INTRODUCTION

One of the important areas of application of parallel structure mechanisms is medicine, which is due to qualities of such mechanisms such as higher positioning accuracy and rigidity compared to mechanisms with a sequential structure. Most often, mechanisms of this type are used in surgery, in particular, neurosurgery and minimally invasive abdominal surgery [1, 2]. In this case, various structural diagrams of mechanisms can be used, but the most widespread are devices with a remote center of rotation. In these mechanisms, the output link (most often, a surgical instrument) rotates around a constant point, and the intermediate links are always located at some distance from this point. This feature is caused by the fact that the mentioned point, as a rule, is the site of a puncture or incision at which the instrument is inserted into the patient’s body. The two main types of center-remote mechanisms are spherical and flat mechanisms.

Spherical mechanisms are devices that, as a rule, have three degrees of freedom. In the vast majority of cases, such mechanisms are spatial; i.e., their links are not located in the same plane. Examples of medical spherical mechanisms include the CoBRASurge robot [3] with a classic structure including arc links, designed for minimally invasive operations, as well as the ESTELE robot [4] with a similar structure, the main purpose of which is robotic ultrasound examination of the abdominal region.

Flat mechanisms, the links of which lie in the same plane, can also use arc elements, as, for example, is done in the SASSU [5] and Probot [6] robots. However, the most common are mechanisms with pantographs, in which the movement of the drive link (crank) is copied by a tool. Examples include the LARS robot [7], which is one of the first representatives of such devices, and the BlueDragon robot [8], which uses several pantographs, each of which controls a separate tool. The main disadvantage of such mechanisms is their relatively low rigidity, which is unacceptable for medical equipment [9]. This article proposes a possible solution to this problem by introducing additional drive redundancy into a flat mechanism with a pantograph.

BASIC MECHANISM

First, let us look at the basic design of the mechanism (Fig. 1).

Fig. 1.
figure 1

Mechanism with a remote center of rotation: (a) diagram of the mechanism and (b) four-bar linkage as a part of the mechanism.

Full size image

The main part of the mechanism is a pair of flat parallelograms ABCD and CEFG. A rotation joint is located at each point. In this case, the link CD of the first parallelogram and the link of the second parallelogram CE are two parts of a physically single link DE. Similarly, links BC and CG are parts of a single link BG. A tool is attached to the FG link, the axis of which is parallel to this link. Using the well-known Chebyshev formula, one can see that the mobility of the considered mechanism will be equal to unity

$$W = 3n - 2{{p}_{5}} - {{p}_{4}} = 3 \times 5 - 2 \times 7 - 0 = 1,$$

where n is the number of moving links of the mechanism, p5 is the number of single-moving kinematic pairs (5th class pairs), and p4 is the number of two-moving kinematic pairs (4th class pairs).

The kinematic structure of the mechanism ensures the synchronous movement of the links AB, DE, FG, and, accordingly, the tool. In other words, rotation of the drive kinematic pair A leads to rotation of the tool around the point O. In this case, the rotation angle of the input link θA will be equal to the rotation angle of the tool φ1. In turn, the link AD is located on the rotary axis MN, which gives a second degree of freedom to the mechanism—the ability to rotate around an axis characterized by the angle φ2. Thus, the mechanism under consideration has two rotational degrees of freedom. In this case, the center of rotation is located at the point O and is removed from the links of the mechanism itself: only the tool passes through this point during operation. The movement of the tool along its own axis, as well as its rotation around this axis, can be done manually or using separate drives and is not considered within the scope of this article.

It can easily be seen that the mechanism is essentially based on a four-bar linkage ABCD (Fig. 1b). Such a mechanism, in fact, is the simplest representative of the class of parallel structure mechanisms, which has the advantages and disadvantages of such mechanisms. One of the main drawbacks is the presence of so-called special positions (singularities), when hitting which, the mechanism loses its degree of freedom or control [10, 11]. In this case, a four-bar linkage, as was shown by Zlatanov [12], can fall into special positions characterized by degeneracy of bonds [13].

Negative effects, such as a decrease in positioning accuracy, loss of rigidity, and increased load on the drives and structural elements of the mechanism, as a rule, begin to appear upon approaching special positions [14, 15], which reduces the actual working area of the mechanism. Since medicine, namely manipulation of surgical instruments, is proposed as the main area of application for the mechanism under consideration, the inadmissibility of the occurrence of the listed phenomena becomes obvious, since each of them can lead to injury or pose a threat to the patient’s life.

One way to deal with special positions is to introduce redundancy into the mechanism, which consists in using only additional drives (drive redundancy) or additional drives paired with auxiliary mobility (kinematic redundancy). It is easy to see that the appearance of additional mobility in the mechanism under consideration would lead to a change in the structure and, accordingly, a loss of synchronization of the movement of the drive link AB and the tool. It follows from this that the most rational solution in this case would be to use drive redundancy.

A MODIFIED MECHANISM WITH DRIVE REDUNDANCY

The simplest way to introduce drive redundancy is to replace any passive kinematic pair with a drive one. In the mechanism under consideration, it seems most rational to replace the kinematic pair D, since this pair is located on the base of the mechanism, which means its physical drive (electric motor) can be conveniently located there. This allows one to avoid increasing the mass of moving parts of the mechanism and the load on them, since there is no need to move the drive during operation. At the same time, it can be seen that the geometry of the mechanism under consideration is far from optimal from the point of view of ensuring its rigidity. Indeed, removal of the center is ensured by the “extended” geometry of the mechanism, which is, in fact, a cantilever beam embedded along the AD link. Reducing the cantilever beam would potentially increase the rigidity of the mechanism without increasing the cross section and, as a consequence, the mass of the mobile links.

To solve this problem, it is advisable to add the simplest Assur structural group with a drive kinematic pair to the mechanism. In this case, the place where such a group is attached to the mechanism should, on the one hand, be as close as possible to the tool and, on the other, not greatly increase the dimensions of the mechanism, since rotation around the MN axis, in any case, will be carried out for the entire structure. As such a solution, this article proposes to use an additional dyad RRR (R denotes a rotational joint) attached to the point E (Fig. 2a).

Fig. 2.
figure 2

Mechanism with an additional dyad: (a) diagram of the mechanism and (b) new four-bar linkage as a part of the mechanism.

Full size image

The additional dyad A'B 'E is located in the rear part of the mechanism and does not operate its main working part (parallelogram CEFG). In this case, the drive pair A' is located on the base of the mechanism. It is worth noting that, at the point E, three links are connected: B ′E, DE, and EF, so theoretically, there are two single-moving pairs at this point. Taking this into account, for the modified mechanism, the Chebyshev structural formula will have the form

$$W = 3n - 2{{p}_{5}} - {{p}_{4}} = 3 \times 7 - 2 \times 10 - 0 = 1.$$

The introduction of an additional RRR dyad made it possible to create an additional circuit A'B 'ED in the structure of the mechanism, which is, in fact, another four-bar linkage (Fig. 2b).

Since the link DE moves synchronously with the link AB, in the four-bar linkage under consideration, it can be considered a drive type. Since the link A'B ' is also a drive element, the drive redundancy of the mechanism is obvious. In this case, the position of the point E depends on the rotation angles of two drive links, and not one, as in the case of the basic mechanism, which potentially makes the mechanism with a dyad more rigid as a whole.

Let us move on to solving the position problem. Since it is noted that the angles θA and φ1 are equal, the task will be to calculate the angle \(\theta _{A}^{'}\) from the given value of θA or φ1. In this case, the calculation scheme for solving the problem can be simplified by considering only the articulated four-bar linkage A'B 'ED (Fig. 3).

Fig. 3.
figure 3

Calculation scheme for solving the position problem.

Full size image

The coordinates of the point E are determined by the length lDE of the link DE and the angle θA

$$\begin{gathered} {{x}_{E}} = {{x}_{D}} + {{l}_{{DE}}}\cos {{\theta }_{A}}, \\ {{y}_{E}} = {{y}_{D}} + {{l}_{{DE}}}\sin {{\theta }_{A}}, \\ \end{gathered} $$
(1)

where xD and yD are known coordinates of the point D.

The coordinates of the point В ' can be expressed similarly through the coordinates of the point A' and the length lA'B' of the link A'B ', as well as the angle \(\theta _{A}^{'}\).

$$\begin{gathered} {{x}_{{B{\kern 1pt} '}}} = {{x}_{{A{\kern 1pt} '}}} + {{l}_{{A{\kern 1pt} 'B{\kern 1pt} '}}}\cos \theta _{A}^{'}, \\ {{y}_{{B{\kern 1pt} '}}} = {{y}_{{A{\kern 1pt} '}}} + {{l}_{{A{\kern 1pt} 'B{\kern 1pt} '}}}\sin \theta _{A}^{'}. \\ \end{gathered} $$
(2)

Using expressions (1) and (2), as well as the constancy of the length lB'Е of the link В ′E, one can write the connection equation

$${{({{x}_{D}} + {{l}_{{DE}}}\cos {{\theta }_{A}} - {{x}_{{A{\kern 1pt} '}}} - {{l}_{{A{\kern 1pt} 'B{\kern 1pt} '}}}\cos \theta _{A}^{'})}^{2}} + {{({{y}_{D}} + {{l}_{{DE}}}\sin {{\theta }_{A}} - {{y}_{{A{\kern 1pt} '}}} - {{l}_{{A{\kern 1pt} 'B{\kern 1pt} '}}}\sin \theta _{A}^{'})}^{2}} = l_{{B{\kern 1pt} 'E}}^{2}.$$
(3)

In a nonsingular position, Eq. (3) will have two solutions that can be obtained analytically [16]. In this case, obviously, a preferable solution would be the one in which the point B ' will be located higher, which reduces the risk of interference between the links A'B ' and B 'E and the parallelogram ABCD.

The location of the point A', as well as the lengths lA'B' and lB′E, determine the permissible angles of rotation of the link DE, and, as a consequence, the tool. In order for the rotation angle to be theoretically unlimited, the following condition must be met for φ1:

$${{l}_{{A{\kern 1pt} 'B{\kern 1pt} '}}} + {{l}_{{B{\kern 1pt} 'E}}} \geqslant {{l}_{{DE}}} + {{l}_{{AD}}}.$$
(4)

In practice, there may be no need for unlimited rotation of the tool, because it is possible to tilt the entire mechanism in the opposite direction by changing the angle φ2. Therefore, condition (4) is not mandatory in a real device.

COMPUTER SIMULATION OF THE MECHANISM

One of the important stages in the development of a real device is computer simulation of the design. In addition to the fact that such a model serves as a starting point for creating the design documentation necessary for manufacturing a product, CAD modeling allows one to evaluate a number of important parameters, such as the size of the working area taking into account design restrictions, loads on drives, etc. For further development, models of the basic mechanism (Fig. 4a) and the mechanism with drive redundancy (Fig. 4b) were created.

Fig. 4.
figure 4

CAD models of mechanisms: (a) basic and (b) with redundancy.

Full size image

After the initial development, a working prototype of the basic mechanism was created to test the main design solutions and test the control system. At the same time, the design was slightly changed and belt drives were used for drives. In addition, the mechanism was balanced (Fig. 5).

Fig. 5.
figure 5

Prototype of the balanced original mechanism.

Full size image

Several four-axis motion controllers have been developed to control the mechanism. One of them, for use in surgery, imitates a laparoscope (Fig. 6a). At the output of the controller, there are four analog signals proportional to the deviation of the handle from the neutral position.

Fig. 6.
figure 6

Motion controllers for use in surgery: (a) manual, (b) pedal.

Full size image

The possibility of freeing the operator’s hands is provided by the pedal motion controller (Fig. 6b). Four foot switches on the platform allow the operator to work either standing or sitting with his feet on the platform; the design makes it possible to stand on the platform without pressing any of the pedals. The four reversible drives are controlled by selecting a combination of the pressed pedals. In this case, these combinations can be configured individually.

The mechanism is supposed to be used as an alternative to the Soloassist system, produced by the German company ActorMed [17], the purpose of which is to control an endoscopic camera during minimally invasive operations.

CONCLUSIONS

This article discusses a mechanism with a remote center of rotation, the main part of which is a flat pantograph. It is shown that such a mechanism can be considered as a four-bar linkage. Flat mechanisms with a remote center of rotation tend to have relatively low rigidity. To solve the problem, it is proposed to introduce drive redundancy into the mechanism through the use of an additional drive dyad RRR. The addition of a dyad forms an additional circuit, equivalent to a four-bar mechanism with two drive links. For such a mechanism, a solution to the position problem is shown. The basic and modified mechanisms are modeled in CAD. In addition, a prototype of the basic belt drive mechanism is presented, for which balancing has been performed.