follows:

$$D_k{\mathcal{L}}_i=\left\{\begin{array}{cc}{\tau }_{ki}& (i\ge k)\\ 0& (i<k)\end{array}.\right.$$

(8)

Equation (8) can be summarized as shown in

Table 1. Relationship between partial Lagrangian and Lagrangian methods.

	Partial Lagrangian								Lagrangian
	${\mathcal{L}}_1$	${\mathcal{L}}_2$	${\mathcal{L}}_3$	⋯	${\mathcal{L}}_i$	⋯	${\mathcal{L}}_n$	$\stackrel{\sum }{\to }$		$\mathcal{L}$
$D_1$	${\tau }_{11}$	${\tau }_{12}$	${\tau }_{13}$	⋯	${\tau }_{1i}$	⋯	${\tau }_{1n}$	→		${\tau }_1$
$D_2$	0	${\tau }_{22}$	${\tau }_{23}$	⋯	${\tau }_{2i}$	⋯	${\tau }_{2n}$	→		${\tau }_2$
$D_3$	0	0	${\tau }_{33}$	⋯	${\tau }_{3i}$	⋯	${\tau }_{3n}$	→		${\tau }_3$
⋮	⋮	⋮	⋮	⋱	⋮		⋮	⋮		⋮
$D_k$	0	0	0	⋯	${\tau }_{ki}$	⋯	${\tau }_{kn}$	→		${\tau }_k$
⋮	⋮	⋮	⋮		⋮	⋱	⋮	⋮		⋮
$D_n$	0	0	0	⋯	0	⋯	${\tau }_{nn}$	→		${\tau }_n$

. $\mathcal{L}$ also changes, and the equations of motion must be re-derived. However, in the partial Lagrangian method, calculations are performed for each degree of freedom; thus, the results of previous calculations can be reused, and additional calculations need only be performed for the increased degrees of freedom. Therefore, the partial Lagrangian method is effective in reducing computational costs and easily adapts to model extensions and modifications. Furthermore, the partial Lagrangian method also has advantages when determining the motion propagation force. The conventional Lagrangian method can also be used to determine the motion propagation force, as the partial torques can be added together to match the joint torque. However, the partial Lagrangian method, which can be decomposed and analyzed link by link, provides a more detailed view of the terms, making it easier to determine the motion propagation force.

Consider the physical meaning of the partial torque ${\tau }_{ij}$. The partial torque is obtained by differentiating the energy about link j from the parameter about coordinate i. Therefore, partial torque can be interpreted as a torque generated at joint i by the global motion of link j. This can be expressed in terms of robotics as Equation (9).

$$\begin{array}{c}\hfill {\tau }_{ij}={\mathit{M}}_{\mathit{ij}}\ddot{\mathit{\theta }}+V_{ij}(\dot{\mathit{\theta }},\mathit{\theta })+g_{ij}\left(\mathit{\theta }\right),\end{array}$$

(9)

where $\mathit{\theta }={\left[\begin{array}{cccc}{\theta }_1& {\theta }_2& \cdots & {\theta }_n\end{array}\right]}^T$, ${\mathit{M}}_{\mathit{ij}}\in {\mathcal{R}}^{1\times n}$ is the inertia matrix, $V_{ij}(\dot{\mathit{\theta }},\mathit{\theta })$ is a function of the velocity-squared term, and $g_{ij}\left(\mathit{\theta }\right)$ is a function of the gravity effect. Next, the partial torque can be expressed in terms of the auto-derivative concept, as shown in Equations (10)–(12).

$$\begin{array}{cc}\hfill {\tau }_{ij}& =\sum _{s=1}^i{\displaystyle \frac{\partial {\tau }_{ij}}{\partial {\ddot{\theta }}_s}}{\ddot{\theta }}_s+{\displaystyle \frac{1}{2}}\sum _{s=1}^i{\displaystyle \frac{\partial {\tau }_{ij}}{\partial {\dot{\theta }}_s}}{\dot{\theta }}_s+\sum _{s=1}^i{\displaystyle \frac{\partial {\tau }_{ij}}{\partial {\dot{\theta }}_s^2}}{\dot{\theta }}_s^2+g_{ij}\left(\theta \right)\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill & \simeq \sum _{s=1}^i{\displaystyle \frac{\partial {\tau }_{ij}}{\partial {\ddot{\theta }}_s}}{\ddot{\theta }}_s+{\displaystyle \frac{1}{2}}\sum _{s=1}^i{\displaystyle \frac{\partial {\tau }_{ij}}{\partial {\dot{\theta }}_s}}{\dot{\theta }}_s+\sum _{s=1}^i{\displaystyle \frac{\partial {\tau }_{ij}}{\partial {\dot{\theta }}_s^2}}{\dot{\theta }}_s^2+\sum _{s=1}^i{\displaystyle \frac{\partial {\tau }_{ij}}{\partial {\theta }_s}}{\theta }_s\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill & =\sum _{b=0}^2{\mathit{c}}_{\mathit{b}}·{\mathit{\theta }}^{\left(b\right)}.\hfill \end{array}$$

(12)

Here, ${\mathit{c}}_{\mathit{2}}\in {\mathcal{R}}^{1\times n}$ is the coefficient matrix for the acceleration, ${\mathit{c}}_{\mathit{1}}\in {\mathcal{R}}^{1\times n}$ is the coefficient matrix for the velocity, and ${\mathit{c}}_{\mathit{0}}\in {\mathcal{R}}^{1\times n}$ is the coefficient matrix for the position. This shows that the partial torque ${\tau }_{ij}$ is the torque generated at joint i by the motion of link j in the world system. Furthermore, it includes the effect of the motion of link j from the link constraints owing to the motion of other links as well as the joint motion of link j. Therefore, by calculating Equation (1), only the joint motion of the link can be extracted from the partial torque, and the motion propagation torque can be derived.

Partial torque is usefulfor identifying the joint torque caused by the global motion of a link. However, if the joint motion effect of a link on other links is to be determined, the motion propagation torque is effective because the joint motion effect of other links is also included, and the partial torque must be extracted further.

2.3. Extraction of Motion Propagation Force Using Automatic Differentiation

The computational methods for ordinary differential equations are considered. Numerical integration methods such as the Euler [23,24] and Runge–Kutta methods [25,26] are known as calculation methods for ordinary differential equations. The Euler method employs a single formula, and it is easy to calculate. However, the method has a large number of errors. The Runge–Kutta method compensates for this disadvantage, and although it requires additional equations, it can be calculated with a smaller error. However, neither method can be solved analytically because they are calculated by substituting numerical values. Therefore, calculations are required to obtain the motion propagation torque, which is solved by further decomposing the partial torque, resulting in expensive calculation costs. Automatic differentiation is therefore used to solve methods analytically and efficiently.

Automatic differentiation analytically solves the derivative of a composite function by using a computational graph that represents the calculation process regarding nodes and edges. As an example, the computational graph when sin$\theta \left(t\right)$ is differentiated by $\theta \left(t\right)$ is shown in Figure 2. By expressing the angular velocity $\omega \left(t\right)$ as the multiplication of the angle $\theta \left(t\right)$ and the micro time $\Delta t$, and the angular acceleration $\dot{\omega }\left(t\right)$ as the multiplication of the angular velocity $\dot{\omega }\left(t\right)$ and the micro-time $\Delta t$, realizing differential calculations of angles and angular velocities is possible.

The four arithmetic operations of the variables are substituted into new variables and repeated until only one variable is left. The derivatives are then differentiated from the reverse and returned to the original variable. This allows the differential of the composite function to be solved without numerical differentiation. Although the automatic differentiation of analytically calculated derivatives also introduces calculation errors, an error reduction method has also been studied [27]. The reasons for using this method in this study are as follows.

Additionally, it is compatible with the properties of the partial Lagrangian method. The partial Lagrangian method consists of a combination of an energy equation and a derivative. Although the energy equation can be easily obtained, it is difficult to differentiate for a complex model. Therefore, the derivation of the equations of motion and the calculation of the equations of motion can be performed easily by combining the partial Lagrangian method with automatic differentiation, which can easily differentiate synthetic functions. This allows inverse dynamics analysis to be performed quickly and efficiently.

Moreover, the method retains its ability to derive analytical solutions. In numerical differentiation, the calculation process remains only in the form of values, whereas in automatic differentiation, the computational graph remains, as shown in Figure 2, to ensure that the intermediate process can also be retained as an expression. Therefore, the joint torque can be obtained simultaneously with the further decomposition of the joint torque, such as the motion propagation torque.

We consider a method for extracting the motion propagation force using the partial Lagrangian method and automatic differentiation. Since the partial torque can be expressed as in Equation (12), it is substituted into Equation (1). Then, it can be transformed, as shown in Equations (13)–(15).

$$\begin{array}{cc}\hfill {\tau }_{ij}^p\left({\theta }_k^{\left(m\right)}\right)& ={\displaystyle \frac{\partial {\sum }_{b=0}^2{\mathit{c}}_{\mathit{b}}·{\mathit{\theta }}^{\left(b\right)}}{\partial {\theta }_k^{\left(m\right)}}}{\theta }_k^{\left(m\right)}\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill & =\sum _{b=0}^2{\displaystyle \frac{\partial {\mathit{c}}_{\mathit{b}}·{\mathit{\theta }}^{\left(b\right)}}{\partial {\theta }_k^{\left(m\right)}}}{\theta }_k^{\left(m\right)}\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill & ={\displaystyle \frac{\partial {\mathit{c}}_{\mathit{0}}·{\mathit{\theta }}^{\left(0\right)}}{\partial {\theta }_k^{\left(m\right)}}}{\theta }_k^{\left(m\right)}+{\displaystyle \frac{\partial {\mathit{c}}_{\mathit{1}}·{\mathit{\theta }}^{\left(1\right)}}{\partial {\theta }_k^{\left(m\right)}}}{\theta }_k^{\left(m\right)}+{\displaystyle \frac{\partial {\mathit{c}}_{\mathit{2}}·{\mathit{\theta }}^{\left(2\right)}}{\partial {\theta }_k^{\left(m\right)}}}{\theta }_k^{\left(m\right)}.\hfill \end{array}$$

(15)

Here, terms other than the term of interest ${\theta }_k^{\left(m\right)}{|}_{m\ne b}$ become zero and cancelled out. Then, the term of interest ${\theta }_k^{\left(m\right)}{|}_{m=b}$ can be mechanically extracted as in Equations (16) and (17).

$$\begin{array}{cc}\hfill {\tau }_{ij}^p\left({\theta }_k^{\left(m\right)}\right)& =\left({\left.{\displaystyle \frac{\partial {\mathit{c}}_{\mathit{b}}·{\mathit{\theta }}^{\left(b\right)}}{\partial {\theta }_k^{\left(m\right)}}}\right|}_{b=m}\right){\theta }_k^{\left(m\right)}\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill & =c_{mk}{\theta }_k^{\left(m\right)}.\hfill \end{array}$$

(17)

In this case, the coefficient $c_{mk}$ (the k-th value of the vector ${\mathit{c}}_{\mathit{b}}{|}_{b=m}$) corresponds to the gradient obtained by automatic differentiation, so it can be extracted while retaining the form of the remaining equation as a computational graph. Using Equations (13)–(17), we can mechanically extract the motion propagation force defined in Equation (1). Therefore, we can analyze only arbitrary terms of interest in the complex motion of equations.

A simpler representation of this method is shown in Figure 3. The calculation graph represents the computational process shown in Figure 2. Since the calculation process remains as a computational graph, any term can be extracted by using Equations (13)–(17).

The method combining partial Lagrangian and automatic differentiation is an algorithm that can analytically solve multi-linked systems while preserving the formula structure by solving the derivative calculation as a gradient calculation. Because the equations of motion are extremely large in complex systems when extracting terms such as interaction torque, it has been difficult to determine specific terms from the analytically solved equation structure. This method is innovative in that it can extract arbitrary terms without the need for recalculation.

3. Simulation and Results

Experiments and inverse dynamics analysis simulations were performed to inspect how the motion of one link propagates to the other links using the aforementioned method. Note that Equation (1) was defined for the generalized force, whereas here, the generalized force represents the torque since the rotary manipulator is the target. First, an experiment was conducted using a six-DoF articulated robot CRANE-X7 [28], shown in Figure 4a, to obtain the input values (angle, angular velocity, and angular acceleration) for the inverse dynamics analysis. Simulations were carried out using the model shown in Figure 4b with the obtained values as input values.

Angular angles were obtained by manually controlling the position of the joints using the CRANE-X7 user interface. The angular velocity and angular acceleration data were obtained by applying a low-pass filter to the obtained angular data after calculating the derivatives using a five-point formula. The target motion was a motion in which link 3 was moved until approximately 4 s, and link 2 was moved thereafter, as shown in Figure 5. The purpose of targeting this type of motion was to examine whether the effects of the joint motion of the links could be extracted. Moreover, it is easier to examine whether the effects can be extracted when all but one link is not in motion.

Using the abovementioned data obtained on the actual machine, a simulation was performed to analyze the motion propagation torque. The parameter values used in the simulation are shown in

Table 2. Parameter values of $l_a$ (the length of link a), $l_{ga}$ (the distance to center of gravity of link a), $m_a$ (the mass of link a), and $I_a$ (the moment of inertia of link a) used in the simulation ($a=1,2,3$).

	Parameter Values
$l_a$	0.250 m
$l_{ga}$	0.125 m
$m_a$	0.600 kg
$I_a$	$5.00\times {10}^{-3}\mathrm{kg}·{\mathrm{m}}^2$

.

The results of the inverse dynamic analysis using the partial Lagrangian method under the above conditions are shown from Figure 6, Figure 7, Figure 8 and Figure 9.

Figure 6 shows the joint torque ${\tau }_i$ from link 1 to link 3. Figure 7 shows the partial torque ${\tau }_{1i}$ at joint 1. Figure 8 shows the motion propagation torque ${\tau }_{1i}^p\left({\theta }_i^{\left(1\right)}\right)$, focusing on the velocity transmitted to joint 1. Figure 9 shows the motion propagation torque ${\tau }_{1i}^p\left({\theta }_i^{\left(2\right)}\right)$, focusing on the acceleration transmitted to joint 1.

Simulations were also conducted to analyze the motion propagation torque using manually generated data as input values, in addition to the data obtained in the experiments. The target motion was a motion in which only link 3 moved for up to 3 s, and link 2 and link 3 moved thereafter, as shown in Figure 10. The purpose of using this type of motion was to confirm whether only the effect of the moving link could be extracted when the other links were not moved at all and also to confirm whether the effect on the other joints could be extracted even when multiple joints were moved simultaneously. The parameters and models used in the simulation and the indices shown in the graphs are the same as in the previous simulation.

The results of the inverse dynamics analysis using the partial Lagrangian method under the above conditions are shown from Figure 11, Figure 12, Figure 13 and Figure 14. The parameters and other conditions are the same.

Focusing on joint 1, the partial torques ${\tau }_{11}$, ${\tau }_{12}$, and ${\tau }_{13}$ in Figure 12 are added together to match ${\tau }_1$ in Figure 11. This is also the case for the partial torques and joint torques in Figure 6 and Figure 7. Figure 13 shows the results of extracting the velocity component of the motion propagation torque at ${\tau }_{12}$ and ${\tau }_{13}$ in Figure 12. Figure 14 shows the results of extracting the acceleration component of the motion propagation torque at ${\tau }_{12}$ and ${\tau }_{13}$ of Figure 12. The sum of all components of the motion propagation torque also corresponds to the partial torque. Therefore, the motion propagation torque represents the component of joint torque that is of interest.

Computational costs can be reduced by using automatic differentiation to extract the elements involved in motion propagation. As an example, Figure 15 shows the reduction rate of the calculation cost when extracting the elements related to the angular acceleration of the root joint from the joint torque applied to the root. The reduction rate of the calculation cost is defined as the ratio of the number of terms in a calculation that are eliminated by performing automatic differentiation to the number of terms in the total calculation.

This result shows that automatic differentiation can reduce the computational cost by the approximation $y=A(1-e^{-\lambda x})$. Where x is the DoF, y is the reduction rate, and A and $\lambda$ are the coefficients. In this case, $A=0.95$, and $\lambda =0.44$ when using the results from $x=1$ to $x=10$. The larger DoF x is, the more the reduction ratio y and the coefficient A approach 1 asymptotically.

4. Discussion

From Figure 7, ${\tau }_{12}$ and ${\tau }_{13}$ are the partial torques exerted on joint 1 by the motion of link 2 and link 3, respectively. However, their waveforms do not correspond to the joint motion of link 2 and link 3. This is because, as explained earlier, the partial torque includes all terms on the right-hand side of Equation (9). Therefore, the centrifugal and Coriolis forces act on the latter part of ${\tau }_{13}$; hence, the effect of Link 2 is also superimposed.

In contrast, the difference between when the link’s own joints are in motion and when they are not in motion can be seen in Figure 8 and Figure 9. When the link is not in motion, the value is close to zero. The waveforms of the torque when the joint of the link itself is in motion, such as the second half of ${\tau }_{12}^p\left({\theta }_2^{\left(2\right)}\right)$ (blue line) and the first half of ${\tau }_{13}^p\left({\theta }_3^{\left(2\right)}\right)$ (green line) in Figure 8 and Figure 9, are close to the joint motion. The value is not zero because the CRANE-X7 robot was operated manually when the experiment was conducted; thus, joint motion was generated, albeit slightly. Moreover, considering Figure 13 and Figure 14, the motion propagation torque ${\tau }_{12}^p\left({\theta }_2^{\left(1\right)}\right)$ (blue line) and ${\tau }_{12}^p\left({\theta }^{\left(2\right)}\right)$ (blue line) for link 2 without angle change, the first half of which is zero. Furthermore, the latter part, where several links are in motion, also corresponds to the motion of each link.

Therefore, by further decomposing the partial torque, the torque given by the joint motion of the link itself is visible. In the conventional Lagrangian method, ${\tau }_{12}$ and ${\tau }_{13}$ were added together and expressed as ${\tau }_1$; thus, the meaning of the ${\dot{\theta }}_2$ term could not be distinguished directly from the analytical solution of ${\tau }_1$. However, by using the partial Lagrangian method, the torque applied to a joint can be determined in more detail, and the joint motion that gives rise to the torque can also be determined.

The torque of motion propagation focusing on acceleration and velocity was obtained in this study. However, the torque of motion propagation focusing on position is only influenced by gravity, which cannot be expressed as a linear sum of $\theta$. Gravity can also be regarded as an external force rather than an internal force. Therefore, treating the effect of gravity as an external force creates symmetry in the extraction of the motion propagation torque and presents better handling.

Figure 15 shows that the calculation process for obtaining the motion propagation torque can be significantly reduced by utilizing automatic differentiation. Calculating the motion propagation to be extracted from the equations after all the analytical solutions have been obtained involves several redundant calculations, which increase with the number of links. When extracting the elements related to the angular acceleration of the root joint from the joint torque applied to the root, it is necessary to perform approximately three times more redundant calculations for a 5-DoF analysis and approximately 16 times more redundant calculations for a 10-DoF analysis. The coefficient A and reduction ratio y in the approximate equation asymptotically approach 1 as the DoF increases. This shows the usefulness of utilizing automatic differentiation when extracting motion propagation in multi-degree-of-freedom models.

One example of how motion propagation torque can be used is in impedance control. In conventional methods, the target impedance of a paw is described as follows:

$$\begin{array}{c}\hfill {\mathit{M}}_{\mathrm{e}}·\Delta \ddot{\mathit{X}}+{\mathit{B}}_{\mathrm{e}}·\Delta \dot{\mathit{X}}+{\mathit{K}}_{\mathrm{e}}·\Delta \mathit{X}={\mathit{F}}_{\mathrm{ext}}.\end{array}$$

(18)

However, ${\mathit{F}}_{\mathrm{ext}}\in {\mathcal{R}}^n$ is the external force, whereas ${\mathit{M}}_{\mathrm{e}}$, ${\mathit{B}}_{\mathrm{e}}$, and ${\mathit{K}}_{\mathrm{e}}\in {\mathcal{R}}^{n\times n}$ are the target inertia matrix, target viscosity matrix, and target stiffness matrix of the paw, respectively. $\Delta \mathit{X}=\mathit{X}-{\mathit{X}}_{\mathit{d}}\in {\mathcal{R}}^n$ is the deviation between the target tip position ${\mathit{X}}_{\mathit{d}}$ and the current position $\mathit{X}$. Here, impedance control is applied to the partial torque using the following equation:

$$\begin{array}{c}\hfill {\mathit{M}}_{\mathrm{e}ij}·\Delta \ddot{\mathit{X}}+{\mathit{B}}_{\mathrm{e}ij}·\Delta \dot{\mathit{X}}+{\mathit{K}}_{\mathrm{e}ij}·\Delta \mathit{X}=F_{\mathrm{e}\mathrm{x}\mathrm{t}ij}.\end{array}$$

(19)

In this case, ${\mathit{F}}_{\mathrm{e}\mathrm{x}\mathrm{t}ij}$ is the partial external force, whereas ${\mathit{M}}_{\mathrm{e}ij}$, ${\mathit{B}}_{\mathrm{e}ij}$, and ${\mathit{K}}_{\mathrm{e}ij}\in {\mathcal{R}}^{1\times n}$ are the target partial inertia matrix, target partial viscosity matrix, and target partial stiffness matrix of the paw, respectively. These impedances correspond to the acceleration, velocity, and position of the motion propagation torque, respectively. Therefore, it is considered that the appropriate impedance can be adjusted from the motion propagation torque. By controlling the impedance using the motion propagation torque, it is possible, when moving a multi-link robot arm, to cancel the effect of the motion of one link on the root while giving no particular control over the effect of the motion of the other links on the root, so that only the motion that has a large effect on the root is cancelled. Partial control can be used to cancel out the movements that have a large impact. The effects of fluctuations in the external load can also be cancelled.

Traditionally, a computational graph has been considered an equation structure, but now we can think of it as a connection between nodes and edges. The graph can then be viewed as having a connection matrix connecting the coefficient vectors and the motion vectors. This can be described as Equations (21) and (22) by transforming Equation (12).

$$\begin{array}{cc}{\tau }_{ij}\hfill & =\sum _{b=0}^2{\mathit{c}}_{\mathit{b}}·{\mathit{\theta }}^{\left(b\right)}=\sum _{b=0}^2\left(\sum _{s=1}^i{c}_{bs}{\theta }_s^{\left(b\right)}\right)=\sum _{s=1}^i\left(\sum _{b=0}^2{c}_{bs}{\theta }_s^{\left(b\right)}\right),\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill & =\sum _{s=1}^i\left(\left[\begin{array}{ccc}{c}_{0s}& c_{1s}& c_{2s}\end{array}\right]\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}{\theta }_s\\ \dot{{\theta }_s}\\ \ddot{{\theta }_s}\end{array}\right]\right)=\sum _{s=1}^i\left(\left[\begin{array}{ccc}{c}_{0s}& c_{1s}& c_{2s}\end{array}\right]\sum _{m=0}^2{\mathit{M}}_{\left(m\right)}\left[\begin{array}{c}{\theta }_s\\ \dot{{\theta }_s}\\ \ddot{{\theta }_s}\end{array}\right]\right),\hfill \end{array}$$

(21)

$$\begin{array}{c}\hfill {\mathit{M}}_{\left(m\right)}=\left[\begin{array}{ccc}\delta (m-0)& 0& 0\\ 0& \delta (m-1)& 0\\ 0& 0& \delta (m-2)\end{array}\right],\end{array}$$

(22)

where $\delta \left(x\right)$ is Kronecker’s delta; $\delta \left(0\right)=1$ only when $x=0$, and $\delta \left(x\right)=0$ otherwise when $x\ne 0$. The extraction of motion propagation force is used to extract an arbitrary term from the connection matrix as a mask matrix. In other words, the variable m is selected according to the term of interest and assigned to the mask matrix ${\mathit{M}}_{\left(m\right)}$. This can be written mathematically as in Equation (23). This form is equivalent to Equation (17) and treated as an equation structure.

$$\begin{array}{cc}\hfill {\tau }_{ij}^p\left({\theta }_k^{\left(m\right)}\right)& =\left[\begin{array}{ccc}{c}_{0k}& c_{1k}& c_{2k}\end{array}\right]{\mathit{M}}_{\left(m\right)}\left[\begin{array}{c}{\theta }_k\\ \dot{{\theta }_k}\\ \ddot{{\theta }_k}\end{array}\right],\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill & =c_{mk}{\theta }_k^{\left(m\right)}.\hfill \end{array}$$

(24)

Therefore, the extraction of motion propagation force can be done not only as an equation structure as in Equations (13)–(17) but also as a graph structure as in Figure 3 and Equations (21)–(23).

Since the method is based on the Lagrangian equation, it can handle nonlinearity when applied to complex systems. In addition, since the numerical calculations are performed using automatic differentiation, it may be possible to perform real-time processing using computing power and AI technology. Future work will include addressing these potential issues and evaluating the usefulness of this method by comparing it to experimental data and other methods. In addition, the proposed method, as currently formulated, is applicable to open-chain systems such as serial manipulators, where link-to-link interactions can be analyzed without considering loop-closure constraints. However, we believe that the framework may be extended to closed-chain or parallel mechanisms by incorporating internal reaction forces and kinematic constraints into the partial Lagrangian formulation. Future work will explore this extension, including the theoretical modeling of internal constraint dynamics and its computational feasibility.

5. Conclusions

In this study, we proposed a method for analyzing motion propagation force in multi-DoF systems using a partial Lagrangian approach combined with automatic differentiation. The proposed method allows for the analytical derivation of joint torque components generated by the motion of other joints, which we define as motion propagation forces. By leveraging the computational graph structure inherent to automatic differentiation, our method enables the selective extraction of desired torque components without the need to recalculate the entire system dynamics.

The effectiveness of the proposed approach was demonstrated through simulations using a three-DoF robotic arm model. The results confirmed that the method successfully decomposes joint torques into analytically meaningful components, providing insight into the physical propagation of motion across linked joints. Additionally, the method is compatible with symbolic computation and has the potential to support real-time applications using modern computing hardware and AI-based processing.

These simulations confirmed that the partial torque includes all inertial forces, centrifugal forces, Coriolis forces, and gravity. The proposed method successfully extracts motion propagation torque components that are otherwise difficult to isolate analytically in conventional approaches. The proposed concept of motion propagation force complements well-established notions such as interaction torque by enabling the analytical decomposition of joint torques. This clarification helps bridge the method with standard terminology and reinforces its novelty within the field of robot dynamics.