RPQ: ROBOTIC PROXIMITY QUERIES

Development and Applications

Albert Hernansanz, Xavier Giralt

Research Group On Intelligent Robotics and Systems, Technical University of Catalonia, 08028 Barcelona, Spain

Alberto Rodriguez

Robotics Institute, Carnegie Mellon University, Pittsburgh, PA 15213, USA

Josep Amat

Institute of Robotics and Industrial Informatics, Technical University of Catalonia, 08028 Barcelona, Spain

Keywords:

Collision detection, Proximity queries, Surgical application.

Abstract:

This paper presents a robotic proximity query package (RPQ) as an optimization of the general collision

library PQP (Proximity Query Package) for the detection of collisions and distance computation between

open kinematic chains such as robotic arms. The performance of the o ptimizations to non speciﬁc collision

query packages are explained and evaluated. Finally, a robotic assisted surgical application is presented which

has been used as a test bed for the proximity package.

1 INTRODUCTION

One of the most important problems to solve in

robotics is the collision avoidance between a robot

and its environment. A robot should perceive the risk

and have a reactive behavior before an imminent col-

lision occurs. Path planning is a hard computational

problem, so having a fast tool to calculate collisions is

a key factor to decrease the necessary time to gener-

ate safety trajectories. In applications where no path

planning exists, for instance in manual guidance or

teleoperation, a real time collision detector is needed

so as to avoid collisions and to be able to interact with

the environment, for example sliding over a surface.

The knowledge of minimum distances between

robots or objects that share a workspace enables

robots to behave in a predictive way. In the human-

robot interaction ﬁeld, virtual ﬁxtures can be used

both to prevent collisions and help the human oper-

ator by increasing his performance (Stanisic et al.,

1996). In this kind of applications minimum distance

and collision detection must be known in real time.

A new library: Robotic Proximity Queries (RPQ)

package (Giralt and Hernansanz, 2006) has been

developed to deal with these requirements, using

PQP (UNC, 1999) as the a proximity query engine.

The original package has been used to optimize the

queries when working with open kinematic chains,

like robots. These optimizations have been done with

the aim of improving the time performance of the

generic package and simplifying its use in robotic en-

vironments. A system composed of two robots has

been used as a test bed to show the performance of

the RPQ library.

Finally a robotic assited surgical application that

beneﬁts from RPQ performance is presented. The

application consists of the execution of an assisted

cut of a rigid tissue. The surgeon guides freely the

driller held by a slave robotic arm that avoids unde-

sired drillings by means of virtual protections. With

this application not only proximity queries are shown,

but also the graphical interface and the use of a virtual

robot based on RPQ. More information and videos are

available at http://grins.upc.edu

2 RELATED WORK

During the last years, great efforts have been devoted

to the development of efﬁcient collision detection al-

gorithms due to their wide range of applications, such

as CAD/CAM, manufacturing, robotics, simulation

Hernansanz A., Giralt X., Rodriguez A. and Amat J. (2007).

RPQ: ROBOTIC PROXIMITY QUERIES - Development and Applications.

In Proceedings of the Fourth International Conference on Informatics in Control, Automation and Robotics, pages 59-66

DOI: 10.5220/0001629100590066

 SciTePress

and computer animation. A wide study of the perfor-

mance and applicability of such methods can be found

in (B.Geiger, 2000). Proximity query algorithms vary

in terms of their range of applicability and the type

of queries they can solve, mainly collision detection,

minimum distance computation and interpenetrations

modelling. Although most algorithms allow as input

triangulated meshes of 3D points, they differ in the

way those points are pre-processed and represented

internally in order to speed up speciﬁc queries.

There is a wide set of methods that rely on Lin-

Canny or Gilbert-Johnson-Keiethi like algorithms for

computing minimum distances between pairs of ob-

jects as I-Collide, Swift, Swift++, SOLID, DEEP. . .,

but they are only applicable to convex polytopes

(S.Ehmann and Lin, 2000; S.Ehmann and Lin, 2001;

Kim et al., 2002; Bergen, 2002). This restriction

makes them inappropriate for RPQ purposes, due to

the need to deal with more general geometric models.

More general collision detection methods usu-

ally base their efﬁciency on pre-computed represen-

tations by means of hierarchies of bounding volumes.

Their differences rely on the speciﬁc type of bound-

ing volumes used, ranging from binary space decom-

positions, spheres trees to oriented bounding boxes

(OBB).

Among this set of algorithms, RAPID and PQP

turn to be those that have both, fewer restrictions

in the range of allowable geometric models and an

easier application programming interface (API). Both

of them use oriented bounding boxes for performing

collision tests, and have similar time performances.

However, the fact that PQP offers a wider range

of queries, including minimum distance computation

and tolerance tests makes PQP the best option for the

proximity queries engine of RPQ, the Robotics Query

Package presented in this paper.

3 LIBRARY DESCRIPTION

The goal of the Robotic Proximity Queries (RPQ) li-

brary is to offer an easy, modular and fast proxim-

ity query package oriented to robotics. As explained

above, the aim of the project was not the development

of a new collision detector, but specialize an existing

one into the robotics ﬁeld.

As described in section 2, there is a wide set of

general purpose proximity query packages. The crite-

rions used to choose PQP as the best candidate for the

development of RPQ are:

1. Types of proximity queries available.

2. High time performance on proximity queries.

3. Ability to use geometrical models based on trian-

gulated meshes of points.

4. Lack off restrictions on possible geometrical mod-

els.

The PQP library has been developed by UNC Re-

search Group on Modelling, Physically-Based Simu-

lation and Applications and offers three different kind

of queries:

- Collision detection: detecting whether two mod-

els overlap, and optionally, give the complete list

of overlapping triangle pairs.

- Distance computation: computing the minimum

distance between a pair of models.

- Tolerance veriﬁcation: determining whether two

models are closer or farther than a given tolerance

distance.

RPQ has been implemented in C++ language

and its graphical interface has been developed using

OpenGL. The RPQ library can be easily integrated

into any software application.

The library interface allows non expert program-

mers to use it in an easy manner. The graphical inter-

face is a separate module, allowing the programmer

to decide whether using it or not. Fig. 1 shows the in-

tegration of the library and its graphical interface into

a generic application.

3.1 Class Description

The RPQ library is based on the Object Oriented

paradigm. Focused on this paradigm, and based on

robotic environments, three main classes have been

developed: Scenario, Object and Robot.

3.1.1 Scenario

Scenario is the workspace where the objects cohabit.

Concerning its implementation, Scenario is a class

that contains all the objects (Robots and generic ob-

jects), a global reference frame, and all the methods

necessary to generate the proximity query.

3.1.2 Object

An Object is the minimum entity that exists in a Sce-

nario. There are two types of Objects: simple and

complex. A simple Object is represented by a geo-

metrical model composed of a set of triangles referred

to a frame tied to the Object. The Object has also a

transformation matrix to refer itself to the world refer-

ence frame. A complex Object is an Object composed

of a set of geometrical models with joints (rotational

ICINCO 2007 - International Conference on Informatics in Control, Automation and Robotics

or prismatic) between them. Thus, a complex Ob-

ject is an open kinematic chain composed of sub ob-

jects. The transformation matrix M

refers subobject

to subobject

i−1

. The transformation matrix M

refers

the object base (subobject

) to the world. The object

stores its own geometrical model. Concerning its im-

plementation, an Object is a class containing the geo-

metrical model, the transformation matrix and a set of

methods to position and to orient itself in space. This

class also contains methods to calculate the different

detail representations of its geometrical model.

3.1.3 Robot

A Robot is a particularization of a complex Object

where each of its links is represented by a simple Ob-

ject. A Robot has a set of functions to make a com-

plex Object as similar as possible to a real robot. For

instance, the spatial relationship between links is de-

scribed using the Denavit-Hartenberg notation. Direct

and inverse kinematics can be calculated considering

the robots own restrictions (joint limitations, conﬁg-

urations, etc). Concerning implementation, the class

Robot is derived from the class Object. Robot adds all

the functions that are necessary to control a robot. For

instance joint positioning of the robot (direct kinemat-

ics), position and orientation of its tool center point

(inverse kinematics), change of the robot conﬁgura-

tion, joints overshoot . . . These added functions with

respect an Object are very helpful when a new robot is

created or used in robotic applications like simulators,

path planners, etc.

Figure 1: Schema of integration of RPQ into a generic ap-

plication.

3.2 Optimizations

PQP is a generic package that does not use the knowl-

edge of the object’s kinematics. In contrast, RPQ is

oriented to robotics, and the knowledge of robot’s

kinematics is the base of the optimizations that spe-

cialize it. RPQ is designed to answer proximity

queries between two robots or between a robot and

any kind of rigid object.

Three optimizations have been developed and

tested to improve the performance offered by PQP:

• Different resolution levels of object’s representa-

tion

• Collision queries sorted using a Weight Matrix

• Collision Matrix

3.2.1 Different Resolution Levels of Object’s

Representation

Objects can be represented in very different resolu-

tion levels. The idea of this optimization is to use the

simplest representation models (minimum number of

triangles) to discard collisions. The lower the number

of triangles of the geometric model are, the faster the

collision queries are executed.

Three resolution levels are used to represent

robots and two for the rest of objects. The highest res-

olution level is the complete geometrical model. The

second level is the oriented bounding box (OBB) of

each sub object in which a complex object is divided.

The lowest resolution level is the bounding box of the

whole complex object. This level is only deﬁned for

complex objects with more than a sub object, as in

robots with several links. There are other possible in-

termediate resolution levels that can be used, for in-

stance the convex hull. It offers a good ratio between

resolution and the quantity of triangles, although it

does not reduce it as drastically as the low resolution

levels chosen.

This optimization is useful in two different situa-

tions. First, in applications where no high precision is

required, for instance when the precision of the OBB

or the convex hull of each link is enough. The second

situation occurs when the different resolution levels

are used in a complementary manner.

Figure 2: Robot with three resolution level representation:

The geometrical model of each link (L3), the OBB of each

link (L2) and the whole bounding box(L1).

When a collision query is performed, a low to high

resolution level list of collision queries is generated.

RPQ: ROBOTIC PROXIMITY QUERIES - Development and Applications

Starting with the lowest resolution level, queries are

generated until any collision can be completely dis-

carded. For instance, if a possible collision between

two 6-DOF robots is studied, the ﬁrst query is done

between the bounding boxes of each robot. If the col-

lision can not be discarded, then the bounding box of

each link is used. If at this level collisions still can not

be discarded, the geometrical models of each link are

checked. As shown in Fig. 2 a Robot with its three

resolution levels of representation.

A test has been developed for the evaluation of

the performance of the optimizations. It consists on

a couple of virtual robotic arms (Staubli RX60B) that

are placed one close to the other in a scenario. The

geometrical models used for the test are high resolu-

tion (composed of 23012 triangles each). The robots

are placed at a variable distance between them and

the scenario is checked for collisions for an equidis-

tributed set of joint positions in their 6 dof.

This test allows us to study the dependency on the

performance of the proposed improvements in terms

of the probability of collision. This is because, as

shown in Table 1, in the designed test, the closer the

robots are, the greater the number of joint conﬁgura-

tions that result in collision.

Table 1: Dependence of the amount of colliding conﬁgura-

tions on the distance between robots.

Dist. Joint conﬁg. Collis. Not Collis.

robots(m) checked

0,4 9216 4864 4352

0,5 9216 3530 5686

0,6 9216 2500 6716

0,7 9216 1121 8095

0,8 9216 140 9076

Fig. 3 shows the consequences of using differ-

ent resolution levels. When the distance between the

robots increases, the queries solved with the bounding

box of the robot increases, and consequently the time

to solve a collision query between the robot decreases.

If the distance decreases, the best combination is us-

ing levels two and three or only level three.

3.2.2 Collision Queries Sorted using a Weight

Matrix

This optimization is based on two assumptions. The

ﬁrst one is that the goal of the query is just to know

whether there is a collision or not, but not the number

of them. The second assumption is that the kinematics

and the morphology of the robots are well known.

Given these assumptions, the objective is to ﬁnd

quickly whether there is collision or not, by means

of minimizing the number of partial collision queries.

Figure 3: Time to solve a collision query between two

Staubli RX60B robots using different resolution levels. L3:

Geometrical model of each link. L2: The OBB of each link.

L1: Bounding box of the robot.

The knowledge of the kinematics and the morphol-

ogy of the robots gives us the possibility of assign-

ing an a priori collision probability to each link of

the robot with respect to the rest of robots and ob-

jects present in the same workspace. During exe-

cution time, these probabilities are automatically up-

dated depending on the result of the collision queries

(Probability increases in case of detecting a collision

and decreases otherwise). Therefore, a weight matrix

C is generated combining the probability of collision

between each pair of objects in the workspace. Each

component c

∈ C veriﬁes c

= P

+ P

where

and P

are the assigned probability of collision of

Object

and Object

respectively. These weights de-

termine the order of the collision queries, that is if

> c

the collision query between Object

and

Object

will be generated before Object

and Object

A simple way to assign a collision probability to

the links of a robot is to assign higher probability to

those links that are farther in the kinematic chain, with

respect to the base of the robot.

3.2.3 Collision Matrix

The Collision Matrix is a binary matrix that reﬂects

the possibility of collision between two objects. If

the collision matrix indicates that a collision between

two objects is impossible, its correspondent collision

query is not performed. Of course, a matrix query is

much less expensive than a collision query in compu-

tational terms.

This optimization improves the performance of

the system when a high number of collision queries

are discarded by the Collision Matrix. Computation-

ally, this condition can be expressed as in equation

(1):

ICINCO 2007 - International Conference on Informatics in Control, Automation and Robotics

n · QC > m · QM + k · (QC + QM ) (1)

with n = m + k

where:

n Total number of queries.

m Queries resolved with the Collision Matrix.

k Queries resolved with the Query Collision.

QC Average time to solve a Query Collision.

QM Time to solve a query with the Col. Matrix.

The performance of the Collision Matrix has been

studied using the same test designed for testing the

use of different levels of representation. The results

are shown in Fig. 4. The farther the robots are,

the lower is the number of links that can collide, as

seen in Table 2. Therefore, the higher the number of

queries that are solved with the Collision Matrix. As

it is shown in the ﬁgure, using the Collision Matrix

the number of collision queries decreases, so does the

time to solve the query.

Table 2: Dependence of the percentage of collision queries

solved by the collision matrix on the distance between

robots.

Dist. between robots (m) Pairs solved with CM

0,4 4,94

0,6 9,88

0,9 24,69

1,2 64,20

Figure 4: Time necessary to solve a collision query between

two Staubli RX60B robots using the Collision Matrix (CM)

or not (NoCM).

Each one of the optimizations proposed improves

the performance of the original library, PQP. How-

ever, the combined use of all of them improves even

more the global performance of the application.

The complete algorithm with all three optimiza-

tions is shown in Fig. 5. First of all, a query colli-

sion between the whole bounding box of both robots

is performed. If at this level the collision cannot be

solved then it is necessary to study collisions among

the whole set of links of both robots. The order in

which these queries must be performed is given by the

Weight Matrix. The query ﬁnishes as soon as a colli-

sion appears between a pair of links either in the sec-

ond or third level, or when all pairs have not reported

any collision. For each pair of links, the second and

third representation levels are studied consecutively,

so if a collision is detected in the second level, the

third level has to be studied as well.

Figure 5: Algorithm for collision detection between two

robots using all the optimizations.

4 APPLICATION

One of the advantages of making RPQ generic (al-

though it is optimized for robots) is that this library

can be applied in a wide set of applications. In this

paper a robotic assisted surgical application is pre-

sented. This application has the aim of helping the

surgeon to make a cut on a rigid tissue, for example

in the cranium, avoiding undesired collisions between

the patient and the driller. The surgeon guides freely

the driller that is held by the robot acting in a pas-

sive mode as seen in Fig. 6, allowing all movements

except those which produce undesired collisions.

The system is composed by a Staubli RX60B

robotic arm and a driller. Between the robot and the

RPQ: ROBOTIC PROXIMITY QUERIES - Development and Applications

Figure 6: Robotic assisted surgical application.

driller there’s a ATI Multi-Axis Force/Torque Sensor.

The system transforms forces and torques generated

in the driller by the surgeon in new destination points

were the robot must go.

The geometric model of the patient’s cranium is

obtained transforming Computer Tomography data

into a geometrical model composed of triangles.

The surgeon can deﬁne the cutting area in the pre-

operative phase and, as will be explained latter, virtual

ﬁxtures are automatically generated to protect the pa-

tient in the operative phase. The possibility of intro-

ducing virtual objects in the scene and the interaction

between them is one of the key factors of RPQ.

The surgeon has a friendly and easy-use graphical

interface that allows the navigation over the virtual

system. This graphical interface helps the surgeon

not only in the pre-operative phase but also during the

surgical operation, providing an augmented reality vi-

sual feedback (collisions, minimal distance between

the cranium and the tool, different points of view of

the scene, virtual ﬁxtures ).

4.1 Surface Navigation

The library developed is useful not only to avoid colli-

sions but also to navigate over an object’s surface. For

instance, the robot tool slides over the surface of the

virtual shield described in section 4.2. This surface

navigation allows the surgeon to feel smooth move-

ments of the robot when the tool is in contact with the

virtual ﬁxtures. The navigation algorithm helps the

surgeon not only avoiding the forbidden regions de-

ﬁned in the pre-operative phase but also guiding him

to the desired cutting path.

The navigation algorithm modiﬁes the position of

the robot tool, but not its orientation. The algorithm is

based on three steps: Knowing the new desired des-

tination of the robot tool, the ﬁrst step consists of

detecting all collisions between the tool and the ob-

ject’s surface. When all collisions are detected, the

second step consists of projecting the desired point

to the plane of the collision triangle, Fig. 7.a. Fi-

nally the projected point that is closer to the desired

one is selected. A problem can occur when the pro-

jected point falls outside the triangle region. In this

situation it is not possible to ensure that this new pro-

jected point is always be outside the object Fig. 7.c.

In this case the new point is projected to the perime-

ter of the triangle. To accomplish this, the outside

region of the triangle is divided into six new regions

(R1, R12, R2, R23, R3, R31), which are deﬁned by

the normals of the edges of the triangle applied to the

vertices of the triangle. The point is then projected

to the closest point of the triangle border of its region

Fig. 7.b. Now, the new destination point is collision

free Fig. 7.d.

Figure 7: Different contact situations between the tool and

a triangle or set of triangles.

4.2 Virtual Protections

Virtual Protections are constraints that rule the be-

haviour of the robot that are speciﬁcally designed

to prevent motion into a forbidden region of the

workspace. In this work, the surgeon guides freely

a driller held by the slave robot. The main idea is to

develop a system helpful for the surgeon that prevents

undesired drillings.

4.2.1 Strategy Description

The main goal of the system is to give the robot a re-

active behaviour by means of the deﬁnition of virtual

objects in the PQP Scenario with two objectives:

• Protect the patient from the robot and the driller.

• Help the surgeon to locate the desired cutting area.

PQPs ability to check collisions in real time allows

us not only to achieve these objectives but to operate

in an efﬁcient manner.

Throughout a simple interface, in the pre-

operative phase, the surgeon speciﬁes the exact loca-

tion and shape of the cut that must be done. With that

ICINCO 2007 - International Conference on Informatics in Control, Automation and Robotics

information, the system generates a shield that covers

the patient while it adapts perfectly to the target area

deﬁned by the surgeon, as shown in Fig. 8.

The proposed behaviour is achieved by the navi-

gation algorithm exposed in chapter 4.1. The system

gives the surgeon the conﬁdence of knowing that the

robot will never let him approach the patient in a non-

desired area. However, while the surgeon does not

attempt to cross the shield, the reobot can be moved

freely.

4.2.2 Shield Generation

The problem here is to generate a surface that con-

nects the inner polygonal curve deﬁned by the sur-

geon with an outer curve. The connection between

them is done in a smooth way, as Fig. 8 illustrates.

Figure 8: Lateral view of a face and the virtual shield.

The surface, s(u, v), is parametrized as the upper

part of a regular torus, but adapting its equators to the

inner and outer curves, as in (4).

Inn. curve i (t) = (i

(t) , i

(t)) t ∈ 0..2π(2)

Out. curve o (t) = (o

(t) , o

(t)) t ∈ 0..2π(3)

s(u, v) =



sin





(u) + cos





(u) ,

sin





(u) + cos





(u) ,

H sin (v)



u ∈ 0..2π v ∈ 0..π (4)

There are peculiarities of the shield to consider:

• The shield should be smooth, in order to make

navigation over it as comfortable as possible.

Once the surface is generated it is transformed

into a triangle model. By augmenting or decreas-

ing the number of triangles, one can vary the

smoothness of the surface.

• Reaching the inner curve from the surface it has

to be easy and intuitive. This implies certain re-

strictions on the shield design. This property de-

pends on the way inner points are connected to

outer points. First of all inner and outer curves

have to be parametrized, (2) and (3). There are

some restrictions regarding them in order to make

the surface easy and intuitive.

- i (t) and o (t), must be parametrized radially,

from an inner point, with constant radial veloc-

ity, in order to avoid self intersections of the

surface, as in Fig. 9 a) and b).

- The polygon deﬁned by i (t), must be star-

shaped from the parametrization origin, in or-

der to avoid self intersections of the surface, as

in Fig. 9 c) and d).

- If i (t) deﬁnes a non star-shaped polygon a pre-

vious step must be done. An adaptation disc be-

tween the polygon and its convex hull is needed

before constructing the shield, so that the latter

can be parametrized radially from a suitable in-

ner point, as in Fig. 10.

Figure 9: a) Non valid parametrization of curves (self inter-

section of the surface). b) Valid parametrization of curves.

c) Non valid point for radial parametrization of curves.

Polygon is not star-shaped from that point. d) Valid point

for radial parametrization of curves.

Figure 10: Adaptation disc between an non star-shaped

polygon and its correspondent 2D Convex Hull.

• H, the height of the shield, must be chosen in or-

der to protect the patient.

Once the required smoothness for the surface is de-

ﬁned, the parametrization for i (t) and o (t), solved

the problem with non star-shaped polygons and cho-

sen a desired height, the surface is generated by eq.

(4).

Finally, the intersections of the discrete set of

geodesics u =

2π

n and v =

n for n ∈ 1 . . . N is

triangulated in order to obtain the geometrical model.

The parameter N allows us to choose the number of

points and triangles of the surface, and ultimately its

smoothness.

RPQ: ROBOTIC PROXIMITY QUERIES - Development and Applications

4.2.3 Cutting Aid

Besides the shield generated, another virtual object is

proposed in order to aid the surgeon to carry out the

cut with great accuracy. It consists of an inner cover

that prevents the driller from escaping the perimeter

of the shape deﬁned.

Since (2) has the shape of the cut to be done, we

deﬁne

i (t) as an under scaled version of i (t).

i (t)

has the same shape as i (t) but its sides are placed at

a distance ∆x from i (t) sides, being ∆x the desired

width for the cut.

Therefore, a new shield is generated, bs(u, v), hav-

ing a semisphere shape, with

i (t) as its equator,

parametrized by (5).

s(u, v) =

= H cos

(v) cos (u) + sin

(v)i

(u) sin (v),

H cos

(v) sin (u) + sin

(v)i

(u) sin (v),

H cos (v) u ∈ 0..2π v ∈ 0..

(5)

5 CONCLUSION

This paper presents the development of RPQ, a prox-

imity queries library optimized for applications where

robots share a common workspace and interact with

objects. Due to the amount of collision detection

packages, RPQ is built above a generic collision pack-

age, PQP. It is the generic collision package that better

ﬁts RPQ purposes.

The goal of developing RPQ was to ﬁll an exist-

ing gap in computer tools for robotic applications,

where robots interact with objects in their environ-

ment. Three optimizations have been performed to a

generic collision library: working with different reso-

lution levels of representation, the use of a weighted

matrix for choosing the best order for collision check-

ing and the deﬁnition of a binary matrix that deter-

mines the possibility of collision between objects.

RQP has been validated in different applications such

as multirobot collision avoidance, virtual robot con-

troller and surface navigation.

As expected, optimizations improve the time per-

formance of the system, although this improvement is

highly application dependent.

The introduction of different levels of resolution

in the geometric models of the objects and robots gen-

erally decreases the computational time for collision

checking. The use of bounding boxes decreases dras-

tically the number of high resolution queries needed.

This is a really important point taking into account

that they are much more time consuming. There are

cases where low resolution queries do not solve the

whole collision query. This increases the computation

time. However, choosing a suitable order for check-

ing collisions helps to ﬁnd them in a quicker manner.

The precomputation of impossibilities of collision be-

tween different objects (Collision Matrix) increases

the performance of the system in case of having ob-

jects with restricted mobility in the workspace.

The combined use of optimizations generate good

results in workspaces shared by at least two robots

and objects.

RPQ has a wide range of applicability. RQP li-

brary is not only useful for proximity queries but has

also proved to be a good tool for surface navigation

and virtual representations, due to its ability to intro-

duce virtual objects in the shared workspace. The vir-

tual ﬁxtures developed in the paper are examples of

how RPQ can be used to modify robot’s behaviour.

As proved in the application presented, RPQ is not

only useful for developers of robotic applications, but

also for users of robotic applications, i.e. surgeons

that require new robotic tools for improving surgical

procedures.

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