The control concept can be summarized as follows: The control software receives the desired configuration of the tip as an input; this information is converted by the inverse kinematics module into actuator extensions and sent as a command to each syringe. The syringe pump displacement translates directly to an actuator extension with a known transmission ratio.

The first step for the implementation of this control system is the definition of the inverse kinematic equations. Given the mechanical design of the instrument, a relationship needs to be established between the actuator extensions, hence the joint rotations, and finally the tool tip position. The equations will be derived by applying inverse kinematic theory for both mechanisms: the serial chain and the internal mechanisms. The relationship between tool tip position and joint rotations can be established by applying to the serial chain the inverse serial kinematic method of Denavit-Hartenberg (DH). The relationship between the joint rotations and the actuator extensions can be determined by applying a geometrical method to the internal mechanism. The details of the derivations are described in the following paragraphs.

Figure 2 Image showing the instrument’s tip from two sides: a frontal face and a view of the instrument rotated by 90 degrees. The frame’s position and orientation, placed on the joints accordin to the Denavit-Hartemberg convention, are also defined.

Table 1 Denavit-Hartemberg Parameters, defined for finding the relationship between instrument tool tip position and joint rotation.

In order to find the relationship between tool tip position and joint rotations, reference frames of the mechanism’s joints were affixed according to the Denavit-Hartemberg convention [10], as shown in Fig. 2 and the DH parameters were defined as shown in where:

– *a* is the distance between z_{i} and z_{i+1} measured along x_{i};

– *α* is the angle from z_{i} to z_{i+1} around x_{i};

– *d* is the distance between x_{i−1} and x_{i} measured along z_{i};

– is the angle from x_{i−1} to x_{i} around z_{i};

Using these parameters it is then possible to define the homogeneous transformation matrix ${}_{30}T$.

$$\begin{array}{rl}{}_{3}^{0}T& {=}_{1}^{0}A{\times}_{3}^{2}A\\ & =\left(\begin{array}{cccc}c{\theta}_{1}c{\theta}_{2}& c{\theta}_{1}s{\theta}_{1}& s{\theta}_{1}& -c\times c{\theta}_{2}c{\theta}_{1}-b\times c{\theta}_{1}a\\ s{\theta}_{1}c{\theta}_{2}& -s{\theta}_{2}s{\theta}_{1}& -c{\theta}_{1}& -c\times c{\theta}_{2}s{\theta}_{1}-b\times s{\theta}_{1}\\ s{\theta}_{2}& c{\theta}_{2}& 0& -cs{\theta}_{2}\\ 0& 0& 0& 1\end{array}\right)\end{array}$$$${}_{1}^{0}A=\left(\begin{array}{cccc}1& 0& 0& a\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)$$(1)$${}_{2}^{1}A=\left(\begin{array}{cccc}\mathrm{cos}{\theta}_{1}& 0& \mathrm{sin}{\theta}_{1}& b\mathrm{cos}{\theta}_{1}\\ \mathrm{sin}{\theta}_{1}& 0& -\mathrm{cos}{\theta}_{1}& -b\mathrm{sin}{\theta}_{1}\\ 0& 1& 0& 0\\ 0& 0& 0& 1\end{array}\right)$$(2)$${}_{3}^{2}A=\left(\begin{array}{cccc}\mathrm{cos}{\theta}_{2}& 0& -\mathrm{sin}{\theta}_{2}& c\mathrm{cos}{\theta}_{2}\\ \mathrm{sin}{\theta}_{2}& 0& \mathrm{cos}{\theta}_{2}& -c\mathrm{sin}{\theta}_{2}\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)$$(3)The last column of the matrix (1) represents the vector that links the tip reference frame and the origin. Its components are defined as:
$$\left\{\begin{array}{l}{p}_{x}=-c\mathrm{cos}{\theta}_{2}\mathrm{cos}{\theta}_{1}-b\mathrm{cos}{\theta}_{1}-a\\ {p}_{y}=-c\mathrm{cos}{\theta}_{2}\mathrm{sin}{\theta}_{1}-b\mathrm{sin}{\theta}_{1}\\ {p}_{z}=-c\mathrm{sin}{\theta}_{2}\end{array}\right.$$(4)

– a is the distance between reference frame and joint 1;

– b is the distance between joint 1 and joint 2;

– c is the distance between joint 2 and the tip frame;

px; py and pz define the desired position in the reference coordinate system of one of the two pliers jaws;

*θ*_{1} and

*θ*_{2} are respectively the rotation angles of joints 1 and 2. The system

equation 5 can be, with the help of

Fig. 2, easily adapted for the second pliers jaw.

Hence from equation (4) it is possible to deduce the relationship between end-effector jaw position and joint rotations:

$${\theta}_{1}=\mathrm{arcsin}\left(-\frac{{p}_{y}}{c\mathrm{cos}{\theta}_{2}}\right)$$(5)$${\theta}_{2}=\mathrm{arcsin}\left(-\frac{{p}_{z}}{c\mathrm{cos}{\theta}_{2}+b}\right)$$(6)With the relationship between tool tip position and joint rotation defined, a relationship can be established between the joint rotation and actuator extensions from the inverse kinematics of the internal mechanism. A schematic of this mechanism is shown in Fig. 3.

Figure 3 Schematic diagram showing the parallel kinematic mechanism.

Defined as: *h*_{p} the x coordinate of the joint; L_{a}, L_{b} and, L_{c} the segment dimensions of the parallel kinematic; *θ*_{b1} and *θ*_{b2} the angle between the first and the second segment of the two branches of the mechanism; *θ*_{c1} and *θ*_{c2} the angle between the third segment of the two mechanism branches and a horizontal line; x_{1} and x_{2} the distance between piston and cylinder base.

A geometrical method is used to establish the following kinematic relationships:

$${\theta}_{b1}=\mathrm{arcsin}\left[\frac{1}{{\text{L}}_{\text{b}}}({\text{L}}_{\text{c}}\mathrm{cos}{\theta}_{\text{c1}}-{\text{L}}_{\text{d}})\right]$$(7)$${\text{x}}_{1}={\text{h}}_{\text{p}}-{\text{L}}_{\text{a}}-{\text{L}}_{\text{b}}\mathrm{sin}\left[\mathrm{arcsin}\left(\frac{{\text{L}}_{\text{c}}}{{\text{L}}_{\text{b}}}\mathrm{cos}{\theta}_{\text{c1}}-\frac{{\text{L}}_{\text{d}}}{{\text{L}}_{\text{b}}}\right)\right]-{\text{L}}_{\text{c}}\mathrm{sin}{\theta}_{\text{c1}}$$(8)$${\theta}_{\text{b2}}=\mathrm{arccos}\left[\frac{1}{{\text{L}}_{\text{b}}}({\text{L}}_{\text{c}}\mathrm{cos}{\theta}_{c2}-{\text{L}}_{\text{d}})\right]$$(9)$${\text{x}}_{2}={\text{h}}_{\text{p}}-{\text{L}}_{\text{a}}-{\text{L}}_{\text{b}}\mathrm{sin}\left[\mathrm{arccos}\left(\frac{{\text{L}}_{\text{c}}}{{\text{L}}_{\text{b}}}\mathrm{cos}{\theta}_{\text{c2}}-\frac{{\text{L}}_{\text{d}}}{{\text{L}}_{\text{b}}}\right)\right]{\text{+L}}_{\text{c}}\mathrm{sin}{\theta}_{\text{c2}}$$(10)$${\theta}_{\text{c2}}+\phi -{\theta}_{\text{c1}}=\pi $$(11)The identities resulting from the serial and internal mechanisms are linked by the following geometrical relationship:

$${\theta}_{\text{c1}}={C}_{\theta}+{\theta}_{1}$$(12)Where *C*_{θ} is a constant angle due to the parallel kinematic structure (*C*_{θ} is the value of *θ*_{c1} when x_{1} = x_{2}). (12) Defines that a rotation of the joint corresponds to an equal change of *θ*_{c1} from its equilibrium position. It can be concluded that once the end-effector position (p_{x}; p_{y} and p_{z}) is defined, *θ*_{1} and *θ*_{2} can be calculated from (5) and (6). Knowing *θ*_{1}, the orientation of joint 1, it is possible to calculate *θ*_{c1} from (12) and then *θ*_{c2} from (11). At last *θ*_{c1} and *θ*_{c2} can be respectively substituted in (8) and (10), obtaining x_{1} and x_{2}, the piston extensions.

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