Influence of pulley kinematics on cable-driven

Influence of pulley kinematics on cable-driven parallel robots Andreas Pott

Abstract In this paper the modeling of a pulley mechanism for cable-driven parallel robots is presented. In many works, the proximal anchor points of the robots are simplified to be ideal points. Real cables achieve reasonable life time only when a minimum bending radius is exceeded. Therefore, pulley mechanisms have to be used which in turn require the extension of the kinematic modeling. In this paper a kinematic model for a pulley mechanism of the winches is revisited. Then we derive a corrected structure equation and compare the different results from the extended model with the estimation from the simplified standard model with respect to kinematics transformation, workspace, and force distribution. Key words: cable-driven parallel robot, pulley mechanism, kinematics, statics, stiffness, workspace.

1 Introduction A cable-driven parallel robot is a special type of parallel kinematic machine where the rigid struts are replaced by light-weight cables. Therefore, the inertia of the robot is largely reduced allowing for application in large-scale [2], ultra-fast [5], and heavy duty applications [1]. Due to their advantages cable robots attracted increasing attention during the last year. Although some fundamental issues of cable robots are still open, researchers have started to address practical issues related to construction and control of prototypes. The results presented in this paper were driven by the development of the cable robot IPAnema [7], which targets at application in large-scale handling and assembly. Cables are very flexible and versatile construction elements that are used in applications such as bridges, cable-cars, and elevators. Nevertheless, there are important Andreas Pott Fraunhofer IPA, Stuttgart, Germany e-mail: [email protected]

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Andreas Pott z

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design rules when using cables that have to be taken into account such as minimum feasible bending radius. Therefore, one has to integrate elements such as pulleys to allow for acceptable durability as well as safety. Only few authors have addressed the influence of guiding pulleys on the kinematics of cable-driven parallel robots. Bruckmann [3] derived an inverse kinematic algorithm to cope with pulleys. The influence of pulleys was also taken into account for the dynamic simulation of cable robots [6]. This paper aims at studying the influence of a pulley in the winch on the properties of a cable robot. Therefore, a kinematic modeling is presented and the equations for inverse kinematics and statics are derived taking into account the effect of a pulley as guiding element in the winches of the robot. The rest of the paper is organized as follows. In Sec. 2 the basic equations for the modeling of pulley mechanisms are presented while in Sec. 3 the method to determine the workspace properties is briefly explained. The results from the comparison are discussed in Sec. 4 where the paper closes with the conclusions.

2 Kinematics for pulley mechanism For better reference, the kinematic foundation of cable robots are briefly reviewed. We refer to the well-known approach as standard model and we extend it by the guiding pulley in this section. Fig. 1a shows the kinematic structure of a general spatial cable robot, where the vectors ai denote the proximal anchor points on the robot base, the vectors bi are the relative positions of the distal anchor points on the movable platform, and li denote the vector of the cables. The length of the cables is abbreviated by li = ||li ||2 . Applying a vector loop, the closure-constraint reads ai − r − R bi − li = 0 for

i = 1, . . . , m ,

(1)

where the vector r is the Cartesian position of the platform and the rotation matrix R represents the orientation of the platform.

Influence of pulley kinematics on cable-driven parallel robots

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The parameters and coordinate frame used to exactly define the geometry of a guiding pulley are depicted in Fig. 1b. In the rest of this section we omit the index i for the reference points, frames, angles, and lengths for the sake of clarity. In this paper we propose to express the pulley kinematics based on a local coordinate frame KA which largely simplifies the kinematic equation and represents a natural concept of arranging the winches in space. The pulley kinematics realizes a two degree-offreedom motion. The first revolute joint is aligned with the z-axes of frame KA . The second joint is the pulley itself and its joint axis is initially aligned with the y-axis of frame KA . The center of second rotation is initially located in point M. The distance between the two screw joint axis is the effective radius r p and it is assumed that the two joint axis are perpendicular to each other. The fixed point A in the origin of the coordinate frame KA is the characteristic point of the pulley kinematics and considered to be a design parameter a. The cable hits the pulley at point A and wraps around the pulley with an effective radius r p , i.e. the radius that applies to the neutral fibre in the center of the cable. In the following considerations we assume that r p is the effective radius, i.e. the radius resulting from both the actual radius of the pulley and the radius of the cable. Note, that this holds true only if the geometric profile of the pulley and the radius of the cable perfectly match. The cable leaves the pulley at point C and the angle between point A and C is denoted by βu . In Fig. 1c one can see the rotated pulley, where the rotation angle is denoted by γ and is taken in positive direction around the z-axis of frame KA . The definition of the angles βu and γ with respect to KA is crucial for both the formulation of the kinematic codes and the consideration of collisions between the cable and the pulley mechanism. For γ = 0 the pulley is located in the xz-plane of frame KA . Therefore, in this position βu is measured in positive direction around the y-axis of KA . The orientation of KA w.r.t. to K0 is expressed by the rotation matrix RA and assumed to be given. The kinematic equations are now derived in KA as follows. From Fig. 2, the corrected cable length taking into account the pulley radius becomes

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l = βu r p + l f ,

(2)

where βu is the angle around the pulley, r p is the pulley radius, and l f is the free cable length from point C to point B. Considering the two right-angled triangles we receive 2 (bxy − r p )2 + b2z = MB = l 2f + r2p , (3) q where bxy = b2x + b2y and bz are the coordinates of the point B with respect to frame KA in cylinder coordinates. To solve the inverse kinematics we need the angle βu which is computed as follows: Considering the tetragon CMDB, we find two angles to be right-angles. Therefore, we conclude that the enclosed angle β1 + β2 at point B equals the sought complementary angle βu at point M. Using elementary trigonometric functions yields lf bz βu = β1 + β2 = arccos p + arccos p . (bxy − r p )2 + b2z (bxy − r p )2 + b2z

(4)

Thus, we receive a closed-form solution for the cable length l. Further reduction in the computational costs can be achieved using the addition theorem for arccos. It is worthwhile to mention that one can set up similar formulas using either arctan or arcsin where both formulations require a distinction of cases when bz changes its sign. Nevertheless, using arctan and the respective addition theorem gives a very compact expression which is only valid for positive bz . The advantage of the presented formula is that one can get the symbolic derivative for the first-order kinematics without additional efforts. A unique solution for the rotation of the first joint can easily be obtained using the four-quadrant arcus tangens γ = arctan 2(by , bx )). To calculate the normal vector u along the cable in K0 we rotate a negative unit vector −ez along the z-axis with the following transformation matrices u = −RA Rz (γ ) Ry (βu )ez ,

(5)

where Ry (βu ) and Rz (γ ) are the elementary rotation matrix around the y- and z-axis, respectively. Considering the force and torque equilibrium for the platform leads to the wellknown structure equations of the standard model (see e.g. [9]) AT f + w = 0 ,

(6)

where AT is the pose-dependent structure matrix, f is the vector of the positive cable forces, and w is the applied wrench at the platform. When considering a pulley model for the robot, the type of the equation is maintained where we have to use a different unit vector for the direction of the cables as given by Eq. (5). Thus, the columns of the structure matrix become [ATp ]i = [uTi , (bi × ui )T ].


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3 Workspace To compare the results from the workspace calculation with and without pulleys we use a triangulation of the workspace’s hull [7]. Although the triangulation lacks the verified nature of interval computations [4] it can be computed with a very high accuracy at moderate computational times of some seconds. Here we use a method for the determination of the border of the workspace based on discrete investigation of single points and on a line search. In this approach the workspace is represented by triangulation that allows for simple but accurate determination of the volume and surface. Let (va , vb , vc )i be the vertices of the triangles of the border of the b be the projection center of the workspace. Note, m b is a paramworkspace W and m eter in the algorithm for workspace determination describing the point where the projection of a unit sphere is started. It is straightforward to calculate the surface S(W ) and the volume V (W ) of the workspace as follows 1 T ||(va − vb ) × (va − vc )||2 2∑ 1 T b × (vb − m)) b . (vc − m) b V (W ) = ∑((va − m) 6 S(W ) =

(7) (8)

4 Computational Results The geometrical parameters of the robot used for this study are given in Tab. 1. The parameters represent the scale of the archetype IPAnema 1, but no dot exactly match the values of the prototype. For this robot, the diameter of the cable is rC = 0.002 m and the effective radius of the pulleys is r p = 0.05 m. All local frames KA,i of the proximal anchor points Ai and thus the orientation of the winches were parallel aligned with the world frame K0 . The difference between the standard model and the pulley model for inverse kinematics is depicted in Fig. 3, where the diagrams show the difference between both inverse kinematic codes along a quadratic trajectory with 2 m edge length for dif-

Table 1 Geometrical parameters of the investigated robot given as platform vectors b and base vectors a. cable i base vector ai [m] platform vector bi [m] 1 [−2.0, 1.5, 2.0]T [−0.06, 0.06, 0.0]T 2 [2.0, 1.5, 2.0]T [0.06, 0.06, 0.0]T T 3 [2.0, −1.5, 2.0] [0.06, −0.06, 0.0]T 4 [−2.0, −1.5, 2.0]T [−0.06, −0.06, 0.0]T T 5 [−2.0, 1.5, 0.0] [−0.06, 0.06, 0.0]T 6 [2.0, 1.5, 0.0]T [0.06, 0.06, 0.0]T 7 [2.0, −1.5, 0.0]T [0.06, −0.06, 0.0]T T 8 [−2.0, −1.5, 0.0] [−0.06, −0.06, 0.0]T

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Fig. 4 Difference between the standard kinematic model and the pulley model for the forces f 1 . The left diagram shows the absolute cable forces in cable 1 for a pulley radius of r p = {0; 0.01; 0.025; 0.05; 0.1} where the right diagram shows the difference between the standard model and different radii of the pulley.

ferent radii r p of the pulley. One can easily see that the cable length computed from the extended formula is always longer than the standard model. This is clear since the distance around the pulley must be longer. In the right diagram one can see the relation between the radius of the pulley and the differences between the kinematic models. For the considered interval of pulley radii r p ∈ [0.1; 0.01] m the ratio is almost constant. Thus, the dependency between the additional length of the cable and the radius of the pulley seems to be linear in this range. We analyze the difference in the force distribution that arises from the static model taking into account the guiding pulleys. To calculate the force distribution, the following closed-form formula is used [8] f = fm − A+T (w + AT fm ),

(9)

where fm = 21 (fmin + fmax ) is the mean feasible force and w is the applied wrench. For the example we used fmin = 1 N, fmax = 10 N, and an external wrench w = 0. Note that as long as no external wrench is applied only the ratio between fmin and fmax influences the results of workspace and force distribution. Fig. 4 shows the comparison for the forces f1 in cable 1 when moving along a trajectory for different radii r p of the pulley. The differences are again in the range of some percentage and


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Table 2 Comparison of the workspace volume and surface for different radius of the pulley r p . pulley radius r p [m] volume V [m3 ] relative volume [%] surface S [m2 ] relative surface [%] 0.0 5.81 100.0 17.92 100.0 0.001 5.81 99.9 17.87 99.7 0.01 5.80 99.8 17.89 99.8 0.025 5.79 99.7 17.87 99.7 0.05 5.77 99.3 17.75 99.0 0.15 5.68 97.8 17.50 97.6 0.25 5.57 95.8 17.17 95.8 0.35 5.42 93.2 16.76 93.5 0.40 5.32 91.6 16.49 92.0

the magnitude of the difference seems to be linear for typical sizes of the pulley. For the practical use in force control the influence seems to be less important since the error in the cable forces is the scale of the measurement error caused from typical force sensors. In order to study the influence of pulleys on the workspace we compute the hull of the workspace and use the performance criteria surface and volume to compare the results for different radii of the pulleys. Force limits and external wrench were chosen as given above. To check for existence of the workspace for a pose, Eq. (9) was evaluated and the determined force was compared to the force limits as given in the previous section. The parameters of the workspace algorithms were set as follows: The iterations depth for the recursive refinement of the hill were chosen to be six leading to 16386 computed vertices and 32768 triangles. The accuracy for the line search was ε = 10−4 m such that the first four digits of the performance indices shall be meaningful. The computational results from the study are given in Tab. 2. In this evaluation we used even larger radii for the pulleys than before. The relative error of the workspace volume is less than 2% for realistic values of the pulley’s radius.

5 Conclusions In this paper we presented the modeling of a cable-driven parallel robot taking into account the effects of pulleys in the robot’s winches. Using an extended modeling for kinematics and statics the differences of the workspace between the simplified and extended model where studied. For the robot at hand it turns out that the difference of the volume and surface of the workspace is in the range of 1% for typical pulleys, i.e. with a radius of 2.5% of the robots shortest edge length. Considering other unconsidered uncertainties the influence on the workspace may be neglected in many cases. The comparison of the inverse kinematic codes and thus the expected accuracy of the robot unveils more important differences. The deviations between standard and pulley model are found to be almost linear in the considered range for the pulley radius. The shortening of the cables caused by the pulleys may signifi-

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cantly increase the inner tension in the robot and thus disturb the force equilibrium of the mobile platform. Our future research aims at extending the study of pulleys to stiffness, interference, and singularities. Furthermore, we are working towards deriving a real-time capable kinematic code for the forward kinematics of the pulley model. The presented model may serve as a basis especially because it allows for a simpler Jacobian. Anyway, more sophisticated methods are needed to meet the required real-time constraints for use in the controller.

Acknowledgement This work was partially supported by the Fraunhofer-Gesellschaft Internal Programs under Grant No. WISA 823 244.

References [1] Albus, J.S., Bostelman, R.V., Dagalakis, N.G.: The NIST RoboCrane. Journal of Robotic Systems 10(5), 709–724 (1993) [2] Baoyan, D., Yuanying, Q., Fushun, Z., Zi, B.: Analysis and experiment of the feed cable-suspended structure for super antenna. In: IEEE/ASME International Conference on Advanced Intelligent Mechatronics, 2008, pp. 329–334 (2008). [3] Bruckmann, T., Mikelsons, L., Brandt, T., Hiller, M., Schramm, D.: Wire robots part I – Kinematics, analysis and design. In: Parallel Manipulators – New Developments, ARS Robotic Books. I-Tech Education and Publishing, Vienna, Austria (2008) [4] Gouttefarde, M., Daney, D., Merlet, J.P.: Interval-analysis-based determination of the wrench-feasible workspace of parallel cable-driven robots. IEEE Transactions On Robotics 27(1), 1–13 (2011). [5] Kawamura, S., Choe, W., Tanaka, S., Pandian, S.R.: Development of an ultrahigh speed robot Falcon using wire drive system. In: IEEE International Conference on Robotics and Automation, pp. 1764–1850 (1995) [6] Miermeister, P., Pott, A.: Modelling and real-time dynamic simulation of the cable-driven parallel robot IPAnema. In: European Conference on Mechanism Science (EuCoMeS 2010), p. 353–360. Cluj-Napoca, Romania (2010) [7] Pott, A.: Forward kinematics and workspace determination of a wire robot for industrial applications. In: ARK, pp. 451–458. Springer-Verlag, Baz-sur-Mer, France (2008) [8] Pott, A., Bruckmann, T., Mikelsons, L.: Closed-form force distribution for parallel wire robots. In: Computational Kinematics, pp. 25–34. Springer-Verlag, Duisburg, Germany (2009) [9] Verhoeven, R.: Analysis of the workspace of tendon-based stewart platforms. Ph.D. thesis, University of Duisburg-Essen, Duisburg (2004)