Kinetostatics of a Snake Robot with Redundant Degrees of Freedom

Abstract

This paper proposes a kinetostatic approach for analyzing the joint torques of a redundant snake robot. The method is suitable for weightless space environments. With the high degree of freedom and flexible cable actuation, the redundant snake robot is well-suited for utilization in space-weightless environments. This method reduces computational cost by using the multiplication of matrices and vectors instead of inverse matrices. Taking advantage of the velocity screw (twist) and force screw (wrench), this strategy provides an idea for redundant serial robots to achieve the calculation of joint torques. This methodology is straightforward for programming and has good computational efficiency. The instantaneous work performed by the actuation is expressed with the force screw. According to the principle of virtual work, the kinetostatic equation of the robot can be obtained and the torque required for each joint can be determined. Meanwhile, to solve the inertia force generated by joint acceleration, D’Alembert’s principle is adopted to transform the dynamic problem into a static problem. Through kinetostatic analysis of a redundant snake robot, this paper shows the approach of establishing the kinetostatic model to calculate the torque in screw form. At the same time, the actuation distribution of the redundant snake robot is also cracked effectively for practical purposes. Due to the difficulty of achieving weightless space environments, this paper validates the method by using ADAMS simulation without gravity in the simulation.

1. Introduction

With the continuous progress of industry, the workspace of robots has become increasingly complicated. The narrow space is a typical difficult operating environment. Since the environment is unstructured, its constraints are uncertain. Robots need to meet both workspace and obstacle avoidance requirements. Traditional robots confront a lot of difficult operation requirements. Due to these reasons, it is necessary to investigate flexible robotic arms with large aspect ratios, high degrees of freedom (DOFs), and operation flexibility [1,2,3]. Flexible robotic arms are suitable for environments that are dangerous and inaccessible to humans and suitable for exploration, maintenance, rescue, and other operations in complex and unstructured narrow spaces such as aerospace equipment and nuclear power plants [4,5,6]. In the field of aviation, flexible robotic arms are used to perform autonomous assembly, monitoring, and maintenance tasks in a small space [7].

The major research difficulty of the snake robot lies in the kinematics, statics, and dynamic control. It is necessary to establish a kinetostatic model to achieve static control when requiring a quick response speed. The flexible robot is a highly coupled nonlinear system with a lot more redundant DOFs and stronger coupling forces and torques compared with the traditional non-flexible robots, making it difficult to establish an accurate statics model. Inverse kinetostatics is used in torque real-time control based on kinetostatic models [8]. It is within the remit of the control problem to find out the torque of all joints of the real-time servo motors when the external load of a manipulator is prescribed [9]. Therefore, kinetostatics analysis of robots plays an important role in engineering design and robot control [10].

In fact, the inverse kinematics of redundant robots is an important issue and a prerequisite for studying the statics of redundant robot motion. The calculations in this paper are based on the premise of providing joint angular displacement functions, which require solving through inverse kinematics.

Currently, the statics equations for serial robots are usually available in two methods: the static force transfer formula and the force Jacobian matrix method [11,12]. The static transfer formula method is a widely adopted approach for statics analysis [13,14]. This method primarily involves the analysis of forces acting on each rigid body, resolving the mechanical equations for all rigid bodies to determine the forces at all joints. However, employing this method necessitates the force analysis of each rigid body and in the statics modeling of serpentine arms, it entails a considerable computational cost. Another method for statics analysis of mechanisms involves the force Jacobian matrix formula, grounded in the Denavit–Hartenberg (D–H) method [15,16]. For example, Palpacelli et al. derived a redundant device driven by cables using the static transfer formula, solving the static and kinematic problems of industrial robot auxiliary equipment [17]. Lu et al. studied the kinetostatics of a 6-DoF parallel manipulator using the principle of virtual work [18]. Zhang et al. designed a novel parallel robot and analyzed its force solution by using a generalized static transfer equation [19]. However, the current efficiency of solving kinematics and statics of robots cannot meet the actual industrial demands. The screw is a six-dimensional vector that has great advantages in solving a couple of motions and forces [20]. The kinematics and statics of lower mobility series robots are systematically studied but lack research by considering acceleration parameters [21].

In the practical operations of redundant snake-like robots, they are typically employed to transport the end effector to the desired position for tasks such as inspection. Greater emphasis is placed on the precision of the end effector’s positioning rather than on the intermediate dynamic characteristics. Consequently, the actual movement is relatively slow and can be considered quasi-static. Therefore, studying the kinetostatics of these robots is of significant importance.

This paper investigates the kinematics and statics of redundant snake robots by using the screw kinetostatics method. The kinetostatics equations are derived by combining the velocity screw and force screw with the principle of virtual work. Therefore, the computational efficiency of the kinetostatic equation is improved by avoiding the traditional method of matrix inversion, which is of great significance for practical on-time engineering applications. A unified description is established in a coordinate system by using the velocity screw and force screw to describe motion and force, which makes the mechanism analysis more straightforward. The computational efficiency of the traditional static method and the new method proposed in this paper is analyzed and compared. Simultaneously, computer programming that constructs the equations of equilibrium is extended to analyze the kinetostatics of manipulators and is used to simulate the motion process. Due to the difficulty of achieving weightlessness in real space environments, this paper validates the method described in this paper by using ADAMS simulation to remove gravity in the simulation environment.

2. Structure of the Redundant Snake Robot

The structure diagram of the snake robot is illustrated in Figure 1. It consists of six equal segments, connected in series with universal joints between each segment. Each segment is restrained from each other by a universal joint. A motor mounted at the end drives three soft cords for robot attitude control. The snake robot contains six universal joints. Each universal joint has 2 degrees of freedom, so the total number of degrees of freedom of the mechanism is 12. The robot’s end effector is connected to the sixth segment by means of a universal joint. The end effector is directly connected to the gluing unit.

As shown in Figure 2, the single segment of the robot consists of a universal joint, an angle sensor, a structural subject, a cushion clock, and linear bearings. The segment is driven by a cable to move the universal joint.

Past researchers calculated the deformation of the cable. Although the robot is driven by soft cables, the tension of each cable is difficult to know in practice. Measuring the joint moments of the universal joint is one way to obtain the cable tension. By converting the joint moments, the cable tension can be obtained.

The angular displacement and angular velocity of each universal joint can be obtained when the motion of the end-effector is known. This is a kinematic problem that is easy to solve. When the external force acting on the end-effector is known, the joint moment of each universal joint needs to be calculated. Therefore, this paper mainly investigates the kinetostatics of redundant snake robot, which aims to solve the motion and forces at each universal joint in equilibrium.

3. Kinetostatics Fundamentals of Screw Vector

3.1. Theoretical Foundations of the Screw Kinetostatics

To simplify the discussion, let us suppose that a multi-body system in series is connected by $n$ rotational joints. According to the principle of velocityposition, the angular velocity and linear velocity of the joints of a series of connected rigid bodies relative to the absolute coordinate system can be expressed by

$$\{\begin{array}{l}{\displaystyle \frac{\mathrm{d}{}_0{}^i\mathit{r}}{\mathrm{d}t}}={}_0{}^i\mathit{\omega }\times \mathit{r}\\ {}_0{}^i\mathit{\omega }={}_0{}^1\mathit{\omega }+{}_1{}^2\mathit{\omega }+{}_2{}^3\mathit{\omega }+\cdots +{}_{i-2}{}^{i-1}\mathit{\omega }+{}_{i-1}{}^i\mathit{\omega }\end{array}$$

(1)

where ${}_0{}^i\mathit{r}$ represents the position vector of the $i\mathrm{th}$ rigid body and ${}_0{}^i\mathit{\omega }$ denotes the angular velocity of the $i\mathrm{th}$ rigid body.

The velocity screw ${\mathit{\xi }}_i$ at joint $i$ of rigid body $i$ is composed of the angular velocity ${}_{i-1}{}^i\mathit{\omega }$ of the rigid body and its dual vector ${}_{i-1}{}^i\mathit{v}$ as

$${}_{i-1}{}^i\mathit{\xi }=\left[\begin{array}{c}{}_{i-1}{}^i\mathit{\omega }\\ {}_0{}^i\mathit{r}\times {}_{i-1}{}^i\mathit{\omega }\end{array}\right]={}_{i-1}{}^i\mathit{\omega }{}_{i-1}{}^i\mathit{\xi }_u$$

(2)

where ${}_{i-1}{}^i\mathit{\omega }$ is the angular velocity at which the rigid body $i$ rotates relative to the rigid body $i-1$ and ${}_{i-1}{}^i\mathit{\xi }_u$ is the unit velocity screw.

By combining Equations (1) and (2), the velocity screw $\mathit{\xi }$ of the end effector for a series of rigid body systems with $n$ joints can be gained, as follows:

$$\mathit{\xi }={\displaystyle \sum _{i=1}^n{}_{i-1}{}^i\mathit{\xi }}=\mathbf{V}\mathit{\omega }$$

(3)

where $\mathbf{V}=\left[\begin{array}{cccc}{\mathit{\xi }}_1& {\mathit{\xi }}_2& \cdots & {\mathit{\xi }}_n\end{array}\right]$ is the unit velocity screw matrix with $n$ connected rigid body and $\mathit{\omega }={\left[\begin{array}{cccc}{}_0{}^1\omega & {}_1{}^2\omega & \cdots & {}_{n-1}{}^n\omega \end{array}\right]}^{\mathrm{T}}$ denotes the angular velocity of all rigid bodies; the $i\mathrm{th}$ element of which is the relative angular speed of the $i\mathrm{th}$ rigid body relative to the $\left(i-1\right)\mathrm{th}$ rigid body.

When the kinematic parameters of the end effector in the absolute coordinate system are known, the angular velocity of each joint can be solved as

$$\mathit{\omega }={\left[{\mathbf{V}}^T\mathbf{V}\right]}^{-1}{\mathbf{V}}^T\mathit{\xi }$$

(4)

where ${\left[{\mathbf{V}}^T\mathbf{V}\right]}^{-1}{\mathbf{V}}^T$ is the left inverse of the unit velocity screw matrix $\mathbf{V}$.

After the angular velocities of all joints are solved, the angular displacement of each joint can be obtained by using numerical iteration methods. Assuming that the angular displacement of each joint at the initial moment is $\mathit{\theta }\left(0\right)={\left[\begin{array}{cccc}\begin{array}{cc}{}_0{}^1\theta \left(0\right)& {}_1{}^2\theta \left(0\right)\end{array}& \cdots & {}_{n-2}{}^{n-1}\theta \left(0\right)& {}_{n-1}{}^n\theta \left(0\right)\end{array}\right]}^T$, the iteration step is $\Delta t$. When the angular velocity $\mathit{\omega }\left(k\right)$ and displacement $\theta \left(k\right)$ at moment $k$ are known, the angular displacement $\theta \left(k+1\right)$ of each joint at moment $k+1$ is

$$\theta \left(k+1\right)=\theta \left(k\right)+\Delta t\mathit{\omega }\left(k\right)$$

(5)

By employing Equations (4) and (5), the displacements and velocities of all joints can be solved in screw coordinates.

Suppose the differential work performed by the rotational joint because of its infinitesimal displacement $d\mathit{\theta }$ is $d{W}_U$. It leads to

$$d{W}_U={\mathit{M}}^{T}d\mathit{\theta }$$

(6)

Because $d\mathit{\theta }={\left[\begin{array}{cccc}d{\theta }_1& d{\theta }_2& \cdots & d{\theta }_{12}\end{array}\right]}^T=\mathit{\omega }dt$. Equation (6) can be expressed as

$$d{W}_U={\mathit{M}}^T\mathit{\omega }dt$$

(7)

where $\mathit{M}$ is the driving torques of every rotational joint and $dt$ is the differentiation of time.

Based on force screw theory, the virtual work conducted by the external load of the end effector during operation is

$$d{W}_E={\mathsf{\$}}^T\aleph \mathit{\xi }dt$$

(8)

where $\mathsf{\$}=\left[\begin{array}{c}\mathit{F}\\ \mathit{M}\end{array}\right]$ is the force screw, $\aleph =\left[\begin{array}{cc}0& \mathbf{I}\\ \mathbf{I}& 0\end{array}\right]$ and $\mathbf{I}=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$.

Equation (8) can be expressed as follows

$$d{W}_E={\mathsf{\$}}^T\aleph \mathbf{V}\mathit{\omega }dt={\mathsf{\$}}^T\aleph \mathbf{V}d\theta$$

(9)

According to the principle of virtual power, it yields

$$d{W}_U+d{W}_E=0$$

(10)

Therefore, the driving torques for all joint motors are demonstrated in the vector form as

$$\mathit{M}=-{\mathbf{V}}^T\aleph \mathsf{\$}$$

(11)

To make it easier to understand the structure of the method and the computational process, Figure 3 shows the computational flow of the screw kinetostatics method.

3.2. Kinetostatic Analysis of a Planar Robotic Grasper

To illustrate this method, we first give a case study. Figure 4 illustrates a planar robotic grasper. The external load is P acting on the end effector. We need to analyze the individual joint torques ${\mathit{M}}_1$, ${\mathit{M}}_2$, and ${\mathit{M}}_3$ required to balance the external load.

First, an absolute coordinate frame is established, as shown in Figure 4. The origin of the absolute coordinates is at the point $A_1$ and the z-axis is perpendicular to the plane. In the absolute coordinate system, the twist matrix of the end effector $\mathit{\xi }=0$. Then, the twist matrix is obtained as follows

$$\mathbf{V}=\left[\begin{array}{ccc}{\mathsf{\$}}_1& {\mathsf{\$}}_2& {\mathsf{\$}}_3\end{array}\right]$$

(12)

In the absolute coordinate frame, the velocity screw of the end effector can be obtained. V is the screw matrix composed of unit screws. Unit screws of rotational joints are demonstrated as follows

$${\mathsf{\$}}_i=\left[\begin{array}{c}{\mathit{e}}_i\\ {\mathit{r}}_{O_i}\times {\mathit{e}}_i\end{array}\right]$$

(13)

According to the geometric relationship, it yields

$$x_1=0,y_1=0$$

(14)

$$x_2=l_1\cos {\theta }_1,y_2=l_1\sin {\theta }_1$$

(15)

$$x_3=l_1\cos {\theta }_1+l_2\cos \left({\theta }_1+{\theta }_2\right),y_3=l_1\sin {\theta }_1+l_2\sin \left({\theta }_1+{\theta }_2\right)$$

(16)

Equation (12) can be described by the geometrical parameters as follows

$$\mathbf{V}={\left[\begin{array}{llllll}0\hfill & 0\hfill & 1\hfill & y_1\hfill & -x_1\hfill & 0\hfill \\ 0\hfill & 0\hfill & 1\hfill & y_2\hfill & -x_2\hfill & 0\hfill \\ 0\hfill & 0\hfill & 1\hfill & y_3\hfill & -x_3\hfill & 0\hfill \end{array}\right]}^T$$

(17)

With reference to

Table 1. Structural parameters in the initial conditions of the planar grasper.

$l_1\left(\mathrm{mm}\right)$	$l_2\left(\mathrm{mm}\right)$	$l_3\left(\mathrm{mm}\right)$	$P\left(\mathrm{N}\right)$	${\theta }_1\left(\mathrm{rad}\right)$	${\theta }_2\left(\mathrm{rad}\right)$	${\theta }_3\left(\mathrm{rad}\right)$
200	200	260	20	$\pi /3$	$\left(11\pi \right)/6$	$\left(4\pi \right)/3$

, the initial structural parameters and all angular displacements at the initial time are set.

Based on the analysis above, it is known that the external load and the twist matrix are at the initial time. As the system is in equilibrium conditions, the torques are directly obtained as follows:

$$\mathit{M}=-{\mathbf{V}}^T\aleph \mathsf{\$}={\left[\begin{array}{c}P{l}_1\cos {\theta }_1+P{l}_2\cos \left({\theta }_1+{\theta }_2\right)\\ P{l}_2\cos \left({\theta }_1+{\theta }_2\right)\\ 0\end{array}\right]}^T$$

(18)

At this time, the twist matrix $\mathbf{V}\left(t_0\right)$ is substituted into Equation (17) and the torque of each joint at time t = 0 can be obtained. Then, through numerical iteration, the twist matrix $\mathbf{V}\left(t_0+\Delta t\right)$ at $t=t_0+\Delta t$ is updated. Substituting $\mathbf{V}\left(t_0+\Delta t\right)$ into Equation (16) and after iteration, the curve of the input torques of the motors in a period can be calculated. By programming the algorithm, the input torque curve (see Figure 5) of each joint was plotted.

From Figure 5, we find that both motors at joints $A_1$ and $A_2$ change cyclically. The torque of motor 3 at the joint $A_3$ is constantly zero. This is true in Figure 5 since the weight of the grasping object passes through the joint $A_3$.

4. Kinetostatics of the Redundant Snake Robot

4.1. Kinematic Analysis of a Redundant Snake Robot

For the sake of simplicity, we decompose each universal joint into two orthogonal rotational joints and establish an absolute coordinate system on the base. We establish a joint coordinate system at the center of each rotational joint so that the axis of the joint coordinate system points in the direction of the angular displacement of the rotational joint. The redundant snake robot involves 12 screws, namely ${\mathsf{\$}}_i\left(i=1,2,\cdots ,12\right)$. The coordinate systems and screws are shown in Figure 6.

The direction vector and position vector of all joints were exhibited geometrically. Suppose ${\mathit{e}}_i\left(i=1,2,\cdots ,12\right)$ represents the unit vector of each rotational joint and ${\mathit{r}}_i\left(i=1,2,\cdots ,12\right)$ denotes the position vector of each rotational joint.

For a universal joint, there are two-unit screws. The twist matrix of this redundant snake robot is

$$\mathit{S}=\left[\begin{array}{cccc}{\mathsf{\$}}_1& {\mathsf{\$}}_2& \cdots & {\mathsf{\$}}_{12}\end{array}\right]$$

(19)

It leads to

$$\mathit{S}=\left[\begin{array}{cccc}\begin{array}{c}{\mathit{e}}_1\\ {\mathit{r}}_{O_1}\times {\mathit{e}}_1\end{array}& \begin{array}{c}{\mathit{e}}_2\\ {\mathit{r}}_{O_2}\times {\mathit{e}}_2\end{array}& \cdots & \begin{array}{c}{\mathit{e}}_{12}\\ {\mathit{r}}_{O_{12}}\times {\mathit{e}}_{12}\end{array}\end{array}\right]$$

(20)

Next, the twist matrix is derived by adopting the rotation matrix method. The transformation matrix from coordinate system {1} to the absolute coordinate system is

$${}_0{}^1\mathit{R}={\mathit{R}}_x\left({\theta }_1\right){\mathit{R}}_y\left({\displaystyle \frac{\pi }{2}}\right)$$

(21)

The transformation from coordinate system {i} to the absolute coordinate system is easy to calculate. Therefore, the position vectors of each universal joint center are

$$\{\begin{array}{l}{\mathit{r}}_1={\left[\begin{array}{ccc}0& -l_1& 0\end{array}\right]}^T\\ {\mathit{r}}_2={\mathit{r}}_1{+}_0^2\mathit{R}{\left[\begin{array}{ccc}0& -l_2& 0\end{array}\right]}^T\\ {\mathit{r}}_3={\mathit{r}}_2{+}_0^4\mathit{R}{\left[\begin{array}{ccc}0& -l_3& 0\end{array}\right]}^T\\ {\mathit{r}}_4={\mathit{r}}_3{+}_0^6\mathit{R}{\left[\begin{array}{ccc}0& -l_4& 0\end{array}\right]}^T\\ {\mathit{r}}_5={\mathit{r}}_4{+}_0^8\mathit{R}{\left[\begin{array}{ccc}0& -l_5& 0\end{array}\right]}^T\\ {\mathit{r}}_6={\mathit{r}}_5{+}_0^{10}\mathit{R}{\left[\begin{array}{ccc}0& -l_6& 0\end{array}\right]}^T\end{array}$$

(22)

Equation (22) provides the position vector of the robot in the absolute coordinate system. Substituting Equation (22) into Equation (20) yields the screw matrix of the redundant snake robot.

To calculate the angular displacement of each joint for motion control, the initial parameter values of the robot are shown in

Table 2. The structure parameter values of the redundant robotic arm.

$l_1\left(\mathrm{mm}\right)$	$l_2\left(\mathrm{mm}\right)$	$l_3\left(\mathrm{mm}\right)$	$l_4\left(\mathrm{mm}\right)$	$l_5\left(\mathrm{mm}\right)$	$l_6\left(\mathrm{mm}\right)$
294	218	218	218	218	218

.

Each joint motor and its angle of rotation are shown in Figure 7.

Sine waves are provided in

Table 3. Drive parameters of the redundant robotic arm.

${\theta }_1\left(t\right)\left(\mathrm{rad}\right)$	${\theta }_2\left(t\right)\left(\mathrm{rad}\right)$	${\theta }_3\left(t\right)\left(\mathrm{rad}\right)$	${\theta }_4\left(t\right)\left(\mathrm{rad}\right)$	${\theta }_5\left(t\right)\left(\mathrm{rad}\right)$	${\theta }_6\left(t\right)\left(\mathrm{rad}\right)$
$0.2\sin \pi t$	$0.2\sin \pi t$	$-0.2\sin \pi t$	$-0.2\sin \pi t$	$0.2\sin \pi t$	$0.2\sin \pi t$
${\theta }_7\left(t\right)\left(\mathrm{rad}\right)$	${\theta }_8\left(t\right)\left(\mathrm{rad}\right)$	${\theta }_9\left(t\right)\left(\mathrm{rad}\right)$	${\theta }_{10}\left(t\right)\left(\mathrm{rad}\right)$	${\theta }_{11}\left(t\right)\left(\mathrm{rad}\right)$	${\theta }_{12}\left(t\right)\left(\mathrm{rad}\right)$
$-0.2\sin \pi t$	$-0.2\sin \pi t$	$0.2\sin \pi t$	$0.2\sin \pi t$	$-0.2\sin \pi t$	$-0.2\sin \pi t$

as a function of motion for each joint. The sine wave is one of the most fundamental waveform contours. In practical applications, other more complex waveforms can be obtained by using the Fourier transform. Therefore, other waveform movements can be seen as the superposition of multiple sine waveform functions.

The relative angular displacements of each of the robot’s joints are plotted in Figure 8. Figure 8 shows that, when a regular motion is given, there are no abrupt changes in the displacement, velocity, and acceleration of the joints over a complete cycle of motion.

4.2. Screw Statics of a Redundant Snake Robot

Based on the kinematic analysis, the screw matrix $\mathbf{S}$ of the robot can be obtained. The moment of each universal joint can be calculated with Equation (1). However, during the movement of the robot, it needs to be equipped with an actuator, which is subjected to a gravity value of $\mathit{G}$. Furthermore, the end effector of the redundant snake robot is utilized to manipulate within a narrow workspace. The wrench matrix of the actuator is

$${\mathsf{\$}}_{\mathit{G}}=\left[\begin{array}{c}\mathit{G}\\ {\mathit{r}}_{O_6}\times \mathit{G}\end{array}\right]$$

(23)

It is worth noting that the torque of each joint we provide is a sinusoid curve with respect to time. This means that each joint will have an angular acceleration during the motion process, so an inertial moment will be generated.

The d’Alembert principle is a theory that transforms dynamic problems into static problems. It treats the moment of inertia acting on each joint as an external load, enabling the use of statics principles and methods to analyze and solve dynamic problems. Specifically, the d’Alembert principle converts the inertia moment and external force in dynamic problems into an equivalent static load, which can be represented by the forces and moments applied to each joint. In this way, we may use the principles and methods of statics to analyze and calculate the forces and moments on each joint, thereby obtaining solutions to dynamics problems.

In a redundant snake robot, the d’Alembert principle is utilized to transform the inertia moment acting on each joint into a static wrench. So, the inertia moment of each joint is

$${\mathsf{\$}}_{{\mathit{M}}_{Ii}}=-\left[\begin{array}{c}\mathbf{0}\\ \mathit{J}{\mathit{\beta }}_{\mathit{i}}\end{array}\right], \left(i=1,2,\cdots ,12\right)$$

(24)

where $\mathit{J}$ is the moment of inertia of each segment and ${\mathit{\beta }}_{\mathit{i}}$ is the angular acceleration of the joint.

All the external loads can be expressed in screw form for the $i\mathrm{th}$($i=1,2,\cdots ,12$) joint, as follows:

$${\mathsf{\$}}_i={\mathsf{\$}}_{\mathit{G}}+{\displaystyle \sum _{j=i}^{12}{\mathsf{\$}}_{{\mathit{M}}_{Ii}}}, \left(i=1,2,\cdots ,12\right)$$

(25)

Hence, the driving torques of the redundant snake robot can be obtained directly. Assuming that the robot does quasi-static motion according to the joint angle function shown in Table 3, the joint torques can be obtained and plotted in Figure 9, where curves 1–12 correspond to the torques of these 12 motors, respectively. The calculation time 6 s can show that the driving function is also a trigonometric curve with periodic reciprocation. The amplitude of the driving moment gradually decreases as it approaches the end position. The inertial parameters of the robot are provided in

Table 4. Inertial parameters of the robot.

$I_1\left(\mathrm{kg}\cdot {\mathrm{mm}}^2\right)$	$I_2\left(\mathrm{kg}\cdot {\mathrm{mm}}^2\right)$	$I_3\left(\mathrm{kg}\cdot {\mathrm{mm}}^2\right)$	$I_4\left(\mathrm{kg}\cdot {\mathrm{mm}}^2\right)$	$I_5\left(\mathrm{kg}\cdot {\mathrm{mm}}^2\right)$	$I_6\left(\mathrm{kg}\cdot {\mathrm{mm}}^2\right)$
$\left[\begin{array}{ccc}6402& & \\ & 6384& \\ & & 674\end{array}\right]$	$\left[\begin{array}{ccc}3609& & \\ & 3567& \\ & & 554\end{array}\right]$	$\left[\begin{array}{ccc}3609& & \\ & 3567& \\ & & 554\end{array}\right]$	$\left[\begin{array}{ccc}3609& & \\ & 3567& \\ & & 554\end{array}\right]$	$\left[\begin{array}{ccc}3570& & \\ & 3529& \\ & & 549\end{array}\right]$	$\left[\begin{array}{ccc}3218& & \\ & 3195& \\ & & 539\end{array}\right]$

.

The maximum driving torque for different motions is shown in

Table 5. Maximum driving torque for different motions (N.mm).

Motion 1	Motion 2	Motion 3	Motion 4	Motion 5	Motion 6
2818.09	2876.88	2119.55	2075.67	1424.83	1454.34
Motion 7	Motion 8	Motion 9	Motion 10	Motion 11	Motion 12
847.05	829.45	390.11	397.99	90.21	88.25

.

To verify this result, a simulation experiment is designed in the ADAMS environment. The driving function shown in Table 3 is added to the motor. A 6-s simulation is conducted. The simulation results are shown in Figure 10.

From the simulation results, it can be seen that the simulation curve has the same period and approximate amplitude as the theoretical calculation curve. In order to further compare and analyze this, the simulation curve and the theoretical calculation curve are plotted in one graph, as shown in Figure 11.

From Figure 11, it can be seen that the theoretical calculation curve is basically consistent with the simulation, which validates the theoretical method. However, there is a slight difference in amplitude between these two results, which is mainly due to two reasons. On the one hand, the parameters of the robot linkage in theoretical calculations are design parameters, while in simulation, the parameters of the robot linkage are measured or calculated through models. Another difference is that friction is taken into account in the simulation while it is ignored in our theoretical calculation. On the other hand, the results of the theoretical calculations are obtained by MATLAB R2021b programming. Floating-point numbers in the computer also lead to the difference between the MATLAB R2021b programming results and the ADAMS 2020 dynamics simulation software results.

In addition, there are differences between the theoretical curves and the simulated ones near the peak of the curves, appearing at times of 0.5 s, 1.5 s, 2.5 s, 3.5 s, 4.5 s, and 5.5 s, respectively. By observing the points corresponding to the response time on the joint angle curves, it can be found that the angular velocities of the joints at the above times are all zeros. This difference is mainly due to the static friction force. Due to the difference in the static friction coefficient and sliding friction coefficient of the object, the friction force will change at the time of 0 velocity, resulting in the difference at the corresponding time curve.

Based on the analysis of joint torque with a redundant snake robot, the numerical algorithms are efficient and comprehensible in calculating the torques of all joints. The joint torque of the spatial serial multi-chain robot can be dynamically controlled on time by applying this kinetostatics algorithm.

5. Discussion

When the equilibrium equations of statics are used to analyze robot kinematics and statics, we need to list the equilibrium equations for each rigid body and ultimately the association of these equations. As shown in references [13,14,17,22,23], the general statics equations can be written as

$$\mathbf{A}\mathit{f}=\mathit{w}$$

(26)

where $\mathbf{A}$ is the structure matrix, $\mathit{f}$ is the joint torque, and $\mathit{w}$ is the external load. From formula (26), it can be seen that solving the joint torque requires calculating the inverse of the structural matrix $\mathbf{A}$.

However, the method proposed in this paper, which expresses the forces and moments through the force screw, is ultimately to solve Equation (11), that is, to solve for a product of matrices and vectors.

According to linear algebra, the multiplication of matrices and vectors can be transformed into a number of dot products of vectors. It has the time complexity of $O\left(n\right)$. The time complexity of solving the inverse of a matrix is $O\left(n^3\right)$. The static equilibrium method requires solving the inverse of the matrix while the screw kinetostatics method only requires the calculation of the product of a matrix and a vector. Therefore, the computational cost of the static equilibrium method is much more than that of the screw kinetostatics method.

In addition, this method is very concise in calculating the joint forces of a redundant serial robot. A screw simultaneously represents the position and direction of a vector, making it more convenient for the motion and force direction of a rigid body. Compared with the traditional methods in the Cartesian coordinate frame, the screw equation is free from complex algebraic manipulation and convenient to obtain a twist matrix from by Plücker coordinates. With the definition of twist and wrench, we associate the kinematics with statics by unit screws.

Lastly, to solve the problem of inertial forces generated by joint acceleration, the d’Alembert principle is adopted to transform dynamics into statics, thereby simplifying the solution process. This approach allows us to more quickly analyze and calculate the forces and moments on each joint, leading to the better design and optimization of mechanical systems.

6. Conclusions

In this paper, taking the space space-weightless scenario as a typical application, a systematic kinetostatics of a redundant serpentine robotic arm is carried out. The approach involves kinematic analysis by using matrix transformation and screw theory, which are very suitable for analysis and has good intuition. Firstly, the mechanical model of the redundant serpentine robotic arm was established. By using the matrix transformation method, the direction vector of each universal joint can be obtained. Furthermore, the twist matrix can be established from Rodrigues parameters. According to the principle of virtual work, the kintostatics equation is derived under the condition of zero virtual power. At the same time, to solve the problem of inertial forces generated by joint acceleration, we adopt the d’Alembert principle to transform dynamics problems into statics, thereby simplifying the solution process. This method enables us to analyze and calculate the forces and moments on each joint more accurately, thereby better designing and optimizing mechanical systems. Finally, the driving torque of each universal joint can be calculated. The results indicate that the proposed method provides a convenient way for the computer programming of a redundant robot. The method can also be utilized in motion control and trajectory planning.