This paper presents a new method to calculate and display an approximated workspace of a surgical robot in nearly realtime. Displaying this information on a screen in the operation room could support the surgeon during intraoperative trocar placement for teleoperated minimally invasive robotic surgery (MIRS). We give a short overview on existing trocar placement procedures in teleoperated MIRS and describe the possibilities and limitations of workspace analysis methods to support the surgeon during trocar placement. Our new method uses MIRS-specific simplifications to reduce the search space and enable the creation of a reduced workspace map. It was implemented for the DLR MiroSurge system. The implementation can create a reduced workspace map and display a mesh representation of the map in less than 2 seconds. We give a short outlook on how this method could be embedded in trocar placement procedures in the operation theater and what our future plans are with this method.
In figure 3a) it can be seen that the TCP can maximally reach the outline of a sphere HS1 if the last three joints (roll, pitch, yaw, see figure 3a)) are restricted to the zero position. This sphere is defined by the length of the instrument Linstr. By allowing only a movement in the last three joints the system can move the TCP maximally on the surface of the spherical sector HS2, defined by the joint limits of the instrument [5]. For planning autonomous tasks, it is advantageous to have a WMrob approximating the whole volume of Wrob translationally and rotationally. A map WMrob(Ttroc) for MIRS does not have to cover all this information. We assume that with some experience a surgeon can approximate the rotational part of Wrob(Ttroc) through the movability of the last three joints. Regarding the translational part the important information is the outer border of Wrob(Ttroc). Therefore we define the search space to create WMrob(Ttroc) as a discrete space of TCP poses TTCP (tTCP, ,RTCP), which meet the following restrictions: (eq.1
To create WMrob(Ttroc) we propose an algorithm that discretizes WMrob(Ttroc) into circular discs S (figure 3b),c)). The discs are defined as orthogonal to z and with a distance of res to each other, their radius measures r(z, Linstr). The steps described in the following are done for all discs Sl (figure 3c)).
In step1, the outer border of Wrob(Ttroc) within disc Sl has to be found. Thereby the algorithm starts from a zero position TTCP,l,zero, searching in the positive x-direction with steps of res until it reaches r. The last change from reachable to not reachable is identified as the outer border of Wrob(Ttroc) and marked as TTCP,l,start. If during step1 two borders are found, Wrob(Ttroc) includes a reachability hole within Sl. In this case the algorithm deletes all previously found TCP poses from WMrob(Ttroc) and continues with the next disc Sl+1. This is done to gain a conservative estimation of the workspace by only representing an outline of a fully reachable volume in WMrob(Ttroc) . To increase the chance of finding reachability holes, step1 can be repeated with different search directions. In step 2 all reachable TCP poses T TCP,l,i along the border of Wrob(Ttroc) within Sl are identified. Thereby the global search direction is anticlockwise, which means that the border of Wrob(Ttroc) is always assumed on the right side. Starting with the TCP pose TTCP,l,i which is left of TTCP,l,start the algorithm will always turn its local search direction 90° clockwise if the current TTCP,l,i is reachable. If the algorithm comes to a TTCP,l,i which is not reachable, it will step back to the last reachable TCP pose TTCP,l,i-1 and turn its local search direction 90° anticlockwise.
The difference in calculation complexity between this method and a brute force approach can be compared as followed. For this method the function to gain the number of necessary inverse kinematics calculations finvkin is defined as f1invkin(n1, n2, n3) = (n1+n2)n 3. Thereby n1 and n2 are the amounts of TCP poses for which the inverse kinematics have to be calculated within the described steps 1 and 2 and n3 is the amount of the discs Sl . For n1 and n3 we can define n1=n3=Linstr /res (as a simplification we set r = Linstr), but for n2 only an approximation for the maximal number of calculations n2,max,approx can be defined. As can be seen in figure 3c), the algorithm has to calculate the inverse kinematics for approximately every TTCP,l,i on the inside and on the outside of the border of Wrob(Ttroc) within Sl. The maximal borderlength is approximated with 2L instr. Because n1, n2 and n3 all depend on Linstr and res, the equation yields: (eq.4) f 1invkin Linstr , res
res 2 2 Linstr res
res
We created workspace maps WMrob(Ttroc) for three different trocar points with the parameters res = 0.01 m, Linstr = 0.2 m. Table 1 shows meshes of the workspace maps, the amount of inverse kinematics calculations ninvkin, the amount of points identified as reachable nWM, and the mean overall time tcalc of one calculation (Prozessor: Intel(R) Xeon(R) CPU W3530, 2.80GHz, 6GB main memory). The mean was determined over 100 repetitions. The number of inverse kinematics calculations stayed under the maximum of 5426, calculated with (eq. 4). The time tcalc was always smaller than 1 s. As a calculation of one inverse kinematics solution of our system takes about 20 s [1], only around 10 % of tcalc is caused by the inverse kinematics calculations. The remainder of tcalc is used by the algorithm for other operations. In the future we hope to optimize this remainder of tcalc to gain faster calculation times. The overall time to create and mesh WMrob(Ttroc) was measured between 1-2 s, which allows to create the map online in the operation room. The mesh to display WMrob(Ttroc) is done with an algorithm that deforms the mesh of a half sphere and which is implemented in open scene graph. As can be seen in table 1, the meshing quality is not optimal and will be improved by a nearest neighbour algorithm. The use of displaying WMrob(Ttroc) during trocar placement, is shown in figure 4. Here, the surgeon uses the robot in the hands-on-mode [4] to measure the position of the desired trocar Figure 4: Installation of pploaiynet.d Tinhea vcairltcuuallatceodpyWoMf rtohb(eTstrcoce)necaton ebvealudaitseone possible scenario: e.g. the overlap of WMrob with a registrated organ. The surgeon checks on a If the overlap is not satisfying for the intended screen if the desired trocar task, the procedure can be repeated until a suitais suitable to reach e.g. a registrated organ. To measure posi- ble trocar point is found. Thereby the risk of comtion of the trocar he uses the robot in the hands-on-mode. ing across workspace borders which are caused by wrongly chosen trocar points might be reduced. We will evaluate the use of the described method for trocar placement with surgeons within the next six months. Moreover we will use the map for setup optimization methods and augmentation of the endoscopic picture.