A Novel Design to Improve Pose Accuracy for Cable Robots

Transcription

1 The 14th IFToMM World Congress, Taipei, Taiwan, October 25-30, 2015 DOI Number: /IFToMM.14TH.WC.OS2.018 A Novel Design to Improve Pose Accuracy for Cable Robots A. Gonzalez-Rodríguez Univ. Castilla-La Mancha Ciudad Real, Spain E. Ottaviano Univ. of Cassino and Southern Lazio, Cassino, Italy F.J. Castillo-García Univ. Castilla-La Mancha Toledo, Spain P. Rea Univ. of Cassino and Southern Lazio, Cassino, Italy Abstract Cable robots are parallel manipulators in which rigid links are replaced by cables, while actuator and transmission systems are responsible for commanding the end-effector by rolling cables in and out by means of drums and changing cable direction by means of pulleys. Commonly developed kinetostatic models assume that the robot frame and end-effector are point-to-point connected. This assumption may induce non-negligible errors in the end-effector pose when performing pick and place operations or trajectory tracking. This paper deals with a new design for connecting the frame and end-effector to reduce any modeling inaccuracies due to the aforementioned assumption. Experimental tests are presented for a planar 4 cable robot. Keywords: Cable robot 1, Kinetostatics, Pose inaccuracies, Trajectory tracking, Experimental tests. I. Introduction Cable robots are parallel manipulators [1] in which rigid links are replaced by cables that connect the fixed frame and end-effector/payload [2]. Cables are commanded by actuators (commonly DC motors) which are used to extend or retract each cable. In this way, actuators can change cables length between frame and end-effector and therefore the endeffector can perform translation motion [3] or spatial motion, being suspended in a crane like configuration [4], providing a 2D or 3D workspaces [5, 6]. Some of the main advantages of cable robots are: They are able to move heavy payloads with a high efficient use of energy. As end-effector is connected to fixed frame only by a set of cables, whose mass is usually negligible with respect to the payload, actuators power is used to move the robot payload. They can be designed to provide a large workspace. Conventional manipulators can be designed only to achieve limited workspaces owing to the physical limitation related to the mass of links. In cable robots this limitation reduces drastically due to the negligible mass of the cables and, therefore, cable robots can be designed to operate into large workspaces [7, 8]. On the other hand, using rigid links and transmissions for designing and building robots is an extended solution because the measurement of motors positions allows a direct and accurate tip position estimation and therefore its control. For classical robots with rigid links tip position, Q, can be estimated by using Forward Kinematics (FK), Λ, i.e. Q = Λ (Θ) (1) where Θ is the joints position, which can be directly measured. Inverse Kinematics (IK), Λ, is used to generate joint trajectories for a desired tip trajectory Θ = Λ (Q) (2) In the case of cable robots, the statics/dynamic equilibrium should be verified for each pose to make it feasible. For the Static equilibrium, it holds that F 1 L 1 Ln P + 0 r L r L M 1 1 n n F n in which F i is the cable tension applied to cable length unit direction; P and M are the resultant force and torque (giving a wrench W) which are exerted on or by the environment. Rearranging (3) in matrix form yields T A (3) F W (4) where A is the so called structure matrix. The feasible force distribution for cable robots is a major challenge as regards design and control issues. The basic problem is to find positive solutions for (4), which can be summarized as F W 0 with 0 fmin fi fmax (5) T A where f min denotes the minimum force in the cables e.g. to tense it; and the maximal load f max is owing to limitations in the actuator or the breaking load of the cable. It is possible to determine the least-square solution F of (5) by means of the inverse of Moore-Penrose matrix, as described in [2]. This paper deals with the design of cable robots, the antonio.gonzalez@uclm.es ottaviano@unicas.it fernando.castillo@uclm.es rea@unicas.it

2 applicability of kinetostatic model described above, and proposes a new design solution, which allows improving the pose accuracy and trajectory tracking. The paper is organized as follows: Section II presents the cable robots design and its kinetostatic model. Section III details the proposed solution for improving pose accuracy. Section IV presents experimental results and, finally, Section V summarizes main conclusions. Although this work is oriented to planar cable robots the formulation and results can be extended to spatial cases. II. Cable robots design and kinetostatic model A. Cable robots design Cable robots are composed of: Fixed frame, which supports the robot. End-effector, whose pose needs to be controlled. A set of cables, which connects frame to end-effector. Actuator and transmission systems, responsible of commanding each cable and changing its length and direction. Figure 1 illustrates all these functional elements for the case of a planar cable robot. Actuator and transmission systems are composed by: DC motors, for providing rotational movement. Drums, for rolling in and out the cables. Pulleys, for changing cable direction. Anchor points/point joint, for connecting cables to the end-effector/payload. Figure 2 shows actuator and transmission systems. This mechanical solution, consisting of connecting frame and end-effector by means of drum, pulleys and point joint, is the most extended one as reported in [9-11]. Although recent works studied the influence of frame pulleys on the kinematics problem [12-14], the most extended solution does not take them into account. As an example, in [12], results describing the kinematics errors were computed, for a pulley radius of r=0.1m, the difference of cable length owing the pulley presence can be more than 0.15m. B. Cable robots kinetostatic model: a planar case Let s assume a planar cable robot in Figs. 1 and 2 with endeffector pose Q = [x, y, δ] and joints angular position Θ = [θ, θ, θ ], being x and y, the Cartesian coordinates of the center of mass of the end-effector on X and Y axis, respectively, δ the rotation of end-effector respect to an axis perpendicular to XY plane, and θ the angular position of motor/gearbox set i (see Fig. 2). As it was mentioned in Section I, FK is denoted as Λ and IK as Λ. In the following, Statics can be solved by assuming no other forces than gravity effects. The commonly used kinetostatic model assumes that end-effector is commanded by varying cables lengths. Furthermore, used simplification assumes that cables lengths are measured from a fixed point at the frame to a fixed point at the end-effector [15-19]. Let us denote this as point-to-point model. Under this assumption the kinetostatic model for the case understudy can be developed as follows. Let s assume that frame and end-effector are connected by means of n cables which are commanded by n motors. These n cables connect frame, from n fixed points Q to its respective n points at end-effector Q (i = 1,2, n). Let s also denote the n cables lengths as L and the n cables angles as α. Figure 3 clarify this nomenclature. The IK in (2) can be used to obtain the motors angular position, Θ, for a desired end-effector pose, Q. Let s therefore define a known initial end-effector pose Q at which all connection points are also known Q (they only depends on the geometry of the end-effector) as in Fig. 4. Fig. 1. Functional elements of a cable robot. Fig. 3. A scheme for frame and end-effector points, cables angles and cables lengths. Fig. 2. Functional elements of the actuator and transmission systems. Fig. 4. A scheme for the end-effector initial pose.

3 At that initial pose we consider that motors angular position are zero, Θ = 0. The geometry of end-effector defines all Q that can be determined by initial pose Q, d and γ (see Fig. 4). Since initial pose and end-effector geometry are defined we can obtain inverse kinematics Λ. Let s assume an endeffector pose Q = [x, y, δ], motor angular position yields Θ Θ = 1 r (L L ) (6) being L = [L, L,, L ] and r the pulley radius. Θ and L are known parameters, which only depend by the endeffector geometry and initial pose. On the other hand, L can be determined by end-effector connection points Q = [x, y ] and frame connection points Q = [x, y ] as L = x x + y y (7) the frame, Q, but when pulleys are included in the mechanical design cables start at their correspondent tangent points, which are not fixed points but depend by the endeffector pose. Figure 5 illustrates the above-mentioned issue and shows a comparison between the point-to-point model and a realistic model for connecting frame and end-effector. If we overlap Fig. 5a) and 5b) it results Fig. 6. It can be observed that cables directions are different between the two models. The cables lengths are also different, as well as the cable forces, and therefore Kinetostatics in (9) to (11) will not provide the desired end-effector pose, with evaluated cables' tensions inducing inaccurate pose and inaccurate trajectory tracking. Section IV illustrates these issues by providing an example. Although some authors have tackled this problem by modifying kinetostatic model taking into account pulleys effects [20, 21] the most extended solution is still applying (9) to (11) owing to its simplicity. Q are fixed known values since Q can be written as Q = R Q (8) being R the rotation matrix through δ. Combining (6), 7) and (8) IK is given by θ (x, y, δ) = θ + 1 r xd cos ( γ + δ) x + yd sin(γ + δ) y x d cos ( γ ) x + y d sin(γ ) y (9) for i = 1, n and where [x, y, δ] is the desired endeffector pose, d and γ are determined by the end-effector geometry and x and y by the frame connection points. The Statics can be written as (4) where P = [0, -mg, 0] T and M = 0 for the case understudy. The structure matrix A T in (4) is the transpose of the Static Jacobian matrix, which can be written as follows Fig. 5. Point-to-point model VS realistic connection points on the frame. J J J J J J J J J J n s n n (10) where J J 1i 2i cos sin i i J cos x x sin y y 3i i ei i ei (11) The above-mentioned formulation is general of the planar cable robot in Fig. 2. Although this Kinetostatic model is widely recognized and used it has been developed assuming a point-to point model, i.e. cables are assumed to start from fixed points placed at Fig. 6. A comparison between point-to-point model VS realistic model for connection points on the frame. III. A New design solution In order to avoid the aforementioned problems, this paper proposes a novel mechanical design to modify the physical connection between the frame and end-effector.

4 Our proposal consists by connecting the cables to endeffector by adding fixed pulleys (with the same radius of frame pulleys) at the end-effector. These fixed pulleys are locked and they cannot rotate. Fig. 7 illustrates this concept by comparing the new proposed way for connecting frame and end-effector to the point-to-point model. If we overlap Fig. 7a) and 7b) it results Fig. 8. It can be observed that, with the proposed design solution, it is possible to use the Kinetostatic model in (6) to (11) since it does not originate inaccurate pose or trajectory tracking when operates a real cable robot with pulleys in its mechanical design. The following Section provides a comparison between numerical and experimental tests for the proposed solution. IV. Example: a 4 cable planar robot For illustrative purposes let s assume the 4 cables planar robot in Fig. 9 with the design parameters summarized in Table I. For the example end-effector geometry is given by d = d = d = d = mm γ = γ = rad, γ = γ = rad (12) The robot workspace ensuring positive cables tension is shown in Fig. 10. Let s assume initial pose: Q = [1020mm 1025mm 0rad] at which motors are referenced as Θ = 0 rad. Denoting the desired spatial trajectory as Q, the example trajectory for end-effector, from Q to Q, is represented in Fig. 11. Fig. 7. Model VS new proposal for connection points on the frame. Adding pulleys at end-effector connection points makes that real cable lengths (see points A-B in Fig. 8) are the same than point-to-point model ones (see points C-D in Fig. 8) owing that both pulleys, frame and end-effector ones, are of same radius, and therefore, points A-B-C-D form a parallelogram. As it is shown in [12], the conventional solution induces difference between real lengths (see points A-B in Fig. 7) and model ones (see points C-D in Fig. 7). Cable directions are exactly the same, and therefore cables lengths, as well as cables tensions are also the same, resulting in accurate modeling. It is important to mention that adding pulleys at end-effector can be found in some works in which they are used with the objective of providing to actuator system the capability to keep positive cable tension [22]. Fig. 9. An example: a 4 cables planar robot. Fig. 10. Robot workspace. TABLE I. The 4 cables planar robot design parameters. Fig. 8. Point-to-point model VS realistic model for connection points on the frame. w h W H 240 mm 300 mm 2040 mm 2050 mm

5 Fig. 11. Example trajectory. We can apply (6) to (11) to obtain motor position Θ for describing the desired trajectory Q. A laboratory prototype has been built for an experimental validation of the proposed novel design concept. Once that motor position Θ has been computed, we are going to compare experimental results if we apply those joint references Θ on the laboratory prototype having the parameters of Table I and pulley radius of 30mm. The pointto-point model with its formulation is used to operate the two end-effector solutions reported in Fig. 12, one with point-topoint connection at end-effector (Fig. 12a) and one with fixed pulleys at the end-effector (Fig. 12b). a) b) Fig. 13. Compared experimental results: a) point-to-point; b) fixed pulley. a) b) Fig. 12. Compared connections for the end-effector prototype: a) point-to-point connection; b) fixed pulleys at the end-effector. Applying the aforementioned motor position reference Θ in both cases the experimental results are shown in Fig. 13. The dotted red line represents the desired end-effector reference trajectory (see Fig. 13) and black continuous line represents the obtained one. Experimental results show that with the same reference trajectory that is obtained from applying (9) to (11) to the desired reference Q, the cable robot with end-effector pulleys connections is much more accurate than the conventional one with point end-effector connections. In particular, the maximum deviation from desired to real one obtained with the point connection solution (see Fig. 13a) is about 34 mm. The maximum deviation obtained with the pulley connection solution (see Fig. 13b)) is about 2 mm. Details about experimental set-up of the prototype, such as actuators type, end-effector payload, time profile trajectory are summarized in Appendix A. A mapping of the overall deviation is reported in Fig. 14, in which the length deviation of cable i has been defined as ϵ = L L (13) where L is cable length assuming point to point connection and L is cable length with the real pulley to point connection. Figure 13 reports the deviation for cables 1 and 2 of the robot under study within its workspace. The deviation of cable i increases when the end-effector is close to its connection point at frame. This deviation goes from 0mm when end-effector is far from its connection point at frame to 80mm when is close to it. a) b) Fig. 14. Realistic connection: cable length deviation: a) cable 1; b) cable 2.

6 V. Conclusions Cable robots are used for a number of industrial and nonconventional applications. The widely used and accepted model for Kinetostatics assumes a point-to-point connection between cables to end-effector although mostly used mechanical designs makes use of pulleys. This dichotomy may cause large pose deviations and inaccurate trajectory tracking. In this paper a new design is proposed to solve the aforementioned problem and consists of adding fixes pulleys (that do not rotate) at the end-effector. A planar 4 cable robot is considered and numerical and experimental tests are reported to show the effectiveness of the solution. Appendix A: Experimental platform details Figure 15 shows the laboratory set-up. The actuators consist of four 0.97KW Synchronous Servomotor Bonfiglioli BTD3 controlled by means of control unit Active cube closing a velocity control loop. Quadrature encoders were added in order to get an angular position control loop by means of needle PD-type controllers [23]. For adapting motor speed and torque a 1:10 worm gearbox was provided to each motor. The position control loop of each motor was closed by means of a conventional computer connected to two MultiDAQ board NI-USB The sample time is 8 ms. References Fig. 15. Laboratory set-up for the planar 4 cable robot. [1] Merlet J.-P., Parallel Robots (Second Edition), Springer, Dordrecht, [2] Bruckmann T., Mikelsons L., Brandt T., Hiller M., Schramm D., Wire Robots Part I Kinematics, Analysis & Design, in Ryu J.-H. Parallel Manipulators, New Developments, I-Tech Ed. and Pub., Vienna, [3] Castelli G., Ottaviano E., Gonzalez A.. Analysis and simulation of a new Cartesian cable-suspended robot. Proc. of the Inst. of Mechanical Engineers, Part C: Jnl of Mechanical Eng. 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