Visual Servoing using a Parallel Robot: Preliminary Results A. Traslosheros, J. M. Sebastián, L. Ángel F. Roberti R. Carelli Abstract— This paper describes the visual control of a parallel robot called “RoboTenis”. The system has been designed and built in order to carry out tasks in three dimensions and dynamical environments; the system is capable to interact with objects which move up to 1m/s. The control strategy is composed by two intertwined control loops: The internal loop is faster and considers the information from the joints, its sample time is 0.5 ms. Second loop represents the visual servoing system, is external to the first mentioned and represents our study purpose, it is based on predicting the object velocity which is obtained form visual information, its sample time is 8.3 ms. The trajectory planner lets that the system achieves the target without exceeding the maximum acceleration and speed. Lyapunov stability analysis, system delays and saturation components has been taken into account.
T
I. INTRODUCTION
he vision systems are becoming more frequently used in robotics applications. The visual information makes possible to know about the position and orientation of the objects presented in the scene and the description of the environment with a relative precision. Although the above advantages, the integration of visual systems in dynamical works presents many topics which are not solved correctly yet, thus many important investigation centers [1] are motivated to investigate about this field, such as in the Tokyo University ([2] and [3]) where fast tracking (up to 2 m/s) strategies in visual servoing are developed. In order to study and implementing the different strategies of visual servoing, the computer vision group of the UPM (Polytechnic University of Madrid) has decided to design the ROBOTENIS. Actually the implemented controller make possible to achieve high speed dynamical works. In this paper some experiments made to the system are described, the principal purpose is the high velocity (up to 1 m/s) tracking of a small object (black ping pong ball) with three degrees of freedom. It should be mentioned that the system is designed with higher attributes in order to response to future works.
Typically the parallel mechanisms possess the advantages of high stiffness, low inertia and large payload capacity; however, the principal weakness is the small useful workspace and design difficulties. As shown is Fig. 1, mechanical structure of RoboTenis system is inspired in DELTA robot [4]. Kinematical model, Jacobian matrix and the optimized model of the RoboTenis system have been presented in other previous woks [5]. Dynamical analysis and the joint controller have been presented in [6] and [7]. The dynamical model is based upon Lagrange multipliers thus, is possible to use forearms of non-negligible inertias in the development of control strategies. Two control loops are incorporated in the system: the joint loop is a control signal calculated each 0.5 ms, at this point dynamical model, kinematical model and PD action are retrofitted. The other loop is considered external since is calculated each 8.33 ms; this loop uses the visual data and is described in detail along this work.
Fig. 1. RoboTenis System
This paper is organized as follows. After the introduction above mentioned, section II describes the RoboTenis system; section III introduces the description of the visual servo control structure; section IV presents the experimental results and finally, in Section V some concluding remarks are given and commented. II. DESCRIPTION OF THE SYSTEM
This work is a part of the research project called: Teleoperation architectures in modelling dynamical environments (DPI 2004-07433C02-02), in which the economical resources come from the “Ministerio de Ciencia y Tecnología” (science and technology department). Likewise, thanks to the Spain international cooperation agency (AECI) because, by means of the “interuniversity cooperation 2005” program has been possible the participation of foreign students and investors. Actually J. M. Sebastian and A. Traslosheros are with the Universidad Politécnica de Madrid, DISAM, Madrid, España. (e-mail:
[email protected] and
[email protected]). L. Angel is with the Universidad Pontificia Bolivariana, Bucaramanga, Colombia (e-mail:
[email protected]). R. Carelli and F. Roberti are with the Instituto de Automática, Universidad Nacional de San Juan, San Juan, Argentina. (e-mail:
[email protected] and
[email protected] ) .
1-4244-1264-1/07/$25.00 ©2007 IEEE
This section describes the experimental environment, the elements that are part of the system and the functional characteristics of each element. For more information: [8]. A. Experimentally Environment The objective of the system resides in the tracking of a ping pong ball along 600 mm. Image processing is conveniently simplified using a black ball on white background. The ball is moved through a thread (Fig. 2), the ball velocity is close to 1 m/s.
B. Vision System RoboTenis system has a camera located on the endeffector as the Fig. 3 shows. This camera location merge two important aspects; when the robot and the object are relatively faraway each other, the field of view of the camera is wide but some error are generated in the ball position measurement; when the ball and robot are near each other, the field of view of the camera is narrow but the precision of the ball position measurements is increased. The above mentioned will be usable in future works as catching or hitting to the ball. Thus some characteristics of the components are the following. 1) Camera. The used camera is the SONY XC-HR50 and its principal characteristics are: High frame rate (8.33ms) and 240 x 640 pixels resolution; the used exposure time was 1 ms; progressive scan CCD; relatively small size and little weight: 29x29x32mm and 50gr, see Fig. 2 and Fig. 3.
Fig. 2. Work environment
Fig. 3. System camera 2) Data Acquisition Card. The data acquisition system is composed by the Matrox Meteor 2-MC/4 card and is the responsible of visual data and is used in double buffer mode, the current image is sampled meanwhile the previous image is being processed. 3) Image Procesing. Once images are acquired; the visual system segments the ball on white background. The centroid and diameter of the ball are calculated using a subpixel precision method. The 3D position of the ball is possible to calculate through preliminary camera calibration. The control system requires the velocity of the ball which estimated through Kalman filter ([9] and [10]) from the ball position.
C. System of Positioning Control The positioning system is composed by DSPACE 1103
card, this card is the responsible of: the generation of the trajectories, the calculation of kinematical and dynamical models and the implementation of control algorithmic. The motion system is formed by AC brushless servomotors, Ac drivers (Unidrive) and gearbox, (More information in [8]) D. Characteristics and Functions The visual controller is conditioned by some characteristics of the system and there are others inherited by the own application; some of them are: - High uncertainly in data from the visual system. The small sample data (8.333 ms) makes bigger the errors of the velocity estimation. For example, if the ball is located 600mm from the robot (camera) then the diameter of the ball it will be measured in 20 pixels and, if the estimation of the position has an error of 0.25 pixels, then the estimated distance error will be 8mm approximately, in the same order the estimation of the ball velocity error will be 1 m/s. The errors above cited origin an elevated discontinuity and the control action required exceeds the robot's capacities; the Kalman filter implemented helps to solve the problem. -The target speeds for the robot should be given continually. In order to avoid high accelerations, the trajectory planner needs continuity in the estimated velocity. For example, an 8 mm error in one sample rate (8.333ms) will demand acceleration equivalently to 12 times the gravity. The system guarantees a soft tracking performance by means of a trajectory planner; the trajectory planner is specially designed for reference shifting,, thus the target point can be changed at any moment. -Some additional limitations have to be considered in the system. The delay between the visual acquisition and the joint motion is estimated in two visual servoing sample rate (16.66ms). This delay is attained to some principal reasons as image capturing, image transmission and image processing. The maximum velocity the end-effector of RoboTenis is 2.5m/s. III. VISUAL CONTROL OF ROBOTENIS SYSTEM The coordinated systems are shown in the Fig. 4; they are Σw, Σe, and Σc which represent the word coordinate system, the end-effector robot system and the camera coordinate system respectively. Other notations defined are: cpb represents the position of the ball in the camera coordinate system, wpe represents the position of the robot end effector in the word coordinate system; wpe is obtained by means of the direct kinematical model. The rotation matrix are constant and known as wRe , wRc , eRc; eTc is obtained by means of the camera calibrations. By means of the trajectory planner and the Jacobian matrix, all the joint motions are calculated. “k” is a variable
that indicates the sample time considered.
e& ( k ) = − c Rw w
( v (k ) − w
w
b
vc ( k ) ) .
(5)
Substituting the Equations (3) and (5) into (4), we obtain: (6) vc ( k ) = w vb ( k ) − λ c RwT ⎡⎣ c pb* − c pb ( k ) ⎤⎦ Where w vc ( k ) and w vb ( k ) represent the camera and ball
velocities respectively. Since w ve ( k ) = w vc ( k ) the control law can be expressed as:
[
u (k )= wvb (k ) − λc RwT c pb* − c pb (k )
pb* ( k ) e(k ) +
-
u (k )
Jacobian matrix
Controller c
pb ( k )
and the other contains the tracking error c * c ( ⎡⎣ pb − pb ( k ) ⎤⎦ ).The ideal control eq. (7) requires a
q& (k )
perfect knowledge of all its components, which is not possible. A more realistic approach consist in generalizing the previous control as (8) u (k )= w vˆ (k ) − λc RT c p* − c pˆ (k )
ROBOT
Vision system
b
Fig. 5. The
ball position is used in visual servoing.
Although several alternatives exists [11], the controller selected is based in the ball position. Schematic control can be appreciated in the Fig. 5, the error function is obtained though the difference between the referenced position (
c
pb* (k ) ) and the measured position ( c pb (k ) ) this
difference must be constant because the goal is the tracking of the ball. Once the error is obtained, the controller calculates the velocity of the end effector. By means of the trajectory planner and the Jacobian matrix, all the joint motions are calculated. “k” indicates the sample time considered. A. Visual Servoing Model. In the Fig. 5 can be observed that the position error can be expressed as follows: (1) e ( k ) = c pb* − c pb ( k ) If ball position is referenced in the coordinate system of the camera then the position of the ball can be expressed as: c (2) p ( k ) = c R ( w p ( k ) − w p ( k )) b
w
b
c
If equation 2 is substituted into equation 1 then we obtain: (3) e ( k ) = c p* − c R ( w p ( k ) − w p ( k ) ) b
w
b
The system is supposed stable and in order to guarantee that the error will decrease exponentially, thus we choose: e& ( k ) = −λ e ( k ) con λ >0
constant, we obtain:
(4) c
Rw is
w
[
b
]
b
As is shown in the eq. (8), the variables are estimated and represented by the carets which are used instead of the true terms. The visual servoing control is shown in the Fig. 10. B. . Adjusting The λ Parameter. Fundamental aspect in the performing of the visual servoing system is the adjustment of λ thus, λ will be calculated in the less number of sample time periods and will consider the system limitations. This algorithm is based in the future positions of the camera and the ball; this lets to the robot reaching the control objective ( Pb (k )= Pb (k ) ). The future position (in the k+n instant) of the ball in the word coordinate system is: c
w
c
*
(9)
pˆ b (k + n)= w pˆ b + wvˆb (k )Tn
Where T is the visual servoing sample time period (8.333 ms) and the future position of the camera in word coordinate system in the k+n instant is: w
(10)
pc (k + n)= w pc + wvc (k )Tn
The control objective is reaching the target position in the shorter time as be possible, thus if we substitute the equation (9) (in the future position k+n) into (2), we have: c
p b* − c R w
[
w
]
(11)
pˆ b (k + n)− w p c (k + n) = 0
Substituting (9) and (10) into (11), eq. (12) is obtained.
c
Deriving the equation (2) and supposing that
(7)
The equation (7) is composed by two components: a component which predicts the position of the ball ( w vb ( k ) )
Fig. 4. Considered coordinated systems.
c
]
c
pb* = cRw
[
w
pˆ b (k )− w vˆb (k )Tn − w pc (k )− wvc (k )Tn
It has been indicated that considered the control law is:
u (k )= w vˆb (k ) −
[
w
]
(12)
ve (k ) = w vc (k ) , if (2) is
1c T c * c Rw pb − pˆ b (k ) Tn
]
(13)
If (8) and (13) are compared, we can obtain λ as: (14) 1 Tn The eq. (14) gives a criterion for adjust λ as a function of the number of samples required (n) for reaching the control target.
λ=
represents the initial and final time of the motion, and jmax is the maximum allowed jerk and will be defined later. From (16) we integrate the Jerk in a convenient way towards to obtain the acceleration, thus: (17) Tj max
a (τ ) =
[1 − cos(ωτ )]
ω
C. Implemented Algorithm The visual servoing architecture proposed in Fig. 10, considers the physical limitations of the RoboTenis System such as the delays and the maximum system operation components. Thus we propose the visual control structure −r
as follows. The term z represents a delay of “r” periods for the control signals. It is considered that the information c
supplied by vision system ( pb ( k ) ), has a delay of 2 sampling times (r=2) with respect to the information of the robot (joint position system). Consequently, at an instant k+n, the future position of the ball can be obtained by (15) w
pˆ b (k + n) = w pˆ b (k − r )+ wvˆb (k − r )T ( n + r )
(15)
And the future position of the camera is given by equation (10). The eq. (14) lets to adjust the λ parameter in the control law and considers the following aspects: • The wished velocity of the end-effector is represented by (13). In physical systems the maximal velocity is necessary to be limited. In our system the maximal velocity has a direct effect in the minimal number of sampled time periods (T) which the target function can be achieved. The number of sampled time periods (T) is not the same in all joints, thus this data in the more restrictive joint (the bigger time) will be used in the λ parameter calculation. • Is desirable that the velocity of the robot can be continuous.
D. Trajectory planner. Previously was mentioned that the joint loop has a smaller sample time than the visual loop thus, in order to obtain soft trajectories a sine trajectory planner was designed, Fig 6. The trajectory planner decides which trajectory the robot can do. The calculation of the trajectory planner parameters has to be easy and fast. Continuous paths [12] have to be considered and the speed and maximum acceleration of the robot should not be exceeded. (16) k
j (τ ) =
Where
T3
sin(ωτ ) = jmax sin(ωτ )
ω = 2π , τ ∈ [0,1] = t − ti
t f − ti
Fig 6. Structure of the trajectory planning
The velocity is attained in similar way but we consider an initial speed vi
v(τ ) = vi +
T 2 jmax ⎡ sin(ωτ ) ⎤ τ− ω ⎢⎣ ω ⎥⎦
Finally the position time history and the initial position pi are considered as:
p(τ ) = pi + Tviτ +
= t f − ti then
dτ 1 = , t represents the real time clock, ti and t f dt T
T 3 jmax ⎡τ 2 cos(ωτ ) 1 ⎤ (19) + − 2⎥ ω ⎢⎣ 2 ω2 ω ⎦
From (16) and (17) we can calculate the maximum acceleration amax as:
a max =
Tj max
ω
[1 − cos(ωτ )]
T 1 2 2
⇒ jmax =
τ= =
ωamax
(20)
2T
Thus from (2) and (3) vmax is:
vmax = vi +
T 2 jmax T 2 jmax ⎡ sin(ωτ ) ⎤ = + v τ − i ω ω ⎢⎣ ω ⎥⎦ τ =T =1
(21)
Substituting (20) in (21): (22) Ta vmax = vi + max 2 During the operation of the trajectory planner the first on-line step is to define the initial position pi and the final position p f , thus
, if T
(18)
in
order to
achieve the target
( p (τ ) = p f ) the maximum acceleration and velocity have not to be exceeded and two comparisons are implemented as is shown in the Fig 7, where MAS and MVS are the maximum velocity and acceleration allowed respectively,
both of them are our robot fixed parameters. The acceleration is compared with MAS and with MVS by mean of the eq. (23) and (24); thus if the acceleration is exceeded the planner uses the minimum of the both comparisons.
MAS > amax MVS > vmax =
200
150
100
(23)
vi Tamax + T 2
It is desirable to obtain
amax
50
(24)
0
-50
in terms of known parameters,
substituting (20) in (19).
p(τ ) = pi + Tviτ +
Sine trajectory planner
250
-100
Position Velocity Acceleration Jerk Target
-150
T 2 amax ⎡τ 2 cos(ωτ ) 1 ⎤ − 2⎥ ⎢ + 2 ⎣2 ω2 ω ⎦
(25)
-200
-250
0
5
amax
[
]
15
20
25
30
35
Time (ms)
And finally:
4 = 2 p f − pi − Tvi T
10
T
(26)
…
T
T
Fig 8. Time history of Robotenis trajectory planner.
Sine trajectory planner
250
200
150
100
50
0
-50
Position Velocity Acceleration Jerk Target
-100
-150
-200
0
5
10
15
20
25
30
35
Time (ms)
Fig 9 Generated trajectory if the maximum acceleration is exceeded. Fig 7 Restriction of the maximum velocity and acceleration
The trajectory planner is based in a cycloid motion [13] but it is specially designed towards bounding the maximum velocity and maximum acceleration. The ai , a f , ji and
j f are supposed cero, this is shown in the Fig 8 and Fig 9. When the necessary acceleration to achieve the target is bigger than the maximum allowed, the time history of the trajectory is described in the Fig 9, it can be observed that the position is not reached but the psychical characteristics of the robot actuators are not exceeded. In the Fig 8 can be observed that the first point is not reached, this is because the acceleration, in order to reach the fist point from the initial position v = 0 , has to be bigger.
i
E. System Stability. By means of Lyapunov analysis is possible to probe the system stability; it can be demonstrated that the error converges to zero if the ball velocity and the dynamic system model are completely known; otherwise it can be probed that the error will be bounded under the influence of the estimation errors and unmodeled dynamics. For the stability analysis we consider (5), and (7), to obtain the close loop expression thus:
e&(k ) + λe(k ) = 0
(27)
We choose a Lyapunov function as:
1 V = eT (k )e(k ) 2 V& = eT (k )e&(k ) = −eT (k )λe(k ) < 0
(28) (29)
Equation (29) implies that e( k ) → 0 when k → ∞ ;
ρ e > λ
but if ve ≡ u is not true then: w
w
ve ( k ) = u (k ) + ρ (k )
(30)
(34)
If we consider that ρ ( k ) → 0 (this has two means: The velocity controller is capable to make the system to reach
Where ρ (k ) is the velocity error vector which is produced by the bad velocity estimations and unmodeled dynamics. We consider (29) and, (27) can be written as:
ve (k ) → u and there are not error in the velocity estimations) then e( k ) → 0 ; otherwise we can conclude
e&(k ) + λe(k )=cRw ρ (k )
that when (34) is not fulfilled, the error does not decreases and remains finally bounded by:
(31)
w
We consider again a Lyapunov function as:
V=
ρ e < λ
(32)
1 T e ( k )e ( k ) 2
V& = eT (k )e&(k ) = −eT (k )λe(k ) + eT (k )cRw ρ (k )
(33)
(35)
The sufficient condition for V& < 0 is: c
pb* ( k )
e(k )
+
-
u (k )
− λ cRwT Vc (k − r )
+
q& (k )
Jacobian matrix
+
ROBOT
w
c
z −r
Vˆb ( k − r )
w
+
pˆ b ( k )
c
Rw
Forward kinematics
Velocity estimator
Vˆb ( k − r )
c
w
r ⋅T
pb ( k − r ) +
+
c
+
c
w
c
pc (k − r )
+
w
z
−r
RwT c pe
pc (k )
+
w
pe (k )
RwT
pˆ b ( k − r )
Position filter
c
Vision pb ( k − r ) system
Fig. 10. Visual servoing control architecture proposed.
I. EXPERIMENTAL RESULTS In this section, the experiments results are related to visual tracking object up to 1 m/s. The results show the performance of the visual servoing algorithm proposed for the RoboTenis system. The control objective consists in preserving a constant relationship between the camera and the moving target. The distance was set to [0, 0, 600]T mm and has to stay constant. The ball is subject through thread to the structure of the robot and it moves by means of a manual drag (see Fig. 2). Different arbitrary trajectories have been performed. The experiments have been made using speeds up to 1000 mm/s.
As example, in Fig. 11 is represented an experiment in which is shown the space evolution of the ball.
A. Performance Tracking Indexes. In this section, the experiments results are related to visual tracking object up to 1 m/s. The results show the performance of the visual servoing algorithm proposed for the RoboTenis system. The control objective consists in preserving a constant distance ([0, 0, 600]T mm) between the camera and the moving target. The ball is hanged to the structure and is moved by manual drag (see Fig. 2). Different arbitrary trajectories have been performed. As example, in Fig. 11 is represented an experiment in which the space evolution of the ball is shown.
Fig. 11. 3D ball movements
Two indexes are. • Tracking Relation: (table I) is defined in (36) as the relation between the average of the tracking error module and the average of the estimated velocity module of the ball. This relation is expressed in mm/1/mm/s and allows isolating the result of each experiment from particular features of motion of the ball made in other experiment.. (36)
1 N ∑ e( k ) N k =1 Tracking relation = 1 N w ∑ vˆb (k ) N k =1
• Average of the tracking error in function of the estimated ball velocity. In order to makes an easier comparison; the estimated ball velocity has been divided in 5 groups which the tracking error is shown in the table II.. B. Control Laws Studied And Compared Two control laws have been used: Proportional control law: It does not consider the predictive component of the equation (8), thus:
[
]
u (k ) = −λ cRwT c pb* (k )− c pˆ b (k ) .
(37)
• Predictive control law: It considers the predictive component of the equation (8), thus:
u (k )= w vˆb −λ cRwT
[
c
]
pb* (k )− c pˆ b (k ) .
(38)
The Table I and Table II show the indexes result when the two control laws are applied. The Numerical results presented have been obtained from the average of 10 experiments of each control algorithm. TABLE I. TRACKING RELATION USING A PROPORTIONAL AND PREDICTIVE CONTROL LAWS
ALGORITHM Proportional Predictive
Tracking relation 40.45 20.86
The results show a better performance of the system when the predictive control algorithm is used, thus a smaller tracking relation error and smaller grouped average tracking error is observed. Fig. 12 and Fig. 13 shows a
specific system experiment and can be observed than the predictive controller introduces a significant improvement, the error is reduced to the half in comparison with the obtained error when we use proportional controller. In the Fig. 12 and Fig. 13 a specific system experiment is exposed. For the proportional control law, the maximum tracking error is 34.50 mm and the maximum ball velocity is 779.97mm/s. When the predictive control law is used, the maximum tracking error is 18.10mm and the maximum velocity of the ball is 748.16 mm/s. In this experiment can be observed how the predictive control law improves the tracking robot behavior. TABLE II. GROUPED AVERAGE TRACKING ERROR
Velocity/ Algorithm Proportional Algorithm Predictive Algorithm
V< 200 6.3 4.2
200 400 13. 7
400 600 20. 1
8.1
9.5
600 800 26. 2 11. 3
V> 800 32.5 13.5
II. CONCLUSIONS In this paper a novel visual servoing structure is presented, this control strategy is used in a parallel robot in order to reach a high velocity tracking of an object which moves in unknown trajectories. RoboTenis has been the first parallel robot known which uses a visual tracking control system in dynamic environments. The control strategy is based in obtaining the smaller number of time samples in which the control target function is able to achieved. The existing delays in the system, the saturations in the velocity and acceleration are considered in de control model thus, this consideration makes possible that RoboTenis can move faster than its characteristics allow it. The trajectory planner allows soft position and velocity trajectories, limits the maximum velocity and acceleration (also the Jerk can be limited), and guarantees the continuity of the position, velocity, acceleration and jerk at any moment. The time history attained is the desired for the robot although not all the desired positions are reached. An important disadvantage is that the trajectory planner can not change the target point at the middle of the trajectory, if this occurs, the trajectory will present important discontinuities in the acceleration and jerk; in order to avoid the above mentioned, the target point is calculated from the visual controller before the trajectory planner reaches the last indicated target. The Lyapunov stability of the proposed system was probed under some ideal and non ideal conditions. When the conditions are ideal, it was probed that the error converges to zero otherwise, if the conditions are non ideal, the error will be finally bounded. The experiments were carried out in order to illustrate the high performance of the system was; a ball tracking was
successfully achieved and the error was smaller than 20mm, the visual loop has a sample time of 8.33 ms. In future Works, new control strategies are considered in order to attain a tracking velocity of 2m/s. With the same purpose, the interaction of the system with the environment and (or) works as caching or hitting the ball are desirable. For more information consult: http://www.disam.upm.es/vision/projects/robotenis/
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13] Fig. 12. Proportional control law. Tracking error
Fig. 13. Predictive control law. Tracking error
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