Mechanical Design and Kinematic Analysis of an Autonomous Wrist with DC Motor Actuators for Space Assembly

Abstract

This paper deals with the mechanical design and kinematic analysis of an autonomous wrist for space assembly (AWSA) whose actuators are activated by DC motors and ball screw drives. This robotic wrist was developed and built as a prototype to investigate in-space robotic operations, including maintaining and repairing spacecraft of the US National Aeronautics and Space Administration (NASA), such as the International Space Station (ISS) or satellites. Despite its disadvantages, such as a small workspace and low maneuverability, a parallel structure instead of a serial structure was selected for the design of the AWSA due to several advantages it has over a serial robot manipulator (SRM), including higher payload, greater stiffness, and better stability. The present paper also introduces a hybrid concept for robotic space operations, which combines an SRM performing gross motion and a parallel robot manipulator (PRM) performing fine motion. It then discusses the design and construction of the DC motor actuators and ball screw drives and presents the kinematic equations developed for the AWSA. This paper provides a closed-form solution to the inverse kinematics of the AWSA and a numerical solution using the Newton–Raphson method for its forward kinematics.

1. Introduction

Space robotics has been a very active research area in the past decade due to its potential and important applications in on-orbit servicing (OOS), which implies maintenance of space systems in orbit [1,2]. Maintenance tasks include the assembly, refueling, repair, and upgrading of spacecraft after being deployed. However, NASA already recognized the importance of robotics for space operations as early as the 1980s [3,4] and considered using robot manipulators and astronauts to construct the ISS as the first step to establish a permanent human presence in space [5]. In light of the potential danger astronauts could encounter while constructing the ISS in space, NASA’s intention to build the ISS sparked an intensive and controversial debate in the space community about whether or not the ISS construction should be carried out fully robotically or by an optimal balance of astronauts and autonomous robots. In the end, the ISS was built through a collaboration between astronauts and robot manipulators, such as the Canadarm, well-known as the Shuttle Remote Manipulator System (SRMS), and the Canadarm2, well-known as the Space Station Remote Manipulator System (SSRMS) [6]. The SRMS and the SSRMS were used to deploy, retrieve, and maneuver large modules and other heavy pieces of equipment in the shuttle’s cargo bay or from the ISS for installation at a location on the ISS. They were also employed to carry and position astronauts during spacewalks and to provide them with a stable working platform. Meanwhile, astronauts, after being placed at a desired location by the above manipulators, carried out tasks including joining modules, installing equipment, and making crucial connections for power, data, and fluid lines, sometimes manually or with robotic assistance. In addition, astronauts inside the space shuttle or the ISS teleoperated the above manipulators to maneuver modules, deploy elements, and berth visiting spacecraft. NASA’s initial vision of using fully autonomous robots to construct the ISS was not realized due to the lack of advanced technology in the 1990s for the development of sophisticated robots to perform complicated tasks required for the construction.

During the last decade, advancements in AI and related areas, including smart sensors, smart materials, soft actuators, machine learning, hand exoskeleton systems, and intelligent control, have substantially motivated researchers to develop robot manipulators capable of autonomously performing complicated tasks for space operations without human intervention [2]. In order to be autonomous and effective, manipulators should be equipped with position and vision sensors to be aware of their locations and force sensors to effectively interact with their working environment [7,8]. In addition, robotic controllers should be developed to take sensor information and work in a closed-loop mode to automatically control the position and force of the manipulator to achieve the task objective. Furthermore, advanced hand exoskeleton systems [9,10] can be applied to the master hand in a teleoperation to provide force-reflecting capability from compliant motion of the slave hand or to enhance the manipulation strength of astronauts performing extravehicular activities.

Traditional robot manipulators whose joints and links are actuated in series are traditional serial manipulators or SRMs [11]. SRMs generally have a long reach and a large workspace and are capable of entering small spaces because of their compactness. However, their cantilever-like structure causes them to have low stiffness and, consequently, undesirable dynamic characteristics, especially at high speeds and large payloads. In addition, they have low strength-to-weight ratios due to the fact that the payload is not uniformly distributed to the actuators. Finally, the fact that a relatively large position error occurs at the last link because the joint errors are accumulated throughout the manipulator structure suggests that SRMs are not suitable for high-precision tasks, such as part assembly. Meanwhile, robot manipulators whose actuators are arranged in parallel are classified as PRMs and generally have a relatively small workspace and low maneuverability. In return for these disadvantages, they possess a high positioning capability produced by their high structural rigidity and noncumulative actuator errors. PRMs also have higher strength-to-weight ratios compared with SRMs because the payload is proportionally distributed to the links. Finally, the inverse kinematics of PRMs have simple closed-form solutions.

Implementation of the parallel structure concept first appeared in the Stewart Platform [12], which was originally designed as an aircraft simulator. A typical Stewart Platform consists of two platforms driven by a number of parallel actuators. The invention of the Stewart Platform has attracted tremendous attention from robot designers, and its mechanism has been used in many robotic applications [13,14,15,16,17,18,19,20]. Dieudonne [21] and his coworkers derived an actuator extension transformation and presented experimental results of a Stewart Platform-based simulator built at the NASA Langley Research Center to train aircraft operators. A finite element program was used by Hoffman and McKinnon [22] to simulate the motion of the Stewart Platform, whose mechanism was later applied by McCallion and Truong [23] to design an automatic assembly table. Hunt [24] conducted a systematic study of in-parallel-actuated robot arms and presented the structural kinematic problem of this type of manipulator. Sugimoto and Duffy [25] developed a general technique to describe the instantaneous link motion of a single closed-loop mechanism by employing linear algebra elements to screw systems. Kinematic problems and practical construction of the Stewart Platform were later considered by Fichter [26]. Sugimoto [27] studied the kinematics and dynamics of parallel manipulators, and Lee [28] derived dynamical equations for a three-degree-of-freedom (DOF) CKC manipulator. Nguyen and Pooran [29] developed a learning control scheme for a two-DOF CKC manipulator performing repetitive tasks.

An autonomous wrist for space assembly (AWSA) was developed and built as a prototype to investigate the feasibility of robotic assembly in space, which required the wrist to have the ability to autonomously perform fine and precise compliant motion [30]. With the rapid advancement of AI, AI-based control strategies could soon be developed to provide AWSAs with autonomy. A parallel structure instead of a serial structure was selected for the design of the AWSA due to its accurate positioning capability, despite its small workspace and low maneuverability, which does not pose an issue for part assembly. The AWSA, with its six linear actuators driven by DC motors and ball screw drivers, enables an end-effector attached to it to carry out the delicate motions required for assembly tasks. The control aspect of the AWSA was addressed in [31], where a decentralized adaptive control scheme was developed to successfully control the motion of the AWSA. Under the application of the above adaptive control scheme, whose control gains were adjusted in real-time, the AWSA was able to track several motions with high accuracy despite sudden changes in payloads. The scope of the present paper is the implementation of the Stewart Platform [12] using linear actuators, which comprise DC motors and ball screw drivers and are equipped with linear voltage differential transducers (LVDTs), providing information on the actuator lengths for feedback control. It will focus on the mechanical design of the linear actuators and the LVDTs to meet specified payload requirements and on developing solutions for the forward and inverse kinematics of the AWSA.

This paper is organized as follows. Section 2 gives an overview of the robotic construction of the ISS and Section 3 describes the main components of the AWSA and their functions. The design and component selection of the linear DC actuator for the AWSA will be presented and discussed in Section 4. Section 5 derives a closed-form solution for the AWSA’s inverse kinematics and provides an algorithm to obtain an iterative numerical solution for the AWSA’s forward kinematics. Section 6 discusses a joint-space control scheme proposed for motion control of the AWSA. Concluding this paper, Section 7 provides a summary of this paper’s contents and offers future research directions.

2. Robotic Construction of the International Space Station

In 1984, President Reagan approved the construction of the ISS with international partners, including Canada, Japan, Russia, and many nations of the European Space Agency [6]. The design of the ISS took place between 1984 and 1993, and its components were built all over the US, Canada, Japan, and Europe, starting in the late 1980s. The ISS, a large spacecraft orbiting the Earth at an altitude of about 248 miles, with a speed of about 17,500 mph, was originally planned to be a laboratory and observatory and to serve as a low-orbit staging base for future missions to the Moon and Mars. NASA planned to use the SRMS of the Space Shuttle for the construction of the ISS in space. While the SRMS, a serial robot manipulator, was developed to assist astronauts in deploying, retrieving, and handling big payloads in free space (Figure 1), it was not suitable for carrying out motions needed in typical tasks of the ISS construction, where the manipulator would be constantly in contact with the environment. Typical tasks included connecting electrical and fluid lines, solar array installation, and replacing small equipment and defective components, and they all required high-precision motions. Thus, there was a need to develop a mechanism to be mounted between the end of the SRMS and an end-effector to enable it to perform fine and precise motions while complying with the contact force generated by the environment it was in contact with. This type of compliant motion is often required in an assembly task of the construction of the ISS. Such an idea was conceived for the old Canadarm (SRMS), now retired, but can now be applied to Canadarm2 (SSRMS), which is a larger, more advanced robotic arm developed for the ISS. As a matter of fact, the above type of mechanism was already incorporated into the special-purpose robot Dextre (a serial manipulator) as a tool to perform tasks that require high precision, such as assembly of parts or repair of circuit boards (Figure 2). The specifications of Canadarm, Canadarm2, and Dextre are tabulated in

Table 1. Specifications of Canadarm, Canadarm2, and Dextre.

Manipulator	Canadarm (SRMS)	Canadarm2 (SSRMS)	Dextre (Two Arms)
Operator	NASA	Canadian Space Agency	Canadian Space Agency
Builder	MDA (Spar Aerospace)	MDA (Spar Aerospace)	MDA (Spar Aerospace)
Length (m)	15.2	17	3.5
Mass (kg)	410	1497	1662
Number of DOFs	6	7	15
Payload (kg)	29,000	266,000	600

.

3. The Autonomous Wrist for Space Assembly (AWSA)

Based on the discussion in Section 2 about the need for the development of a robot manipulator system to perform tasks required for the construction, maintenance, and repair of the ISS, Figure 3 illustrates a proposed hybrid concept of combining an SRM with a PRM to accomplish the above tasks. Depending on the tasks to be accomplished, the SRM and PRM are sized accordingly. In general, the size of the PRM is small compared with that of the SRM. One end of the SRM is connected to either a space shuttle or a location of the ISS, and its other end is connected to the PRM. An end-effector (tool) is mounted onto the PRM to carry out robotic tasks. With a serial structure where the manipulator links are coupled serially, the SRM has a long reach and a large workspace, but also suffers from severe actuator backlash, making the manipulator incapable of carrying out precise motions. Meanwhile, despite its small workspace, the PRM, in a parallel arrangement of its actuators, is capable of performing precise motions due to its low backlash and non-accumulating link errors. The proposed concept illustrated in Figure 3 considers the strengths and weaknesses of the SRM and the PRM and assigns them to perform the type of motions that they are most suitable for. It is noted that, in general, the PRM parameters are highly impacted by the SRM kinematics. However, in the above hybrid configuration of the SRM and PRM, after the SRM hands the task over to the PRM, the SRM configuration (position and orientation) is locked and updated by sensors either internally using manipulator joint sensors or externally using vision and proximity sensors. The updated configuration of the SRM is then provided to the control system of the PRM so that the configuration of the PRM is updated with respect to the global frame. Since, after the PRM takes over the task, the configuration of the SRM cannot be changed anymore, the SRM kinematics can no longer influence the PRM configuration. Furthermore, space conditions can affect a robot’s structure by causing thermal expansion, material degradation, and lubrication failure, which require robot control systems to adapt to new physical and operational principles. Robotic engineers must select specialized materials and create robust thermal management systems to ensure a robot can survive and function outside of Earth’s atmosphere.

There are two types of motions a robot manipulator performs, namely, gross motion and fine motion [14]. While gross motion refers to the overall movements a robot manipulator makes to bring itself or an object to within its workspace, fine motion deals with precise and controlled movements required for tasks like part assembly, where accuracy of position and force is essential. To carry out a particular task, such as assembly of a part on the ISS, it is envisioned that the SRM performs the gross motion autonomously or by teleoperation to move the end-effector into the workspace of the PRM. During the gross motion phase, collision avoidance strategies may be employed to avoid colliding with other structures or spacecraft around the workspace. After the SRM finishes its gross motion, it renders motion control to the PRM that then carries out the fine motion to complete the assembly task. Since the end-effector will be constantly in contact with the structure of the ISS, compliant motion control may be employed to avoid damage to the end-effector and the ISS structure by excessive contact forces and torques.

Figure 4 shows a prototype of the PRM, named AWSA, that was designed and built to study the feasibility of using parallel robot manipulators for in-space robotic operations, including maintaining and repairing NASA spacecrafts, including the ISS or satellites [30]. The wrist is composed of two platforms, the tool platform (TP) and the robot platform (RP). Referring to Figure 3, an end-effector is connected to the TP of the AWSA via a force sensor, and one end of the SRM is connected to the RP of the AWSA. The force sensor can acquire six-dimensional forces/torques encountered by TP. As shown in Figure 4, the TP is coupled to the RP through six links. Each link constitutes a linear actuator comprising a DC motor and a linear ball screw drive. The ends of the links are connected to the TP and RP through ball joints. The AWSA possesses six degrees of freedom because it has six linear actuators coupled together in a parallel structure through the TP and RP. The linear actuators are equipped with LVDTs that measure the actuator lengths and provide them for joint position feedback control. The structure of the AWSA is similar to that of a dexterous robotic hand based on the Stewart–Gough parallel manipulator [32]. However, this mechanism consists of three parallel linkage fingers with six prismatic actuators that control manipulation, while the AWSA has six prismatic actuators.

As its name implies, the AWSA is intended for autonomous operations using position and force feedback from LVDTs and force sensors. A control system can be developed to provide the AWSA with positioning and force control capability to carry out space assembly tasks autonomously. However, for complex tasks in inaccessible and high-risk space environments, the AWSA can be used in a teleoperation in which a wearable hand exoskeleton [9,10] is employed in the master hand so that the master hand exoskeleton captures the motion of human fingers and reproduces the contact force between the AWSA and its holding object.

Table 2. Parameters of AWSA.

Parameters	Value
Robot platform (RP) radius (m)	0.36
Tool platform (TP) radius (m)	0.27
Initial height (m)	0.5
Robot platform (RP) offset angle (deg)	2.5
Tool platform (TP) offset angle (deg)	10
Mass of the tool platform (kg)	4.92
Mass of the leg cylinder (kg)	10.29
Inertia coefficient of the TP, Ixx (kg × m2)	0.09
Inertia coefficient of the TP, Iyy (kg × m2)	0.09
Inertia coefficient of the TP, Izz (kg × m2)	0.18

tabulates the parameters of the AWSA.

4. The Linear DC Motor Actuator

Figure 5 shows the details of the linear actuator of the AWSA, which is mainly composed of a DC motor and a ball joint drive comprising a ball screw and a ball nut. An LVDT is mounted along the length of the actuator to acquire its length for feedback control.

We now discuss the design and selection of the linear actuator components that are divided into two categories: the structural components and the dynamic components. The structural components are the main load-bearing components responsible for holding the actuator together. The dynamic components are those providing the actuator with movement capability and, thus, comprise the DC motor, the ball joint drive, and the LVDT. In order to minimize costs and system downtime, parts of the above components are selected to be off-the-shelf items. The ball screw is selected by considering its maximum lead, $L_{max}$, which is calculated by [33]

$$L_{max}={\displaystyle \frac{\alpha }{\beta }}$$

(1)

where $\alpha$ and $\beta$ are the desired actuator resolution and maximum resolution of the motor, respectively. The minimum lead $L_{min}$ can be calculated by

$$L_{min}={\displaystyle \frac{V_{max}}{{\omega }_{max}}}$$

(2)

where $V_{max}$ and ${\omega }_{max}$ are the desired maximum linear velocity and the assumed maximum rotational velocity of the selected motor, respectively. It is important to select a ball screw whose lead lies within the range of the lead specified by $L_{max}$ and $L_{min}$. The maximum allowable rotational velocity ${\omega }_{max}$ of the ball screw must be calculated to assure that its resonant frequency is greater than the operating frequency, namely,

$${\omega }_{max}={\displaystyle \frac{f d {10}^7}{L^2}}$$

(3)

where f is the coefficient specified by the ball screw mounting method, d is the inside diameter of the screw, and L is the length of the screw. We are also aware that the above range satisfies only the motion requirements of the ball screw but does not dictate the behavior of the ball screw as a load-bearing element. The maximum axial load $F_{max}$ for the selected ball screw can be determined by

$$F_{max}={\displaystyle \frac{m d^4 {10}^3}{L^2}}$$

(4)

where m is a coefficient determined by the way the ball screw is supported. Now, based on Equations (1)–(4), a suitable ball screw can be properly selected. If a motor is not available for the selected ball screw, the above process of ball screw selection must be repeated with some different ball screw parameters until a ball screw can be selected for an existing motor.

The motor selection is performed with a compromise between size, weight, torque, and speed. The selected motor should have a maximum operating rotational velocity ${\omega }_{max}$ to allow the actuator to travel at the desired linear velocity by the selected ball screw. However, the selected motor should have a resolution $\beta$ to achieve the desired actuator resolution $\alpha$. The lifting capacity of the selected motor is determined by its average operating torque, $T_{avg}$, which is given by

$$T_{avg}={\displaystyle \frac{R_{max} L }{k}}$$

(5)

where $R_{max}$ denotes the maximum reaction load of the AWSA, L denotes the lead of the selected ball screw, and k (=5.654) denotes the product of the unit conversion factor and the efficiency factor of the ball screw.

Although a motor may be capable of supplying ample torque to sustain a load, it may not be able to supply the torque at the desired speed. Therefore, the torque–speed curve of the selected motor must be analyzed to determine if the selected motor can provide the required torque. However, if the torque–speed curve is not available, the delivered torque $T_{oper}$ at a specific rotational velocity $\omega$ can be computed by

$$T_{oper}={\displaystyle \frac{\omega }{1000 R_T}} \left[K_T \left(V-K_E\right)-K_{D }{R}_T\right]-T_F$$

(6)

where $K_T$ denotes the motor’s torque constant; $V$ denotes the motor’s terminal voltage; $K_E$ denotes the motor’s back EMF; $K_D$ denotes the motor’s viscous damping constant; $R_T$ denotes the motor’s terminal resistance; and $T_F$ denotes the average frictional torque of the motor.

Using Equation (6), the deliverable torque at a given rotational velocity for a specific motor can be calculated. However, to determine if a motor has the minimum power $P_{min}$ to sustain $R_{max}$ at a given velocity, we use

$$P_{min}=T_{avg} {\omega }_{max}$$

(7)

Based on specifications expressed by Equations (1)–(7), the PMI motor, model SM6HI, and NSK Ball Screw with a 2.5 mm lead and 10 mm diameter were selected. The motor is hard-coupled to the ball screw to reduce vibration. The SM6HI was a DC brush, moving-coil servo motor designed for applications requiring rapid acceleration and precise positioning. It was part of the PMI “Super ServoDisc” line, known for its unique flat disc armature design. The moving-coil design gives it a very low inertia, which allows for extremely fast response and acceleration.

The LVDT size should be selected to minimize the chance of colliding with the linear actuator or the TP and the RP during the movement of the AWSA. The selected LVDT should have a resolution that is equal to or smaller than that of the linear actuator. Furthermore, the LVDT’s traverse velocity, defined as the maximum allowable speed of retraction or extension of the LVDT, should be equal to or greater than the actuator’s maximum velocity. Finally, the LVDT should be insensitive to mechanical vibrations and electromagnetic interference due to its planned mounting on the motor housing. Based on the above specifications, the BALLUFF LDT, Series BTL, with a resolution of 0.003 inches, a maximum traverse velocity of 13 inches/sec, and an output voltage range of 0–10 volts, was selected.

5. Kinematic Analysis of the AWSA

This section derives the solutions for the forward kinematics and inverse kinematics of the AWSA.

5.1. The AWSA Inverse Kinematics

Figure 6 shows the frame assignment of the AWSA. To represent the configuration (position and orientation) of the TP with respect to the RP, Frame {B}, whose origin is at the centroid of the RP, and Frame {A}, whose origin is at the centroid of the TP, are assigned as seen in Figure 6. The assignment of the above frames follows the right-hand rule. Points A_i and points B_i are the attachment points of actuator i to the TP and RP, respectively. In addition, each pair of ball joints on the platforms is distributed symmetrically with a separation angle of 120 degrees.

Referring to the vector diagram in Figure 7, the fixed position vectors ${\mathit{a}}_i^A$ and ${\mathit{b}}_i^B$ describe the position of the ball joints at $A_i$ and $B_i$ with reference Frame {A} and Frame {B}, respectively. The radii of the TP and RP are denoted by $r_A$ and $r_B$, respectively.

In Figure 8, let ${\theta }_A$ represent the angle between ball joints $A_1$ and $A_2$, which is the same between $A_3$ and $A_4$, and $A_5$ and $A_6$. Furthermore, if the angle between ${AA}_i$ and the $X_A$-axis is denoted by ${\lambda }_i$ then the position vector ${\mathit{a}}_i^A={\left(a_{ix} a_{iy} a_{iz}\right)}^T$ is computed by the following:

$${\mathit{a}}_i^A={\left[r_A\mathrm{c}\mathrm{o}\mathrm{s}\left({\lambda }_i\right)r_A\mathrm{s}\mathrm{i}\mathrm{n}\left({\lambda }_i\right)0\right]}^T$$

(8)

Similarly, if the angle between ${BB}_i$ and the $X_B$-axis is denoted by ${\Lambda }_i$, as seen in Figure 9, then the position vector ${\mathit{b}}_i^A={\left(b_{ix} b_{iy} b_{iz}\right)}^T$ is given by the following:

$${\mathit{b}}_i^B={\left[r_B\mathrm{c}\mathrm{o}\mathrm{s}\left({\Lambda }_i\right){ r}_B\mathrm{s}\mathrm{i}\mathrm{n}\left({\Lambda }_i\right)0\right]}^T$$

(9)

The angles ${\lambda }_i$ and ${\Lambda }_i$ are determined by the following:

$${\Lambda }_i=60\left(i-1\right) deg; {\lambda }_i=60\left(i-1\right) deg, for i=\mathrm{1,3},5$$

(10)

$${\Lambda }_i={\Lambda }_{i-1}+{\theta }_B deg; {\lambda }_i={\lambda }_{i-1}+{\theta }_A deg, for i=\mathrm{2,4},6$$

(11)

The kinematic equations can be derived from the vector diagram for each actuator in Figure 7. The length vector ${\mathit{q}}_i^B={\left(q_{ix} q_{iy} q_{iz}\right)}^T$, expressing the vector $B_i{A}_i$ with respect to frame {B}, can be computed as follows:

$${\mathit{q}}_i^B={\mathit{a}}_i^B-{\mathit{b}}_i^B$$

(12)

where vector ${\mathit{a}}_i^B$ represents the coordinates of point $A_i$ with respect to frame {B}. Vector ${\mathit{d}}_B$, denoting the position of point A with respect to frame {B}, is given by the following:

$${\mathit{d}}_B={\left(x y z\right)}^T$$

(13)

where x, y, and z are the Cartesian coordinates of point A in frame {B}. If the orientation of the moving frame {A} with respect to the base frame is specified by Z–Y–X Euler angles $(\alpha \beta \gamma )$, then the corresponding Euler angle orientation matrix ${}_A{}^{B}R$ can be computed as follows:

$${{}_A{}^B\mathit{R}=\mathit{R}}_{zyx,\alpha \beta \gamma }=\left[\begin{array}{ccc}C\alpha C\beta & C\alpha S\beta S\gamma -S\alpha C\gamma & C\alpha S\beta C\gamma +S\alpha S\gamma \\ S\alpha C\beta & S\alpha S\beta S\gamma +C\alpha C\gamma & S\alpha S\beta C\gamma -C\alpha S\gamma \\ -S\beta & C\beta S\gamma & C\beta C\gamma \end{array}\right]$$

(14)

Now, vector ${\mathit{a}}_i^B$ can be computed as follows:

$${\mathit{a}}_i^B={}_A{}^B\mathit{R}{a}_i^A+{\mathit{d}}_B$$

(15)

Substituting (15) into (12) yields the following:

$${\mathit{q}}_i^B={}_A{}^B\mathit{R}{\mathit{a}}_i^A+{\mathit{d}}_B-{\mathit{b}}_i^B$$

(16)

Using (16), vector ${\mathit{q}}_i^B$ can be solved for a desired Cartesian configuration of the moving platform, specified by x, y, z, and $\alpha \beta \gamma$ as follows:

$${\mathit{q}}_i^B=\left[\begin{array}{c}{q}_{ix}\\ q_{iy}\\ q_{iz}\end{array}\right]=\left[\begin{array}{c}{r}_{11}{a}_{ix}+r_{12}{a}_{iy}+r_{13}{a}_{iz}+x-b_{ix}\\ r_{21}{a}_{ix}+r_{22}{a}_{iy}+r_{23}{a}_{iz}+y-b_{iy}\\ r_{31}{a}_{ix}+r_{32}{a}_{iy}+r_{33}{a}_{iz}+z-b_{iz}\end{array}\right]$$

(17)

where $r_{ij}$ are the elements of the orientation matrix ${}_A{}^B\mathit{R}$ given by (14). After solving for vector ${\mathit{q}}_i^B$, the square of the required corresponding length of the i^th actuator, q_i can be computed by the following:

$$q_i^2=\left(q_{ix}^2+q_{iy}^2+q_{iz}^2\right) for i=\mathrm{1,2},\dots ,6$$

(18)

Substituting (17) into (18) yields

$$\begin{gathered} q_i^2=x^2+y^2+z^2+a_{ix}^2+a_{iy}^2+a_{iz}^2+b_{ix}^2+b_{iy}^2+b_{iz}^2 \\ +2\left(r_{11}{a}_{ix}+r_{12}{a}_{iy}+r_{13}{a}_{iz}\right)\left(x-b_{ix}\right) \\ +2\left(r_{21}{a}_{ix}+r_{22}{a}_{iy}+r_{23}{a}_{iz}\right)\left(y-b_{iy}\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} +2\left(r_{31}{a}_{ix}+r_{32}{a}_{iy}+r_{33}{a}_{iz}\right)\left(z-b_{iz}\right)-2(x{b}_{ix}+y{b}_{iy}+z{b}_{iz}) \end{gathered}$$

(19)

In Figure 7, since ${\mathit{a}}_i^A$ and ${\mathit{b}}_i^B$ lie on the $x_A-y_A$ plane and $x_B-y_B$ plane, respectively, we have the following:

$$a_{iz}=b_{iz}=0$$

(20)

$$a_{ix}^2+a_{iy}^2+a_{iz}^2=r_A^2$$

(21)

$$b_{ix}^2+b_{iy}^2+b_{iz}^2=r_B^2$$

(22)

Now, in light of (20)–(22), (19) becomes the following:

$$\begin{gathered} q_i^2=x^2+y^2+z^2+r_A^2+r_B^2+2\left(r_{11}{a}_{ix}+r_{12}{a}_{iy}\right)\left(x-b_{ix}\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \hspace{1em}+2\left(r_{21}{a}_{ix}+r_{22}{a}_{iy}\right)\left(y-b_{iy}\right)+2\left(r_{31}{a}_{ix}+r_{32}{a}_{iy}\right)z \\ -2\left(x{b}_{ix}+y{b}_{iy}\right) for i=\mathrm{1,2},\dots ,6 \end{gathered}$$

(23)

where $q_i$ is the length of actuator i. We recognize that Equation (23) provides a closed-form solution to the AWSA’s inverse kinematics. In other words, if a set of actuator lengths $q_1$, $q_2$, $q_3$, $q_4$, $q_5$, and $q_6$ must be determined to yield a desired configuration of the TP with respect to the RP, expressed by the desired position in Equation (13) in terms of Cartesian variables x, y, and z and by the desired orientation in Equation (14) in terms of the elements of the orientation matrix ${}_A{}^B\mathit{R}$, then Equation (23) can be employed to accomplish this objective.

5.2. The AWSA Forward Kinematics

The forward kinematics of the above AWSA deals with the determination of the Cartesian configuration of the TP with respect to the RP, specified by Cartesian positions x, y, z, and the Z–Y–X Euler angles $\alpha , \beta , \gamma$ based on the actuator lengths. Unfortunately, Equation (23) provides six nonlinear simultaneous equations with six unknowns that lead to no closed-form solutions for the forward kinematic problem. As a result, this problem is extremely difficult and needs to be solved by using iterative numerical methods such as the Newton–Raphson method [21]. The Newton–Raphson method first defines a function $\mathbf{f}\left(\mathit{\Phi }\right)$ of the unknowns, including x, y, z, and $\alpha , \beta , \gamma$, where $\mathit{\Phi }$ = (x y z $\alpha \beta \gamma$)^T, so that $\mathbf{f}\left(\mathit{\Phi }\right)=0$. The function $\mathbf{f}\left(\mathit{\Phi }\right)$ can be written as

$$\mathbf{f}\left(\mathit{\Phi }\right)={\left[f_1\left(\mathit{\Phi }\right),f_2\left(\mathit{\Phi }\right),...,f_6\left(\mathit{\Phi }\right)\right]}^T$$

(24)

It then iteratively applies the following formula:

$${\mathit{\Phi }}_{\mathit{k}+\mathbf{1}}\mathbf{=}{\mathit{\Phi }}_{\mathit{k}} - {\mathbf{e}}_{\mathrm{k}}$$

(25)

where ${\mathbf{e}}_{\mathrm{k}}$ denotes the error vector at the kth iteration, which is computed as

$${\mathbf{e}}_{\mathrm{k}}={\mathbf{J}}^{-1}\mathbf{f}\left({\mathsf{\Phi }}_{\mathrm{k}}\right)$$

(26)

where J, the Jacobian matrix of $\mathbf{f}\left(\mathit{\Phi }\right)$, can be determined by

$$\mathbf{J}=\left[\begin{array}{cc}\begin{array}{cc}\begin{array}{cc}\begin{array}{cc}\begin{array}{cc}\begin{array}{c}{\displaystyle \frac{\partial f_1}{\partial x}}\\ .\\ .\\ .\\ .\\ {\displaystyle \frac{\partial f_6}{\partial x}}\end{array}& \begin{array}{c}{\displaystyle \frac{\partial f_1}{\partial y}}\\ .\\ .\\ .\\ .\\ {\displaystyle \frac{\partial f_6}{\partial y}}\end{array}\end{array}& \begin{array}{c}{\displaystyle \frac{\partial f_1}{\partial z}}\\ .\\ .\\ .\\ .\\ {\displaystyle \frac{\partial f_6}{\partial z}}\end{array}\end{array}& \begin{array}{c}{\displaystyle \frac{\partial f_1}{\partial \alpha }}\\ .\\ .\\ .\\ .\\ {\displaystyle \frac{\partial f_6}{\partial \alpha }}\end{array}\end{array}& \begin{array}{c}{\displaystyle \frac{\partial f_1}{\partial \beta }}\\ .\\ .\\ .\\ .\\ {\displaystyle \frac{\partial f_6}{\partial \beta }}\end{array}\end{array}& \begin{array}{c}{\displaystyle \frac{\partial f_1}{\partial \gamma }}\\ .\\ .\\ .\\ .\\ {\displaystyle \frac{\partial f_6}{\partial \gamma }}\end{array}\end{array}\right]$$

(27)

until a certain acceptable accuracy is achieved.

The function $f_i\left(\mathit{\Phi }\right)$ for i = 1,2,…,6 in (24) can be obtained by considering (23) so that

$$\begin{gathered} f_i\left(\mathit{\Phi }\right)=x^2+y^2+z^2+r_A^2+r_B^2+2\left(r_{11}{a}_{ix}+r_{12}{a}_{iy}\right)\left(x-b_{ix}\right)+2\left(r_{21}{a}_{ix}+r_{22}{a}_{iy}\right)\left(y-b_{iy}\right) \\ +2\left(r_{31}{a}_{ix}+r_{32}{a}_{iy}\right)z-2\left(x{b}_{ix}+y{b}_{iy}\right)-q_i^2=0 \end{gathered}$$

(28)

In summary, the solution of the AWSA’s forward kinematics can be obtained by employing the Algorithm 1 as seen below.

Algorithm 1. AWSA Forward Kinematics Solution Using Newton–Raphson Method

Step 1: Choose an initial guess ${\mathit{\Phi }}_1$. Step 2: Compute $f_i\left({\mathit{\Phi }}_1\right)$ for i = 1,2,…,6 using (28). Step 3: Compute $\mathrm{J}\left({\mathsf{\Phi }}_1\right)$ using (27). Step 4: Compute a new guess ${\mathit{\Phi }}_2$ using (25). Step 5: Repeat Steps 1 to 4 until the error between the current and previous guesses as computed by (26) is smaller than a specified tolerance value T such that

$${\left({\sum }_{i=1}^6{e}_i^2\right)}^{1/2}\le T$$

(29)

After applying the above algorithm for several test cases, we have the following remarks:

• In Step 1, if the initial guess is very far away from the true solution, the algorithm will need a considerable number of iterations before convergence, or sometimes it does not converge at all. Unfortunately, there is no general way to select an initial guess to ensure convergence.
• In Equations (25) and (26), an initial guess should be selected so that the inverse of the Jacobian matrix ${\mathrm{J}}^{-1}$ exists, which requires that det $(\mathrm{J}$) is nonzero. If this condition is not satisfied, then we face the singularity problem. In this case, a different initial guess should be selected to avoid a singularity.
• In general, the Newton–Raphson method, being an iterative method, is computationally intensive and is, thus, not suitable for real-time application. However, it could be used in computer simulation studies for control performance evaluation and for visualization of joint space variables in Cartesian space.

6. Motion Control of AWSA

Since this paper focuses on the mechanical design and kinematic analysis of an AWSA, this section discusses its motion control aspect and proposes a control scheme without presenting results of any computer simulations or experimental studies, which are the subject of past and future articles. In robot kinematics, we deal with two types of variables, Cartesian variables and joint variables. Cartesian variables include positional variables x, y, and z, expressing the position of the end-effector with respect to a reference frame, and three orientational variables, such that Euler angles, expressing the orientation of the end-effector with respect to the same reference frame. In serial robot manipulators, joint variables are the angles of revolute joints or distances of prismatic joints. For parallel manipulators, joint variables are typically the lengths of the actuators. Thus, for the AWSA, the joint variables are the lengths of the six actuators, $q_1$, $q_2$, $q_3$, $q_4$, $q_5$, and $q_6.$ A joint-space control scheme (JSCS), illustrated in Figure 10, can be applied to control the joint variables (actuator lengths) of the AWSA. As shown in the figure, the JSCS takes advantage of the existence of a closed-form solution for the AWSA’s inverse kinematics and uses it to convert the desired Cartesian variables expressed in vector ${\mathit{X}}_d\left(t\right)$ to the desired joint variables contained in vector ${\mathit{q}}_d\left(t\right)$. The actual joint variables expressed by vector q(t) are measured by the LVDTs, and then the measured actual joint variables are subtracted from the desired joint variables contained in vector ${\mathit{q}}_d\left(t\right)$ to produce the joint variable errors expressed by vector ${\mathit{q}}_e\left(t\right)$. The joint variable errors are then sent to a controller, which could be a proportional (P) controller, or a proportional–derivative (PD) controller or a proportional–derivative–integral (PID) controller. The controller gains could be set at certain fixed values using standard control techniques, including linearization and pole placement to stabilize the closed-loop control system. The actuators are then driven by the controller outputs so that the joint variable errors are minimized and ultimately converge to zero. We observe that the JSCS controls the joint variables directly, as its name implies. Since it has a closed-form solution for the inverse kinematics, the implementation of its control law is not computationally intensive. Consequently, the JSCS could be considered for real-time applications.

7. Conclusions

This paper considered the mechanical design and kinematic analysis of a robotic wrist, named AWSA, which was designed and built to study the feasibility of using autonomous robot manipulators to perform the precise motion required by assembly tasks in space. A parallel structure was selected in the development of the AWSA because this structure had a better positioning capability than its counterpart, the serial structure. The AWSA comprised six linear actuators, each of which consisted of a DC motor and a ball screw drive. This paper presented the mechanical design of various components of the AWSA, including the DC linear actuators and the LVDTs, etc., and explained their hardware selections. In addition, it derived a closed-form solution for the inverse kinematics of the AWSA and a numerical solution for its forward kinematics using the Newton–Raphson method. It also introduced a hybrid concept for robotic space operations, combining an SRM performing gross motion and a PRM like the AWSA, performing the fine and precise motion required by space operation tasks.

Future work on the AWSA is recommended as follows.

• Computer simulations and experiments should be conducted to obtain the reachable and dexterous workspaces of the AWSA to assess the type of assembly it is suitable for.
• Advanced control schemes, such as the intelligent control schemes in [31,34,35,36,37], should be studied for potential application in motion and force control of the AWSA.
• The AWSA should be considered for other industrial and medical applications, such as a force-reflecting hand controller for space teleoperation, a patient training device for medical physical therapy, and a microgravity hand trainer for astronauts.