CC BY
4Russian Journal of Nonlinear Dynamics, 2022, vol. 18, no. 5, pp. 755-770. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd221209
NONLINEAR ENGINEERING AND ROBOTICS
MSC 2010: 70E60
On the Problem of Position and Orientation Errors of a Large-Sized Cable-Driven Parallel Robot
E. A. Marchuk, Ya. V. Kalinin, A. V. Sidorova, A. V. Maloletov
This paper deals with the application of force sensors to estimate position errors of the center of mass of the mobile platform of a cable-driven parallel robot. Conditions of deformations of cables and towers of the robot are included in the numerical model and external disturbance is included too. The method for estimating the error in positioning via force sensors is sensitive to the magnitude of spatial oscillations of the mobile platform. To reduce torsional vibrations of the mobile platform around the vertical axis, a dynamic damper has been included into the system.
Keywords: cable, robot, additive, printing, position, orientation, errors, force sensors
1. Introduction
Cable-driven parallel robots are mechanical systems with parallel topological structure and wire ropes as tendons which move the end effector of the robot. In general, cable-driven robots consist of a cage which is the body of the robot and a system of cables with pulleys and winches mounted at this cage. Sometimes elements of the robot can be mounted at the walls of buildings or at any spatial structures. For large-sized robots towers or masts can be used instead of cages to mount winches and pulleys at proximal anchor points. Cable-driven robots are easily scalable and usually used in practical tasks of moving payloads in a large volume of workspace of the robot.
Received September 19, 2022 Accepted November 16, 2022
This work was supported by the RSF (Grant No. 22-29-01618).
Eugene A. Marchuk [email protected] Yaroslav V. Kalinin [email protected] Alena V. Sidorova [email protected] Alexander V. Maloletov [email protected] Innopolis University
ul. Universitetskaya 1, Innopolis, 420500 Russia
Typical applications for large-sized cable-driven parallel robots are warehousing, positioning of large objects, video recording at stadiums and so on. Middle-sized cable-driven parallel robots can be used, for example, as exosceletons. Small-sized cable-driven parallel robots are usually used for scientific research in laboratories or as prototypes when designing new systems.
The sources of typical difficulties in designing and modeling cable-driven parallel robots are the nonlinear properties of the cable system. Therefore, most of the problems of cable-driven parallel robots have no exact or closed-form solutions in general form. Nevertheless, some assumptions are used to simplify the models, and advanced techniques to solve these problems with a given accuracy are known. Most of the problems involved in designing and modeling the cable-driven parallel robots are also multiobjective optimization problems.
A specific problem of cable-driven parallel robots is the problem of collisions of cables, which also depends on the configuration of the cable system. Another problem is the sensitivity of the cable system to influences which can cause vibrations of the end effector of the robot. In general, the design and modeling of the cable system of a cable-driven parallel robot is a nontrivial task.
2. Cable-driven parallel robots
Cable-driven parallel robots combine the simplicity of rope hoists and the efficiency of modern automatic control systems. Cable-driven parallel robots are similar to mechanisms of cable cranes which have been used since the end of the 19th century [1]. Moreover, lifting mechanisms as winches were well known in the ancient world and their basics have not changed significantly [2]. The only differences are the sources of power and the lack of automatic control in previous generations. The traditional Korean crane "Geojunggi" that has been known at least since the 17th century is absolutely the same cable-driven parallel mechanism, but is run and driven by men [3]. In the modern world, automatic cable-driven parallel mechanisms have been used in a wide range of practical tasks only since the beginning of the 21st century. One of the first investigations of cable-driven parallel robotic systems appeared in the 1980s [4]. The first example applied to solving practical problems was the Stewart platform with a cable drive. It was used to stabilize the crane of the NIST Robocrane marine platform [5]. Since the 1990s many experimental specimens and prototypes of cable-driven parallel robots have been made, the most famous of these specimens, described in research papers and reports, are: FALCON, WARP, SEGESTA, IPANEMA, CableBOT, and CoGiRo (see [6-11]). Cable-driven parallel robots also differ in the principles of mechanics (underactuated or overactuated), control, operational characteristics (the values of speed and acceleration may reach, respectively, 13 m/sec and 43g), linear dimensions (tens of centimeters and tens of meters), the ability to carry a payload of various weights, and so on (see [12, 13]). Another problem is the modeling of cables. The conventional way is to describe wire ropes and cables as catenary with different properties (see [14]). Several methods are proposed to describe cables in the model of a cable-driven parallel robot, cables can be described as catenaries, or as elastic rods, or as stiff rods, all these methods have advantages and disadvantages (see [15]). The most complete descriptions of cable-driven parallel robots are given in [16] and [17]. The typical ways to improve the accuracy of cable-driven parallel robots are given, for example, in [18] and [19]. These methods are based on readings of force sensors with further processing of the data with Kalman's filter or with ANN and sometimes also with additional observation by cameras.
This work proposes a simple geometrical approach to estimate the errors of a large-sized cable-driven parallel robot with sufficient accuracy via readings of force sensors only of lower cables for a specific configuration of the cable system.
3. Design and modeling
Since we consider an example of a machine for large-sized additive manufacturing, we have to choose suitable parameters for the robot's configuration. More exactly, it is supposed that the robot is a base for a 3D-printer for printing buildings such as town houses, Fig. 1. Because building density assumes typical rectangular forms for lands and houses, the rectangular base for the cable-driven parallel robot has been chosen. In general, the base of the cable-driven parallel robot can be of an arbitrary polygonal form, but in this case the rectangular form provides the most convenient way to set a robotic complex. With respect to the large size of the robotic complex, the towers have been chosen instead of a cage as the robot's body. Proximal anchor points of the robot are specified in such a way that the upper ones are fixed at the tops of the towers and the lower ones move along the lower half of a tower. The end effector of the robot has a cylinder shape and each of its distal anchor points is connected with the nearest proximal anchor point, the upper one with the upper one and the lower one with the lower one. As a result, we have an eight-cable driven parallel robot, and its cable system is considered in detail in this paragraph.
Fig. 1. Model of the building complex of the University of Innopolis
It should be understood that in a real cable-driven robot the dynamics of real cable structures will differ from oscillations of rod systems with unilateral constraints which is supposed in most of the models. To provide an acceptable complexity of the mathematical model and the feasibility of the problem, we ought to abandon a more adequate model of Irvin's cables in favor of the rods. It is assumed that the rod model with unilateral constraints matches the main properties of the high-loaded cable.
3.1. Cable system
Strictly speaking, the cables have properties of structural and geometrical nonlinearity, but in special cases some of them may be supposed to be negligible. It is assumed that the cables are highly stressed and not sagging, so the cables are represented as elastic rods. With this assumption geometrical nonlinearity is cancelled out. Thus, we only have the problem of structural nonlinearity, which is discussed in the context of dynamics. Therefore, the basics of kinematics of the cable-driven parallel robot may be described in terms of linear algebra using conventional methods. We are interested in the following lengths of the cables:
li = a - Rbi - r, (3.1)
where r is the radius vector connecting the origin of the world frame with the origin of the tool frame, ai is the vector connecting the origin of the world frame with the fth proximal anchor point, bi is the vector connecting the origin of the tool frame with the fth distal anchor point, and R is the transformation matrix.
Also, we will use a transposed Jacobian matrix, which is
JT
bi X tttV • • • bo X
1 l|li IU 8 111«
(3.2)
As mentioned above, the cables have the properties of structural nonlinearity and may resist only to stretching, but not to pressing. This property is included in the mathematical model via the activation function whose argument is deformation of the fth cable:
f = f *
1
1 + e
a-bAl
(3.3)
where f is the force of tension in the fth cable, Al is deformation of the fth cable, and a and b are some coefficients.
Deformations of the cables are defined according to Hook's law. Each fth cable is assumed to be a viscoelastic body according to the Voigt model:
f* = ES^+VS^, (3.4)
l0 l0
where E is the Young modulus, n is the dynamic viscosity of the material of the cable, S is the cross-section of the cable, and l0 is the length of an unloaded cable.
Therefore, the dynamics of the cable-driven parallel robot can be described in terms of damping oscillations, where the damping factors are viscosities of the material of the cables and media:
Mq + Dq + gc = -wp + JT f, (3.5)
where M is the mass matrix, D is the damping matrix, gc is the factor of centripetal force and angular momenta, wp is the outer wrench, JT is the transposed Jacobian, f is the vector of forces in the cables, and q is the vector of generalized coordinates.
For solving (3.5) most of the methods of numerical integration work well and ode45 is supposed to be the conventional way.
Is
l1 "2
Is
3.2. Towers as Bernoulli beams
Appropriate models of towers can be given with finite elements, but these methods have very high computational cost. We may give a rough approximation of towers as Bernoulli beams, more exactly, vertical uniform cantilever beams. We do not know the coordinates of the top of the deformed tower, but we may assume that the direction of the vector of the force which is applied to the tower at the proximal anchor point does not change significantly after deformation of the tower. Then we suppose the vector of the force coinciding with a given direction of the cable and find the horizontal projection of the force, and obtain the deflection of the free end of the beam:
L3
Ax =-F, (3.6)
3 EIa ' v '
where L is the length of a beam; E is the Young modulus, IA is the moment of the area which depends on the shape of the cross-sectional area of a beam, F is the force which is applied perpendicular to a beam [20].
We also may assume that the impact of deformation of each tower on the system can be represented via the elongation of the corresponding cable and this elongation is equal to the Ax obtained. These assumptions are included in the numerical model of the robot and are used in simulations.
3.3. Estimation of the errors
The idea is based on supposing the errors in positions of lower proximal anchor points negligibly small. Assuming that the tower has the properties of a vertical cantilever beam, we suppose that the deformations in the lower part of the beam are significantly fewer than the ones in the upper half, especially at the top. The condition of deformations of the towers is included in the model, but for real structures it is hard to obtain a precise position of the upper proximal anchor points. Then the methods of linear algebra which are conventional in designing and modeling the cable-driven parallel robots become unusable. To avoid uncertainty, we may exclude the upper cables from calculations and use only the lower cables. The necessary condition is such that all Tait- Bryan angles for a given orientation of the mobile platform must be equal to zero.
Let us consider a horizontal plane and four lower cables lying in this plane when the mobile platform meets a given position. If the cables deform under the payload, then the plane transforms into a truncated rectangular right pyramid, Fig. 2.
Now we can find the heights of any two trapezoids which are the opposite faces of the obtained truncated rectangular pyramid:
where l%, l%, 1%, l\ are the lengths of segments of deformed cables, a is the distance between two neighboring distal anchor points, and d is the distance between two neighboring proximal anchor points.
(3.7)
(3.8)
Fig. 2. Geometry of the lower cables of the robot
The length of each segment l** is obtained as the sum of the given length of the segment with elongation of the cable which is calculated by Hook's law.
Then we can find the height of the third trapezoid having two edges and the heights of the previous two trapezoids:
where b is the distance between two neighboring distal anchor points and c is the distance between two neighboring proximal anchor points.
The height of the third trapezoid is the estimated position error in the vertical coordinate.
Strictly speaking, this is not a frustum and not a trapezoid. But compared to the lengths of segments of cables, the differences are small and we can suppose that the assumption is correct. In the general case for n lower cables we also suppose a frustum with n trapezoidal side faces, but some additional calculations are required.
In such a way we can calculate only the absolute value of the estimated error and do not know the signs of segments of the curve. How can we know whether the center of mass of the mobile platform is above or below the given position? Of course, if only the gravity force acts on the mobile platform, then the center of mass can be only below. But in the case of overshooting of the automatic control system it can be above too. The ends of the cables which are connected to distal anchor points have to be equipped with electronic levels. In this way we know the direction of each cable relative to their proximal anchor points: upward or downward. The algorithm which is proposed below allows us to define the signs of segments of the curve via positive or negative angles of rising cables in the vertical plane:
(3.9)
n
a = ai,
(3.10)
i=1
f (a) = sign(a), where ai is the angle of elevation of the fth cable.
(3.11)
To smooth out the signum function, we approximate it with the sigmoid step function. It has the form
a(a) =-%-- - 1, (3.12)
where k and c are some constants.
Multiplying the functions, we obtain the estimated error for the vertical coordinate of the center of mass of the mobile platform:
h = f (¡1,1*2,1*3,11), (3.13)
Ah = h ■ a (a), (3.14)
the result is shown in Fig. 3.
Absolut value of estimated position in Z-axis
0.2-
a
xf o.
N! <
0.05 0
-0.05
0 50 100 150 200 250 300 350 400 450 500
t, sec
Real position error in Z-axis vs signum
r\ r> r> r\ n r> r>
J \J 8 Si \
a 0.05
< U
Nf
<1 -0.05
0 50 100 150 200 250 300 350 400 450 500
t, sec
Real and estimated position errors in Z-axis
0 50 100 150 200 250 300 350 400 450 500
t, sec
Fig. 3. Estimation of the position error in the Z-axis
We have been trying to estimate the errors in the XF-coordinates via calculations of the geometry of deformed cables, but have faced too big biases. So, we decided that it is not necessary to estimate errors in the XF-coordinates for compensation errors of positioning in the horizontal plane. It is sufficient to calculate the values of elongation for each lower cable by measuring the tension forces and then to compensate them because during the process of regulation errors converge to zero values. In such a way the proposed method is appropriate only for compensation of the errors, but not for estimation of the errors in the XF-coordinates.
The mobile platform has to be equipped with a gyroscope. Errors in the heights of distal anchor points relative to given positions in the tool frame are defined by applying the rotation matrix to given vectors of distal anchor points:
(3.15)
(3.16)
xb xb
y* = R yb
A zb
Azb = zt — zh
where xb, yb, zb are the coordinates of the fth distal anchor point in the tool frame and R is the rotation matrix.
So, if distal anchor points meet their given vertical coordinates, then the pitch and roll angles also meet the given ones.
The yaw angle is obtained from the data of gyroscope directly.
3.4. Error compensation
The idea of error compensation is based on the following assumptions:
• lower proximal anchor points meet their given positions;
• the configuration of the cable system meets the configuration described above;
• given orientation angles are equal to zero.
Then two processes for upper cables have to be executed simultaneously, the first one is compensation of error in the vertical coordinate of the center of mass of the mobile platform and the second one is compensation of errors in the pitch and roll angles. Examples of position and orientation errors for the motion of the mobile platform without and with error compensations are given in Figs. 4 and 5.
0 50 100 150 200 250 300 350 400 450 500
t, sec
Position errors with compensations 0.051-,-,-,-,-,-,-,-,-,-
<1
-0.05
_0 J-,-,-,-,-,-,-,-,-,-
0 50 100 150 200 250 300 350 400 450 500
t, sec
Fig. 4. Position errors in the XY-plane compared to the error in the Z-axis
The first process is executed via synchronized lifting of all upper cables. It is run with PID regulation. The second process is executed via separated lifting of each upper cable to minimize the error in the height of each distal anchor point. It is also run with PID regulation.
For lower cables their elongations are compensated also with PID regulation for each cable to minimize the error in their lengths after deformation.
Limitations in tension forces have to be given for regulators to avoid breaking of the cable system.
0.05
-a
<8
-0.05
0.05
-s
-0.05
Orientation errors without compensations
50 100 150 200 250 300 350 400 450 500 t, sec Orientation errors
0 50 100 150 200 250 300 350 400 450 500
t, sec
Fig. 5. Orientation errors
3.5. Wind pulsations
We have the model of a cable-driven parallel robot with elastic cables and deformable towers, and now is the time to add a factor of some external disturbance to the model, say, wind pulsations. In this model wind pulsations are supposed to have the properties of white noise. Therefore, the signals of the errors become noisy and the control system works in an unstable manner. The analysis carried out has shown that sufficient configuration of the filter for this model includes two channels with transfer functions as low-pass filters. In such a way output signals have some weights and the output signal of the filtering block is the sum of these weighted signals. This configuration is used for regulation of the error in the vertical coordinate as for compensation of errors in pitch and roll angles. The coefficients and weights for these two different types of regulators should obviously be different.
It is assumed that the wind impacts the towers and the mobile platform. The direct impact of wind on the cable system of the robot in the given context is supposed to be negligible.
The assumptions for laminar drag are:
• drag force for the mobile platform is defined as
Fd = \pV2CDA, (3.17)
where p is the density of media, V is the wind flow speed, CD is the drag coefficient, and A is the cross sectional area of the mobile platform;
then the force FD can be decomposed into orthogonal components FDx and FDy and added to the term wp of (3.5);
• for the towers an impact of wind pulsations is taken into account as follows: previously the towers had been designed and modeled in finite elements with "LIRA" software package;
then the deflections of towers under the pressure of windforce were defined with "LIRA" software; then equivalent coefficients of stiffness were chosen for the models of Bernoulli beams.
A numerical experiment has been carried out. The configuration has been tested by moving the mobile platform in different areas of workspace of the robot and the results are shown in Figs. 6, 7. Position errors with errors in the pitch and roll angles could be compensated for satisfactorily, but almost uncontrollable rotation around the vertical axis provided torsional vibrations and these vibrations have an impact on the entire dynamic system. To get rid of oscillations in the position errors, the torsional vibrations have to be reduced. Forcing the lower cables is not suitable for a given configuration of the cable system because of too large forces of tension in the cables required to reduce the torsional vibrations of the end effector. A possible way to reduce these torsional oscillations is to add a dynamic damper to the mobile platform [21].
Position errors without compensations
-0.1-1-1-1-1-1-1-1-1-1-
0 50 100 150 200 250 300 350 400 450 500
t, sec
Fig. 6. Position errors with wind pulsations
3.6. Dynamic damper
The idea of dynamic damping is to absorb the vibration energy bypassing the primary system. Dynamic dampers reduce vibrations in the specific frequency domain of the oscillating object. The system of differential equations describes dynamic damping for torsional oscillations:
r J0 + bd(<j) - j>d) + c4 + cd(0 - 0d) = , 1 Jd+ bd(<Pd - <P)+ cd(0d - = 0
where Jd is the moment of inertia of the damper, bd and cd are the viscoelastic properties of the damper, 0d is the angle of rotation of the damper, the same letters without indexes mark the values for the mobile platform of the robot, and the right part of the first equation is some external periodic torque.
Orientation errors without compensations
0 50 100 150 200 250 300 350 400 450 500
t, sec
Orientation errors with compensations
0 50 100 150 200 250 300 350 400 450 500
t, sec
Fig. 7. Orientation errors with wind pulsations
A passive dynamic damper with given constants as viscoelastic properties is used. The damper has been designed and modeled according to the recommendations given in [22]. Then the calculated numerical values for the parameters bd and cd were optimized using techniques of sequential quadratic programming. The object of damping and the dynamic damper are assumed to be coaxial cylinders, Fig. 8.
1 — Towers
2 — End-effector
3 — Dynamic damper
4 — Spring
5 — Cables
6 — Axis
Fig. 8. Diagram of a dynamic damper for reducing torsional oscillations of the end effector of a large-sized eight-cable-driven parallel robot
Therefore, the parameters bd and cd have to be given, the moment of inertia Jd of the damper is also given, the variables < and < are inputs, so, solving the second differential equation, we can find <)d and <d.
Graphs for errors with an attached dynamic damper are given in Fig. 9. Graphs for the moments and angles of rotation around the vertical axis are given in Fig. 10. Because of the noisy moment of the mobile platform M, Mf which is a filtered M is also shown, and Md is the moment of the damper.
0.05 0
-0.05
Position errors without compensations _and with dynamic damper_
i i: iltt« iir -^--»».»VSOAA/A
u-^M^tilMjuAdtia-a; M
■A, Ay A,
0 50 100 150 200 250 300 350 400 450 500
t, sec
Orientation errors with compensations and with dynamic damper
x) cS
0 50 100 150 200 250 300 350 400 450 500
i, sec
Fig. 9. Position and orientation errors with damping
4. Simulation
The simulation has been run with the given properties of the cable-driven parallel robot which are listed below:
• H is the height of towers, H = 15 m;
• c, d are the distances between two neighboring towers, c = 20 m, d = 20 m;
• m is the mass of the mobile platform, m = 350 kg;
• md is the mass of the dynamic damper, md = 50 kg;
• J is the moment of inertia of the mobile platform, J = 28 kg • m2;
Angles
0 50 100 150 200 250 300 350 400 450 500
t, sec Moments
0 50 100 150 200 250 300 350 400 450 500
t, sec
Fig. 10. Angles of rotation and moments for damper and platform
Jd is the moment of inertia of the dynamic damper, J = 4 kg • m2;
ES is the Young modulus for the material of towers, steel, ES = 200 GPa;
Ed is the Young modulus for the material of cables, Dayneema, Ed = 130 GPa;
cd is the torsion spring constant of the dynamic damper, cd = 3.0728 N • m • rad"1;
bd is the angular damping constant of the dynamic damper,
bd = 0.6171 Joule • sec • rad 1.
The platform moves along a planar curvilinear trajectory at height 4 m, and the upper limit for the smoothly changing speed of motion is 0.15 m • sec"1. The results of simulation are shown in Figs. 11 and 12.
5. Summary
The main result obtained is a significant reduction of the position and orientation errors of the mobile platform in the model of the robot. It has been shown that the impact of deformable structures of the robot and some external disturbances can be substantially reduced by controlling the cable system and by attaching a passive dynamic damper to the mobile platform.
The next stage is to design robust control for the robot because stable work is a necessary condition for industrial applications in robotics.
0.05
Position errors
-0.05
cö
—0.1 -
3000 t, sec Orientation errors
1000 2000 3000 4000 5000 6000 t, sec
Fig. 11. Position and orientation errors
5000
Forces of tension in cable
0.04
<1 0.02
3000 4000 t, sec Deflections of beams
2000 3000 4000 i, sec
Fig. 12. Tensions in cables and deflections of towers
Acknowledgments
The authors extend their gratitude to Prof. Alexandr Klimchik for useful ideas and to Michael Fadeev for his contribution in developing the software for cable-driven robots.
Conflict of interest
The authors declare that they have no conflicts of interest.
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