A Spatial 5-Bar Linkage Analogous to the 4-Bar Linkage and with Application to Robotic Lifting of Large In-Vessel Tokamak Components Including Interaction Issues

1. Introduction

The development of fusion energy technology could lead to the deployment of high-output, carbon-free tokamak reactors in the future. ITER, the world’s largest experimental tokamak, is currently under construction and the European successor, EU DEMO, is now being designed with the aim of demonstrating the technical and economic readiness of fusion energy while providing net electricity to the grid [1]. In addition to being larger and experiencing greater heat and mechanical loads than ITER, the DEMO tokamak (Figure 1a) will also integrate a self-sufficient tritium lifecycle based on breeding blankets (BBs) lining the inner vacuum vessel [2]. The BBs will be the largest and heaviest in-vessel components, and will require occasioreplacement due to material degradation from neutron irradiation. It is a requirement that this remote maintenance task be carried out safely while minimizing down-time of the plant [3].

Unlike ITER, European DEMO will use a vertical maintenance architecture for the BBs, which has also been considered for earlier tokamaks including INTOR [4]. This entails the maintenance of three 180 t outboard and two 125 t BB segments within each of the 16 tokamak sectors using a mobile cask which docks to a sector’s upper port and contains the necessary remote handling (RH) equipment to remove or replace a BB segment. This concept is illustrated in Figure 1b, depicting a robotic RH tool vertically transporting a BB segment within the confines of the vacuum vessel, upper port, and cask. This tool is the BB vertical transporter (BBVT), which was previously introduced in [5,6]. With minimal simplifications, the BBVT can be considered a redundant serial robotic manipulator with 7 degrees of freedom, enabling full spatial motion of the end effector.

Attachment and support structures for the BB segments, seen in Figure 2a, were introduced in [9] and further developed in [8,10]. These include radial, vertical and toroidal support structures which react against different loading conditions of the BB segments during operation of the tokamak. When the tokamak is powered down, as during maintenance, each BB segment is vertically supported only at the bottom and can be lifted by a countersunk interface in the portion of the top surface exposed in the port [11]. A short vertical translation and toroidal tilt are required to free each BB segment from its lower supports.

Besides the vertical load, the BBVT is also subject to large moments [12]. This is due to the fact that the countersunk interface by which each BB segment is grasped is horizontally offset from the center of mass to be accessible by the BBVT inside the upper port. However, rather than being automatically reacted by the BBVT when initially loading a BB segment for lifting, these moments are primarily reacted by the radial and toroidal BB segment supports. This is a problem, as these loads could damage the supports as the contact surface areas shrink during disengagement. This is of particular concern for the toroidal shear keys. Thus, these horizontal moment components must be actively counteracted by the BBVT to avoid loading the supports during disengagement, as illustrated in Figure 2b.

As can be seen in the schematic diagram Figure 3, the BBVT has two tilting joints: the distal gripper tilting joint, and the centrally located trolley tilting joint. The former is required for tilting the BB segments toroidally up to 8^∘ during handling to avoid collisions, while the latter is necessary for tilting the gripper radially by approximately 0.5^∘ to 1.5^∘ in order to ensure proper alignment of the gripper interface before and during engagement. Due to the need to counteract the reaction forces in the BB segment supports during initial lifting, both tilting joints are required to actively apply the equivalent off-center lifting moments to the BB segment. Space is also at a premium, as the BBVT must fit and maneuver within the constricted upper port.

A five bar mechanism, referred to as the trolley tilting mechanism or TTM, is proposed and designed in this work to address this unique set of challenges. A kinematic model of the linkage is established and applied to analyze and optimze the design, and numerical simulations and simulations and experiments are performed to validate the design and the kinematic model.

The current work is divided into 6 sections. In Section 2, Mechanism Design: the design requirements and considerations for the TTM are detailed, and the current design is presented. In Section 3, Kinematics: the kinematic constraint equations for the TTM are derived and an analytical solution for the forward kinematics is presented, along with the configuration-dependent transmission ratio and mechani. In Section 4, Mechanical Design Optimization: the kinematic model is used as a basis to optimize two of the bar lengths with respect to range of motion as well as maximum static loads on the mechanism output and input during BB segment maintenance. In Section 5, Verification and Preliminary Down-scaled Experiment: the kinematic model is verified by comparison with a numerical simulation in ADAMS, and further validated by initial experiments on a down-scaled TTM prototype. Finally, in Section 6, Discussion and Conclusion: the results and contributions are discussed, and limitations and future work presented.

2. Mechanism Design

The main requirement for the TTM is to provide the ability to adjust the radial tilt of the gripper by a small angle (about 0.5 to 1.5) in order to align it during engagement with the BB segment countersunk interface. In terms of carrying capacity, the TTM must have the ability to “preload” the 180 t outboard and 125 t inboard lateral BB segments with a radial moment to counter the formation of reactions within the support structures during lifting. To match the other BBVT joints, the TTM should be implemented using standard electrical actuators rather than hydraulics.

As indicated in Figure 3, the TTM is placed as the fourth joint of the BBVT, between the links referred to as the trolley and the trunk. The reason for this is that the space adjacent to the gripper itself is taken by the gripper tilting joint, and the trolley is the next-closest link with available space. However, even here the horizontal space is limited as the trolley (essentially a box) must fit within the constricted upper port while the BBVT is engaged with each of the 5 BB segments. As the trolley slides on a large radial rail, the radial faces are nearly inaccessible. Also, the lower vertical face is largely occupied by the trunk connection. Thus, the only practical location for the TTM actuator(s) is on the upper face, and the output radial motion must be applied to a tilting plate sitting between the trolley and trunk. The problem, then, is to connect the actuator(s) and tilting plate while affecting the toroidal and radial dimensions of the trolley assembly minimally. This combination of requirements and restrictions is difficult to solve compactly and reliably using only conventional transmission elements such as gearboxes, belts, chains, or screws.

Thus, a compact 5-bar mechanism providing good mechanical advantage was designed, shown in Figure 4a. It is a spatial linkage with structure RSRRR, where “R” denotes a revolute joint and “S” denotes a spherical joint, with Grübler criterion equal to 1. The four mobile bars have been marked with their lengths a, b, c and d. The first (a) is the eccentricity of a cam connected to the input shaft. This is connected via a radial bearing with spherical housing to the rod b. At its lower end, the rod attaches to a short connector c, which connects the rod and the tilting plate d using pins at right angles. The plate connects back to the trolley at the tilting hinge, which constitutes the TTM output.

The use of the mechanism is similar to a planar four-bar mechanism, where mechanical advantage is gained by relative sizing of the bars connected to the input and output joints, while the connecting bars in between serve primarily to transfer the load. The mechanism wraps around the vertical and toroidal faces of the trolley, which is required in order to not interfere with the radial rail. However, a planar four-bar mechanism cannot wrap around in this manner while having good leverage and avoiding collision with the trolley itself. Instead, the input axis located above the trolley is made to be toroidal, perpendicular to the output axis. Then, an extra bar (c) is added to connect the radial output and toroidal input. The main connecting bar (b) then must be able to tilt both toroidally and radially due to the interconnected motion of the input and output bars, necessitating the use of a spherical connection between a and b, in this case chosen as a radial bearing with spherical housing. This configuration also gives space radially along the top surface of the trolley to place multiple tilting mechanisms in parallel to reduce the load on the input actuator(s).

3. Kinematics

Optimization and control of the TTM require an accurate kinematic model, which will be derived in this section. Although kinematic models of RSRRR 5-bar mechanisms exist, since the TTM is a special case, a model using only 5 relative rotation coordinates and 4 equations (compared with 15 equations in [13]) is derived based on the schematic diagram in Figure 4b. The coordinates are: ${\theta }_1$, ${\theta }_2$, ${\theta }_3$, ${\theta }_4$, and ${\theta }_t$, where ${\theta }_1$ is the joint variable associated with the input of the mechanism and ${\theta }_t$ is associated with the output. The input shaft and output shaft angles are each defined as being zero when their respective bars (a and d) are horizontal. The following assumptions are made: the rotation axes of the joints $\left({\theta }_1,{\theta }_t\right)$ and $\left({\theta }_3,{\theta }_4\right)$ are pairwise perpendicular, and the rotation axes of the joints $\left({\theta }_2,{\theta }_4,{\theta }_t\right)$ are all parallel.

With reference to the schematic (Figure 4b), the following four independent equations constrain the kinematics:

$$\begin{array}{cc}\hfill acos{\theta }_1-bsin{\theta }_3& =p_y\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill dcos{\theta }_t-\left(c+bcos{\theta }_3\right)sin{\theta }_2& =p_x\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill -dsin{\theta }_t-\left(c+bcos{\theta }_3\right)cos{\theta }_2-asin{\theta }_1& =p_z\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill {\theta }_2+{\theta }_4& ={\theta }_t.\hfill \end{array}$$

(4)

where $(p_x,p_y,p_z)$ are the constant coordinates of the mechanism output point relative to the input point. For simplicity, and to ensure that ${\theta }_t=0$ when ${\theta }_1=0$, these will be assumed to take on the values $p_x=d$, $p_y=a$, and $p_z=-b-c$ when the equations are applied throughout this work, although it is not strictly necessary. For example, in the case of replacing the input cam with one of different size $a_n\ne a$ without also adjusting $b+c$ and d, this would not hold.

3.1. Forward Kinematic Solution

Treating the input angle ${\theta }_1$ as given, (1) gives ${\theta }_3$ directly:

$$\begin{array}{cc}\hfill sin{\theta }_3& ={\displaystyle \frac{a}{b}}\left(cos{\theta }_1-{\displaystyle \frac{p_y}{a}}\right)\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill cos{\theta }_3& =\pm \sqrt{1-{sin}^2{\theta }_3}\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill {\theta }_3& =atan2\left(sin{\theta }_3,cos{\theta }_3\right).\hfill \end{array}$$

(7)

By squaring and summing (2) and (3) and utilizing Equation (4), ${\theta }_4$ can be found:

$$\begin{array}{cc}\hfill sin{\theta }_4& ={\displaystyle \frac{p_x^2-d^2+{\left(-p_z-asin{\theta }_1\right)}^2-{\left(c+bcos{\theta }_3\right)}^2}{2d(c+bcos{\theta }_3)}}\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill cos{\theta }_4& =\pm \sqrt{1-{sin}^2{\theta }_4}\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill {\theta }_4& =atan2\left(sin{\theta }_4,cos{\theta }_4\right).\hfill \end{array}$$

(10)

Physically valid and non-extraneous solutions are obtained by only taking the positive solutions for (6) and (9).

Using Equation (4), ${\theta }_2$ can be eliminated from Equation (2) and Equation (3). The system of equations can then be solved for ${\theta }_t$ in terms of known and derived angles (${\theta }_1$, ${\theta }_3$, and ${\theta }_4$) by taking advantage of the fact that a system of equations of the form

$$\begin{array}{cc}\hfill pcos{\theta }_t-qsin{\theta }_t& =r\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill psin{\theta }_t+qcos{\theta }_t& =s\hfill \end{array}$$

(12)

has the solution:

$$\theta =atan2\left(ps-qr,pr+qs\right).$$

(13)

Thus,

$$\begin{array}{c}\hfill {\theta }_t=atan2\left(d+sin{\theta }_4\left(c+bcos{\theta }_3\right)\right)\left(-p_z-asin{\theta }_1\right)-p_{x}cos{\theta }_4\left(c+bcos{\theta }_3\right),\\ \hfill p_x\left(d+sin{\theta }_4\left(c+bcos{\theta }_3\right)\right)+cos{\theta }_4\left(c+bcos{\theta }_3\right)\left(-p_z-asin{\theta }_1\right).\end{array}$$

(14)

To find ${\theta }_2$, the solutions for ${\theta }_4$ and ${\theta }_t$ can be plugged into (4). Alternatively, the same procedure used to find ${\theta }_t$ can be applied, but using (4) to eliminate ${\theta }_t$ from (2) and (3). Also, rather than relying on (13), alternative forms of the solutions for ${\theta }_t$ and/or ${\theta }_2$ can be found using the tangent of the half-angle substitution method.

3.2. Transmission Ratio and Mechanical Advantage

The constraint equations (1) to (3) can be differentiated with respect to time to give the velocity constraints:

$${\overrightarrow{\Phi }}_{\overrightarrow{\Theta }}\dot{\overrightarrow{\Theta }}=\overrightarrow{0},$$

(15)

where $\dot{\overrightarrow{\Theta }}$ is the vector of time derivatives of the coordinates and ${\overrightarrow{\Phi }}_{\overrightarrow{\Theta }}$ is the Jacobian matrix of the constraint equations:

$${\overrightarrow{\Phi }}_{\overrightarrow{\Theta }}=\left[\begin{array}{ccccc}asin\left({\theta }_1\right)& 0& bcos\left({\theta }_3\right)& 0& 0\\ 0& -cos\left({\theta }_2\right)c+bcos\left({\theta }_3\right)& bsin\left({\theta }_2\right)sin\left({\theta }_3\right)& 0& -dsin\left({\theta }_t\right)\\ acos\left({\theta }_1\right)& -sin\left({\theta }_2\right)c+bcos\left({\theta }_3\right)& -bcos\left({\theta }_2\right)sin\left({\theta }_3\right)& 0& dcos\left({\theta }_t\right)\\ 0& -1& 0& -1& 1\end{array}\right]$$

(16)

The system of four equations in (15) can be solved to find the ratio of input and output speeds, or transmission ratio $\eta$:

$$\eta ={\displaystyle \frac{{\dot{\theta }}_1}{{\dot{\theta }}_t}}=-{\displaystyle \frac{d}{a}}\left({\displaystyle \frac{cos{\theta }_{3}cos{\theta }_4}{cos{\theta }_{1}cos{\theta }_{2}cos{\theta }_3+sin{\theta }_{1}sin{\theta }_3}}\right)$$

(17)

By the conservation of energy, and neglecting inertia and friction forces, the input power must equal the output power, leading to the assertion of the mechanical advantage provided by the mechanism:

$${\displaystyle \frac{{\tau }_1}{{\tau }_t}}={\left({\displaystyle \frac{{\dot{\theta }}_1}{{\dot{\theta }}_t}}\right)}^{-1}=-{\displaystyle \frac{a}{d}}\left({\displaystyle \frac{cos{\theta }_{1}cos{\theta }_{2}cos{\theta }_3+sin{\theta }_{1}sin{\theta }_3}{cos{\theta }_{3}cos{\theta }_4}}\right),$$

(18)

where ${\tau }_1$ and ${\tau }_t$ are the torques at the input shaft and output hinge, respectively.

The factor in parentheses in (18) depends on the configuration of the mechanism as well as the chosen bar lengths. This is illustrated by Figure 5, where this factor g is plotted as a function of the input angle for several TTM designs. As can be seen, g is sensitive to a, b, or d but does not depend strongly on c. When $g=0$, the mechanism is in a kinematic singularity where the output angle reverses direction, this generally occurs near ${\theta }_1\approx \pm {90}^{\circ }$ and should be considered when controlling the mechanism.

3.3. Inverse Kinematics

Although an analytical solution of the kinematic equations for ${\theta }_t$ exists when ${\theta }_1$ is known, as described in Section 3.1 above, the inverse is not true. Instead, ${\theta }_1$ can be found numerically using the Newton-Raphson method with the constraint Jacobian (16) i.e. iteratively solving for ${\overrightarrow{\Theta }}_{i+1}$ an equation of the form:

$$\overrightarrow{\Phi }(\overrightarrow{\Theta },t)\approx \overrightarrow{\Phi }\left({\overrightarrow{\Theta }}_i\right)+\overrightarrow{{\Phi }_q}\left({\overrightarrow{\Theta }}_i\right)\left({\overrightarrow{\Theta }}_{i+1}-{\overrightarrow{\Theta }}_i\right)=0,$$

(19)

where $\overrightarrow{\Phi }\left({\overrightarrow{\Theta }}_i\right)$ is the vector of constraint equations (1) to (3) evaluated at ${\overrightarrow{\Theta }}_i$.

4. Mechanical Design Optimization

An initial sizing of the TTM based only on the space available and the geometry of the BBVT trolley has given $a=0.015$ m, $b=1.635$ m, $c=0.14$ m and $d=1.038$ m. However, it may be possible to optimize this using the kinematic model derived in Section 3. Two important properties of interest are the transmission ratio and range of motion of the mechanism.

In the home position, the transmission ratio (17) reduces to the simple ratio of the lengths of the output and input bars:

$${\eta }_{base}=-{\displaystyle \frac{d}{a}}.$$

(20)

We will refer (20) as the base transmission ratio, as the true transmission ratio almost always increases in magnitude when the mechanism is not in home position, due to the geometry factor generally obeying $\left|g\right|<1$ for designs where $a<<b$ and $a<<d$. In order to minimize the sizes of the required actuators and gearboxes, a base transmission ratio of about 50–100 is desired. Figure b plots expected values of $\left|{\eta }_{base}\right|$ for relevant ranges of a and d based on (20), showing that smaller a and larger d values are desirable.

Figure 6. Sizing of the TTM primary dimensions a (cam eccentricity) and d (plate length) requires considering the load at the input axis and the output range of motion.

Figure 6. Sizing of the TTM primary dimensions a (cam eccentricity) and d (plate length) requires considering the load at the input axis and the output range of motion.

Another important property of the mechanism is the output range of motion. Since an exact expression for ${\theta }_{t,min}$ and ${\theta }_{t,max}$ is difficult to obtain, a simplification is used, namely that the rotating cam (a) connects directly to the trolley tilting plate (d). This is valid when $a<<b$ and $a<<d$ since the tilt angles ${\theta }_2$ and ${\theta }_3$ then remain small and contribute negligibly to the change in height of the far end of the tilting plate. In that case, the maximum output angle corresponds to the input angle ${\theta }_1=\pm {90}^{\circ }$, and, using (3),

$$\begin{array}{cc}\hfill dsin{\theta }_{t,max}& \approx a\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill {\theta }_{t,max}& \approx {sin}^{-1}\left({\displaystyle \frac{a}{d}}\right)\approx {\displaystyle \frac{a}{d}}.\hfill \end{array}$$

(22)

Note that the simplification also leads to symmetrical extrema, ${\theta }_{t,min}=-{\theta }_{t,max}$, which is not true of the exact values. Figure c shows how the estimated ${\theta }_{t,max}$ varies for different bar lengths; specifically, smaller d and larger a values lead to increased output range.

It should be noted that both the base transmission ratio and the approximate output range of motion depend exclusively on a and d. Thus, choosing appropriate lengths for the input and output bars is a major design consideration.

Another crucial value to consider is the load on the actuator gearbox connected to the TTM input. This depends not only on the transmission ratio and therefore the internal TTM geometry, but also on the external problem geometry relating to the use-case of lifting the BB segments. This is illustrated in Figure a, which shows the difference in the problem setup between the outboard (OB) left and right BB segments. These are the two which produce the greatest radial moments at the gripper interface due to their offset centers of mass. The radial moment load on the TTM output axis is different in each case due to the variable horizontal arm r, which depends on the chosen tilting plate length d. Figure d shows that a value of $d=0.55$ m gives equal loads on the output axis in the two cases, thus minimizing the overall largest load. By taking the larger of the two loads for a given value of d and dividing by the corresponding ${\eta }_{base}$, the maximum load on the TTM input is found (Figure e). This is not only useful for choosing a and d but also enables optimal sizing of the input actuator(s) and gearbox(es).

The analysis supports the idea of using a centered hinge, thus fixing one of the two main design variables. The centered hinge has the advantage of resulting in fully symmetric loading cases for the lateral segments, and desired values of ${\eta }_{base}$ and ${\theta }_{t,max}$ can be achieved by simply varying a.

5. Verification and Preliminary Down-scaled Experiment

5.1. Verification by Adams Model

The kinematic model is verified by comparison to a numerical simulation in MSC Adams. A TTM design with $a=15$ mm, $b=50$ mm, $c=5$ mm, and $d=50$ mm is used for the verification. The input shaft is prescribed a constant speed of ${\dot{\theta }}_1=1\circ$ per s and the simulation is run for $t=\left[0s,359s\right]$ with $\delta t=1s$. Several configurations of the Adams model are pictured in Figure 7.

The results of the verification are given in Figure 8, where the coordinates ${\theta }_2$, ${\theta }_3$, and ${\theta }_t$, as well as the output shaft speed ${\theta }_t$, are plotted for a full revolution of the input shaft. The root mean square error of the position curves does not exceed 5e-06∘, and the RMSE the output shaft speed curve is just 8.5e-8∘ per s.

5.2. Preliminary Down-scaled Experiment

A 3-D printed down-scaled prototype of the TTM was manufactured and used to further validate the analytical model. The dimensions correspond to the full-scale TTM with $a=0.015$ m, $b=1.635$ m, $c=0.14$ m, and $d=0.55$ m, scaled by a factor $1/8.33$. The design of the prototype is shown in Figure 9a, and the setup of the experiment in Figure 9b. The input shaft is driven by a MKS42C stepper motor, which has an internal absolute encoder with 1024 counts per revolution, and the output shaft is attached to a relative encoder with 14400 counts per revolution. The motor is set to turn one revolution in 60s with 4160 steps while the output shaft position measurements are collected via an Arduino microcontroller connected to a PC.

To estimate the input and output shaft speeds, a linear fit was found for the input shaft position data and a sine curve was fitted to the output shaft position data; the first derivatives of these curves were then taken as the experimental speeds, the ratio of which was used as the measured inverse transmission ratio to compare with the model (see (17)). The numerical differentiation of the position data with a light Butterworth filter applied was also found as a reference for the shaft speeds.

The results of the experiment are given in Figure 10. There was a small error in the kinematic relationship between the input and output positions ${\theta }_1$ and ${\theta }_t$, with a RMSE of 6.6e-2∘ compared to an output range of motion of about $\pm 1.8\circ$. A line with equation ${\theta }_1=6.029t-0.6628$ was fitted to the input position data, and a sine curve with equation ${\theta }_t=1.65sin\left(0.1052t-3.123\right)$ was fitted to the output position data. The derivatives of the fits were used to find the position-dependent (inverse) transmission ratio. The RMSE of the predicted and experimentally measured transmission ratio curves was 1.1 × 10⁻³ compared to a peak of about $\pm 0.3$.

6. Discussion and Conclusion

The predictions by the analytical model of the TTM joint positions and output shaft speed show great agreement with the numerical simulation in Adams, verifying the basic accuracy of the model. Although the experimental results show some small deviations compared with the predicted output shaft motions, the agreement is overall still good, and the discrepancies can be explained by geometrical uncertainties and material flexibility inherent in the plastic additive manufacturing of a down-scaled prototype.

A 5-bar tilting mechanism has been proposed and designed for the trolley tilting joint of the BBVT and a detailed kinematic model has been derived, verified by numerical simulation, and validated by experiment. Thus, the kinematic relationships between the mechanism’s input and output position, speed, and torque have been shown. In addition, the relationships of the base transmission ratio, output range of motion, output axis load, and input axis load to the cam eccentricity a and tilting plate length d have been detailed, allowing for intelligent dimensioning of these parameters together with the joint gearbox(es) and actuator(s). The model has thus been applied in designing and optimizing the mechanism with respect to the very large static loads which it will see during operation within the future DEMO power plant, supporting the feasibility of the TTM as a solution for safely lifting the BB segments.

Due to the 3D-printing process, the down-scaled TTM lacks the manufacturing precision and material stiffness required for a highly accurate validation of the kinematic model. This flexibility also prevents carrying out experiments to validate the mechanical advantage predicted by the model. Thus, a high-precision down-scaled prototype constructed with sturdier materials could be manufactured in the future to enable fully validating the analytical model.

Since the TTM is only foreseen to be used at low speeds, a multibody dynamics model of the mechanism is not considered necessary for now. However, the derived kinematic model can be used as a basis for future modelling, potentially including dynamic effects, as well as control solutions. More detailed design explorations guided by the model can also be carried out.