Prototypes of micro-robots and geometric path planning algorithms are presented in this paper. Geometric motion trajectories are generated by switching contacts, control points and tangents methods. Analysis of high-speed formation of trajectories using switching contacts method revealed that the geometric path depends on orientation angle of a micro-robot. Investigation of formation of high-precision trajectories demonstrated that there is minimal deviation from the given motion trajectory at which micro-robot stops. This paper presents methodologies for evaluation of optimal orientation angle of a micro-robot and minimum deflection from the given trajectory. The methodologies are verified by experiments

Asta Drukteinienė

Genadijus Kulvietis

Ramutis Bansevičius

Vytautas Bakanauskas

English

Journal of Vibroengineering

 ? VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. MARCH 2013. VOLUME 15, ISSUE 1. ISSN 1392-8716 239 
942. Analysis of planned motion trajectories for scalable 
micro-robots 
Ramutis Bansevicius, Asta Drukteinienė, Genadijus Kulvietis, Vytautas Bakanauskas 
942. ANALYSIS OF PLANNED MOTION TRAJECTORIES FOR SCALABLE MICRO-ROBOTS.  
RAMUTIS BANSEVICIUS, ASTA DRUKTEINIENĖ, GENADIJUS KULVIETIS, VYTAUTAS BAKANAUSKAS 
Ramutis Bansevičius1, Asta Drukteinienė2, Genadijus Kulvietis3, Vytautas Bakanauskas4 
1, 4Mechatronics Centre for Research, Studies & Information,  
Kaunas University of Technology, Kaunas, Lithuania 
2, 3Department of Information Technologies, Vilnius Gediminas Technical University, Vilnius, Lithuania 
E-mail: 1bansevicius@cr.ktu.lt, 2astad@it.su.lt, 3genadijus_kulvietis@gama.vtu.lt, 
4vytautas.bakanauskas@gmail.com 
(Received 13 September 2012; accepted 4 March 2013) 
Abstract. Prototypes of micro-robots and geometric path planning algorithms are presented in 
this paper. Geometric motion trajectories are generated by switching contacts, control points and 
tangents methods. Analysis of high-speed formation of trajectories using switching contacts 
method revealed that the geometric path depends on orientation angle of a micro-robot. 
Investigation of formation of high-precision trajectories demonstrated that there is minimal 
deviation from the given motion trajectory at which micro-robot stops. This paper presents 
methodologies for evaluation of optimal orientation angle of a micro-robot and minimum 
deflection from the given trajectory. The methodologies are verified by experiments. 
Keywords: micro-robot, path-planning algorithms, optimal orientation angle, minimum 
deflection. 
1. Introduction 
Most of the high-precision positioning devices that have been developed so far are driven by 
piezoelectric actuators [1-6], which possess high resolution, high stiffness and quick response. 
However, the strokes of piezoceramic transducers are extremely small. 
R. Bansevičius and K. Ragulskis were the first who in 1989 presented 6-DOF and 9-DOF 
micro-robots that consisted of two kinematic pears: piezoelectric cylinder – passive sphere, 
piezoelectric disc – passive plane [7]. 3-DOF piezoelectric motor for motion or rotation by 
passive plane was proposed. It was piezoceramics mounted on metal plate and electrodes 
divided to equal 120? sectors [8]. Rotation is accomplished by means of travelling wave. During 
research work of positioning objects, piezoelectric ring-shaped, cylinder-shaped and 
hemisphere-shaped micro-robots were developed.  
The potential application of these micro-robots is mini/micro lasers, in particular for 
industrial processes. As shown by experimental studies, the minimum dimensions of the micro 
robot can be reduced to a few millimeters (Fig. 1). 
 
a) 
 
b) 
Fig. 1. 3-DOF piezoelectric cylindrical actuators: a) 4 × 3 × 4 mm; b) 8 × 6 × 8 mm (D × d × h) 
Piezoelectric ring-shaped micro-robotic schemes feature – ability to realize larger masses 
positioning systems (Fig. 2). In control of motion direction (without return communication 
942. ANALYSIS OF PLANNED MOTION TRAJECTORIES FOR SCALABLE MICRO-ROBOTS.  
RAMUTIS BANSEVICIUS, ASTA DRUKTEINIENĖ, GENADIJUS KULVIETIS, VYTAUTAS BAKANAUSKAS 
240 ? VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. MARCH 2013. VOLUME 15, ISSUE 1. ISSN 1392-8716  
systems) weight increase up to 500 g almost did not change motion parameters. 
 
a) 
 
b) 
Fig. 2. Cases of increased load capacity: a) piezoelectric ring 140 × 120 × 8 mm (D × d × h),  
frequency – 80 kHz, voltage – 80 V; maximum weight of positioning object – 2 kg;  
b) piezoelectric ring 40 × 10 × 2 mm (D × d × h), 
frequency – 80 kHz, voltage – 50 V; maximum weight of positioning object – 500 g 
Different trajectory planning methods are used for autonomous robots. Generally, 
polynomial of any degree [9], splines [10, 11] and clothoids curves [12, 13] interpolation 
between points. These methods are not used for planning the trajectory of designed micro-robot 
because, in our case, only one electrode segment is active for straight line motion and all 
electrode segments are active for rotary motion. Their motion direction depends on active 
contact point (there are three contact points). Thus their motion trajectory is a broken line. Also, 
they have high resolution motion and wide range of speed control. 
Motion path-planning algorithms for these 3-DOF micro-robots under various contact-
excitation schemes were created: switching contacts, control points and tangents methods. 
Results of research of switching contacts algorithm demonstrated that quantity of switching 
contacts is inversely proportional to deviation from given trajectory and angle between 
orientation angle of a micro-robot (angle between the first contact of micro-robot and x axis). 
Dependence is observed to be nonlinear [14, 15]. In high-speed formation of trajectories it is 
important to minimize the number of switching contacts. Though, the exact dependency between 
quantity of switching contacts and orientation angle of micro-robot is not set in the previous 
research. Consequently one of the research tasks was to specify this dependence N(α), when 
motion trajectory is planned with switching contacts method. 
Analysis of displacement length of micro-robot demonstrated that after the reduction of the 
deviation from the given trajectory (ε), length of displacements starts to decrease rapidly as well 
[15]. However, this study did not examine the minimum limit of the deviation value at which 
displacement length is approaching to the motion resolution of a micro-robot (Pmin). There is the 
smallest deviation (εmin) at which micro-robot can still move. But if ε < εmin, robot stops. 
Knowing the minimum deviation, it is possible to generate the precise motion trajectory. 
Calculations of the smallest deviations for each path-planning method are described and 
analyzed in this paper too. 
2. Overview of path-planning methods 
Micro-robot motion trajectory can be formed, when given motion trajectory is described by 
parametric functions ? by formula: 
?? ? ?
??? ? ????? ????? ???
??? ? ????? ????? ???? (1) 
where ? – parameter of parametric function, ???? ??? – nodes of given trajectory, ? – number of 
node. 
942. ANALYSIS OF PLANNED MOTION TRAJECTORIES FOR SCALABLE MICRO-ROBOTS.  
RAMUTIS BANSEVICIUS, ASTA DRUKTEINIENĖ, GENADIJUS KULVIETIS, VYTAUTAS BAKANAUSKAS 
 ? VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. MARCH 2013. VOLUME 15, ISSUE 1. ISSN 1392-8716 241 
According to the requirements of the task, the axis of symmetry of micro-robot must deviate 
as little as possible from the given trajectory ?. Therefore, must be defined maximum allowable 
deviation ???? from function ?. Coordinates located ???? distance from given trajectory are 
called coordinates of the boundary. 
If one electrode segment of micro-robot is active at a time, then motion trajectory is broken 
line. Switching contacts algorithm described in [14] were developed for this kind of motion 
(Fig. 3). 
 
Fig. 3. Example of motion trajectory formation with switching contacts method 
If all three electrode sectors of micro-robot are excited at a time with phase difference 120?, 
running wave is generated leading to rotary movement. In this case, micro-robot will rotate 
around its central axis. Rotation can be clockwise and counter-clockwise. Control points 
(Fig. 4a) and tangents path (Fig. 4b) formation algorithms were developed using these two 
excitation schemes. 
 
a) 
 
b) 
Fig. 4. Examples of trajectory formation: a) control points method; b) tangents method 
Control points method principle is to divide given trajectory into equal length segments, and 
to make rotation check in points joining these segments evaluating motion resolution. These 
points are called control points. The method is suitable for the motion in continuous curves. This 
method was developed to form a high-precision trajectory [16]. 
Method of tangents is designated to shape the trajectory that is optimal with respect to speed, 
but still maintains the accuracy of motion. First, it is checked, if there is possibility to move from 
current node to another without intersecting boundary coordinates. If boundary coordinates are 
intersected, then it is searched for the tangent point at boundary coordinates and the point of 
motion coordinates is calculated [17]. 
3. Motion trajectory generation parameters 
During research of switching contacts quantity, it was demonstrated that orientation angle of 
942. ANALYSIS OF PLANNED MOTION TRAJECTORIES FOR SCALABLE MICRO-ROBOTS.  
RAMUTIS BANSEVICIUS, ASTA DRUKTEINIENĖ, GENADIJUS KULVIETIS, VYTAUTAS BAKANAUSKAS 
242 ? VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. MARCH 2013. VOLUME 15, ISSUE 1. ISSN 1392-8716  
micro-robot affects this parameter. If right orientation angle of micro-robot is selected, then 
high-speed trajectories could be formed. This angle is called optimal orientation angle ?????? 
and for any kind of given trajectories it could be calculated by weighted average formula: 
???? ?
? ??????
?
???
? ??????
?? (2) 
where ? – quantity of given trajectory segments between nodes, ? – length of segment between 
nodes, ? – angle between segment and ? axis,?? – segment number. 
The smallest deflection from given trajectory (Fig. 5) could be calculated by formula, when 
trajectory formation method is switching contacts: 
???? ? ?????????? (3) 
where ??? ? ???? ? ??min???????????? ? ????, where ? – number of electrode segment, ????? – 
motion resolution, ? – angle between electrode segment and x axis, ? – micro-robot orientation 
angle, ? – angle between two nodes segment and ? axis, ? – number of segments between nodes. 
 
Fig. 5. Scheme of setting of minimum deviation from given trajectory 
When control points or tangents methods are used for trajectory generation, permissible 
generated trajectory deflection from the given trajectory (Fig. 6) could be calculated by formula: 
???? ? ???? ? ???? ? ??? ? ?????? (4) 
where ???? ??? – coordinate at given trajectory, where curvature of curve is maximum, 
coordinate ???? ??? is calculated by solving system of equations: 
??
?
??
?? ? ???
??? ? ???
?
?? ? ???
??? ? ???
?
? ???
?? ?????
? ??? ? ??? ? ?? ? ???
? (5) 
where ????? ????, ????? ???? – coordinates of planned trajectory, ?? – parameter of parametric 
function, where curvature of curve is maximum. 
942. ANALYSIS OF PLANNED MOTION TRAJECTORIES FOR SCALABLE MICRO-ROBOTS.  
RAMUTIS BANSEVICIUS, ASTA DRUKTEINIENĖ, GENADIJUS KULVIETIS, VYTAUTAS BAKANAUSKAS 
 ? VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. MARCH 2013. VOLUME 15, ISSUE 1. ISSN 1392-8716 243 
 
Fig. 6. Scheme for determining maximum deflection 
4. Analysis of planned motion trajectories  
To determine, whether the trajectory type affect optimal orientation angle, broken lines 
trajectories (Fig. 7) and curves (Fig. 8) between the same nodes were formed and planned 
motion trajectories using switching contacts algorithm. For testing optimal orientation angle 
calculation formula (2) three kinds of trajectories were selected: open trajectory (Fig. 7, 8a), 
closed trajectory with different segments length (Fig. 7, 8b) and closed trajectory with the same 
segments length (Fig. 7, 8c). 
Results of optimal angle calculations by using formula (2) is shown in Table 1, where  
? – quantity of switching contacts at calculated angle. 
All the calculation results were compared with the experimentally established dependences 
of orientation angle of micro-robot. Results of experiments are provided in Fig. 9. 
 
a) 
 
b) 
 
c) 
Fig. 7. Given broken line and generated micro-robot motion trajectories with switching contacts method:  
a) No. 1; b) No. 2; c) No. 3 
 
a) 
 
b) 
 
c) 
Fig. 8. Curves and generated micro-robot motion trajectories with switching contacts method: 
a) No. 4; b) No. 5; c) No. 6 
Results of analysis indicate that it does not matter, which kind of trajectory is given. Minimal 
and maximal quantity of switching was at the same orientation angle, when motion resolution is 
constant. 
942. ANALYSIS OF PLANNED MOTION TRAJECTORIES FOR SCALABLE MICRO-ROBOTS.
RAMUTIS BANSEVICIUS, ASTA DRUKTEINIENĖ, GENADIJUS KULVIETIS, VYTAUTAS BAKANAUSKAS 
244 ? VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. MARCH 2013. VOLUME 15, ISSUE 1. ISSN 1392-8716  
Permissible deviation of the generated trajectory from the planned one (Fig. 10, 11), which 
depends on the piezorobot motion resolution and on the type of trajectory, was calculated and 
experimentally defined during the investigation. 
Table 1. Results of optimal orientation angle calculation 
Trajectory No. ????, o ? 
1 ? ?55 ,175 ,295? ? ?  16 4 18 
2 ? ?109 ,229 ,349? ? ?  39 5 42 
3 ? ?15 ,135 ,255? ? ?  35 6 24 
 
 
a) 
 
b) 
Fig. 9. Experimentally established dependencies of orientation angle of micro-robot,  
when given trajectories are: a) broken lines; b) curves 
 
Fig. 10. The smallest deviation dependence on the motion resolution, 
when using switching contacts algorithm 
Results of experiment revealed that the smallest deflection calculation methodology is right 
for each path-planning method.  
Motion trajectory formation with control points method was analyzed according to the 
minimum angular displacement. Three points were selected. Trajectory, which segments are 
close to straight line, was planned. It was demonstrated that quantities of switching contacts 
942. ANALYSIS OF PLANNED MOTION TRAJECTORIES FOR SCALABLE MICRO-ROBOTS.  
RAMUTIS BANSEVICIUS, ASTA DRUKTEINIENĖ, GENADIJUS KULVIETIS, VYTAUTAS BAKANAUSKAS 
 ? VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. MARCH 2013. VOLUME 15, ISSUE 1. ISSN 1392-8716 245 
depend on the inner radius R of a micro-robot. Generated trajectory switching contacts quantity 
T and control point’s quantity T* ratio was calculated, when minimum displacement is 0.5, 0.3 
and 0.1 μm (Fig. 12). 
Results of experiment showed, that changing motion resolution leaves quantity T and control 
point’s quantity T* ratio almost constant. 
 
Fig. 11. The smallest deviation dependence on the motion resolution, 
when using control point’s algorithm and tangents algorithm 
 
Fig. 12. Generated trajectory switching contacts quantity and control point’s quantity ratio  
dependence on inner radius of piezorobot 
5. Conclusions 
In order to increase the motion speed the optimal angle of a micro-robot must be calculated, 
when motion trajectory is generated using switching contacts method. Advantage of this method 
is its simplicity in executing control of a micro-robot. 
Methodology for calculation of minimum deflection from the planned trajectory was created 
for each trajectory generation method. Evaluation of this parameter allows predicting critical 
stops of the micro-robot. Taking into account the geometrical dimensions of the robot, and the 
minimum deviation, it is possible to achieve high-precision trajectory generation.  
Acknowledgements 
This work has been supported by Research Council of Lithuania (Project No. MIP-075/2012) 
and the Lithuanian Agency for Science, Innovation and Technology (Project No. 31V14/12). 
942. ANALYSIS OF PLANNED MOTION TRAJECTORIES FOR SCALABLE MICRO-ROBOTS.  
RAMUTIS BANSEVICIUS, ASTA DRUKTEINIENĖ, GENADIJUS KULVIETIS, VYTAUTAS BAKANAUSKAS 
246 ? VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. MARCH 2013. VOLUME 15, ISSUE 1. ISSN 1392-8716  
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