Dynamic Point to Point Trajectory Planning of a Two DOF Cable Suspended Parallel Robot06730645

XAMPLE V. E T RAJECTORIES AND E XPERIMENTAL V ALIDATION

Fig. 6. Tensions in the cables for the trajectory obtained with the polynomial

A prototype of a two-dof planar cable-suspended robot was devel-

formulation.

oped in order to validate the trajectory planning techniques experi- mentally. The prototype is shown in Fig. 4, where it is performing a circular trajectory that extends beyond the static workspace. The dis-

trajectory is then verified for positive cable tensions. The results are tance between the cable attachment points on the frame is

a = 0.79 m shown in Figs. 5 to 8. Fig. 5 shows the target points and the Cartesian and the mass of the end effector is m = 0.196 kg. Two servo-controlled

trajectory connecting the points, obtained with the polynomial formu- winches are used to control the length of the cables. In [13], periodic

lation. It can be observed that, since a zero vertical acceleration is trajectories were developed for a two-dof planar cable-suspended robot

prescribed at the target points, the trajectory segments tend to curve including vertical oscillations, horizontal oscillations (described previ-

in order to produce soft landings at these points. Fig. 6 provides the ously in Section III), and circular motions. These periodic trajectories

tensions in the cables throughout the trajectory. It can be observed that were implemented on the prototype, and they are demonstrated in the

the tensions are continuous and always positive. Some of the lowest first multimedia extension of this paper (see Extension1-periodic.wmv

cable tensions are encountered in the trajectory segment connecting as supplementary material).

points 2 and 3. This drop of tension is related to the nonsymmetric dis- tribution of the trajectory segment with respect to the central vertical line. In this situation, cable 1 is required to pull the end effector to the

A. Polynomial Trajectories right-hand side of the static workspace starting from a configuration The polynomial point-to-point trajectory-planning technique pre-

(point 2) which lies almost on the vertical from the attachment point sented in this paper was tested on the prototype. An example trajectory

of cable 2. Therefore, cable 2 must reduce the tension applied on the is described as a set of target points in Fig. 5, where the initial and final

end effector in order to let cable 1 provide as much horizontal force as points coincide (point 15). The robot is requested to start from a refer-

possible. Fig. 7 shows the velocity components of the end effector for ence configuration located on the central axis of the static workspace

the trajectory. It can be observed that both components are zero at each (x = 2, y = 0) and to then sequentially proceed to the target points.

of the target points, as prescribed. In addition, it can be observed that Based on the polynomial formulation, the travel times are computed

the velocity component in the horizontal direction ( ˙y) is alternating be- for each trajectory segment using (38), and the fifth-degree polynomi-

tween negative and positive values for successive trajectory segments, als of (14) and (21) are used to ensure continuous accelerations. The

while the sign of the vertical component ( ˙x) may remain the same for

734 IEEE TRANSACTIONS ON ROBOTICS, VOL. 30, NO. 3, JUNE 2014

Fig. 7. Cartesian velocities for the trajectory obtained with the polynomial formulation.

Fig. 9.

Target points and Cartesian trajectory obtained with the trigonometric formulation.

Fig. 8. Cartesian accelerations for the trajectory obtained with the polynomial formulation.

Fig. 10. Tensions in the cables for the trajectory obtained with the trigono- metric formulation.

several consecutive intervals. Finally, Fig. 8 shows the components of the acceleration of the end effector. It can be observed that the acceler-

can be considered an effective exploitation of the dynamics of the ations are continuous, as prescribed. In addition, it can be observed that

robot.

the acceleration in the horizontal direction reaches a maximum when the target points are reached (zero velocity), while the acceleration in the vertical direction is equal to zero at the target points (zero velocity

B. Harmonic Trajectories

and acceleration). An example trajectory based on the formulation using trigonometric Finally, a video demonstrating an example point-to-point tra-

functions is now presented, which was also tested on the prototype. jectory is presented in the second multimedia extension (see

The trajectory is described as a set of target points in Fig. 9, where the Extension2-point-to-point.wmv as supplementary material). The tra-

initial and final points coincide (point 6). Similarly to the polynomial jectory demonstrated in the video was produced using the polynomial

trajectory, the robot is requested to start from a reference configuration formulation described in this paper. It can be observed in the video that

located on the central axis of the static workspace (x = 2, y = 0) and the trajectories are smooth, that the target points are properly reached

to then sequentially proceed to the target points. and that tension is maintained in the cables. Prescribing a zero velocity

The expressions of (39) and (40) are used to plan the trajectory seg- and acceleration in the vertical direction at the target points helps to

ments between the target points. The parameters of the segments are ensure smoothness. In addition, it can be observed that the travel time

determined using (50) in order to ensure the continuity and the feasi- between two consecutive points is adjusted according to the position

bility of the trajectory are verified using inequalities (65) to (72). The of the latter points. Indeed, referring to (35) and (38), it can be inferred

results are shown in Figs. 9 to 12. Fig. 9 shows the target points and that when the robot is operating closer to the attachment points of the

the Cartesian trajectory connecting the points, which are obtained with

the trigonometic formulation. It can be observed that the shape of the the special frequency and, therefore, reduces the travel time. Hence,

cables (higher in the workspace), x 0 becomes smaller, which increases

trajectories differs slightly from that obtained with the polynomials, al- the robot tends to move more swiftly, as observed in the resulting

though the difference is hardly perceptible (visually) on the prototype. trajectory and in the video. This behavior is somewhat intuitive and

Fig. 10 provides the tensions in the cables throughout the trajectory.

IEEE TRANSACTIONS ON ROBOTICS, VOL. 30, NO. 3, JUNE 2014 735

extends beyond its footprint. Two approaches were proposed. In the first approach, polynomial trajectories are designed in order to ensure the continuity of the accelerations throughout the trajectory. The travel time between two consecutive points is adjusted based on the math- ematical analysis of periodic point-to-point motion. This approach, although somewhat heuristic, produces feasible trajectories for a va- riety of possible consecutive target points. In the second approach, trigonometric functions are used to connect the target points while en- suring continuity of the accelerations. The advantage of the second approach is that the feasibility (positive cable tensions) can be verified using simple algebraic expressions and does not require a discretization of the trajectory. On the other hand, the second approach may fail to connect prescribed points that can be successfully produced with the first approach. Therefore, each of the approaches has its advantages and drawbacks. An experimental validation of the polynomial trajectories is demonstrated in the accompanying videos. Although not shown here, the results obtained with the trigonometric approach are visually very

Fig. 11. Cartesian velocities for the trajectory obtained with the trigonometric formulation.

similar. The experiments show that the proposed approaches are feasi- ble, stable, and robust. Indeed, it can be observed that the mechanism is capable of performing point-to-point trajectories very effectively.

This study opens the avenue to using cable-suspended robots beyond their static workspace. Current study includes the extension of the proposed approaches to 3-dof cable-suspended robots, such as those studied in [14], and the development of a constructive method that would allow one to determine the set of target points that can be attained with positive cable tensions from a given point in the Cartesian workspace.

ULTIMEDIA VII. M E XTENSIONS

1) Extension1-periodic.wmv—Experimental demonstration on the prototype of periodic trajectories: vertical oscillations, hori- zontal oscillations, and circular motions.

2) Extension2-point-to-point.wmv—Experimental demonstra- tion on the prototype of point-to-point polynomial trajectories.

Fig. 12. Cartesian accelerations for the trajectory obtained with the trigono- metric formulation.

R EFERENCES

[1] S. Kawamura, W. Choe, S. Tanaka, and S. Pandian, “Development of an The tensions are continuous and always positive. As opposed to the

ultrahigh speed robot FALCON using wire drive system,” in Proc. IEEE Int. Conf. Robot. Autom. polynomial trajectory, no discretization was required in order to guar- , Nagoya, Japan, May 21–27, 1995, pp. 215–220. [2] S. Tadokoro, Y. Murao, M. Hiller, R. Murata, H. Kohkawa, and antee that the trajectory based on trigonometric functions is feasible

T. Matsushima, “A motion base with 6-dof by parallel cable drive ar- (positive tensions). Indeed, only the extrema of the trajectory segments

chitecture,” IEEE/ASME Trans. Mechatron., vol. 7, no. 2, pp. 115–123, need to be tested. Fig. 11 shows the velocity components of the end

Jun. 2002.

effector for the trajectory. The velocity components are both zero at the [3] J. Albus, R. Bostelman, and N. Dagalakis, “The NIST robocrane,” J. Robot. Syst. , vol. 10, no. 5, pp. 709–724, 1993.

target points, as prescribed by construction of the trigonometric trajec- [4] J. Pusey, A. Fattah, S. Agrawal, and E. Messina, “Design and workspace tories. However, as opposed to the case of the polynomial trajectories,

analysis of a 6-6 cable-suspended parallel robot,” Mech. Mach. Theory, the acceleration in the vertical direction ( x-direction) is not prescribed

vol. 39, pp. 761–778, 2004.

to be zero at the target points, which makes the velocity profile look [5] C. Gosselin and S. Bouchard, “A gravity-powered mechanism for extend- ing the workspace of a cable-driven parallel mechanism: Application to

more like a harmonic function. Finally, Fig. 12 shows the components the appearance modelling of objects,” Int. J. Autom. Technol., vol. 4, no. 4, of the acceleration of the end effector. It can be observed that the accel-

pp. 372–379, Jul. 2010.

erations are continuous, as prescribed. In addition, as opposed to the [6] L. L. Cone, “Skycam, an aerial robotic camera system,” Byte, pp. 122–132, polynomial trajectory, the acceleration in the vertical direction ( x) is in ¨

Oct. 1985.

general not zero at the target points. [7] A. T. Riechel and I. Ebert-Uphoff, “Force-feasible workspace analysis for underconstrained, point-mass cable robots,” in Proc. IEEE Int. Conf. Robot. Autom. , New Orleans, LA, USA, Apr. 26–May 1 2004, pp. 4956–

4962. [8] D. Cunningham and H. Asada, “The Winch-Bot: A cable-suspended, This paper addressed the dynamic trajectory planning of two-dof

ONCLUSION VI. C

under-actuated robot utilizing parametric self-excitation,” in Proc. IEEE cable-suspended parallel mechanisms for point-to-point motion. The

Int. Conf. Robot. Autom. , Kobe, Japan, May 12–17, 2009, pp. 1844–1850. trajectories to be performed are prescribed as a set of target points to

[9] S. Lefranc¸ois and C. Gosselin, “Point-to-point motion control of a pendulum-like 3-dof underactuated cable-driven robot,” in Proc. IEEE Int.

be reached with zero velocity. This type of trajectory is of practical Conf. Robot. Autom. , Anchorage, AK, USA, May 3–8, 2010, pp. 5187– interest for applications in which the working range of the mechanism

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[10] D. Zanotto, G. Rosati, and S. K. Agrawal, “Modeling and control of a 3-DOF pendulum-like manipulator,” in Proc. IEEE Int. Conf. Robot. Au- tom. , Shanghai, China, May 9–13, 2011, pp. 3964–3969.

[11] N. Zoso and C. Gosselin, “Point-to-point motion planning of a parallel 3-DOF underactuated cable-suspended robot,” in Proc. IEEE Int. Conf.

(a)

Robot. Autom. , St. Paul, MN, USA, May 14–18, 2012, pp. 2325–2330. [12] G. Barrette and C. Gosselin, “Determination of the dynamic workspace of cable-driven planar parallel mechanisms,” ASME J. Mech. Design, vol. 127, no. 2, pp. 242–248, 2005.

[13] C. Gosselin, P. Ren, and S. Foucault, “Dynamic trajectory planning of a two-DOF cable-suspended parallel robot,” in Proc. IEEE Int. Conf. Robot. Autom. , St. Paul, MN, USA, pp. 1476–1481, May 14–18, 2012.

[14] C. Gosselin, “Global planning of dynamically feasible trajectories for three-dof spatial cable-suspended parallel robots,” in Proc. 1st Int. Conf. Cable-Driven Parallel Robots , Stuttgart, Germany, Sep. 2–4, 2012, pp. 3–

(b)

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Fig. 1. Nonlinear (a) elastic and (b) viscoelastic models of contact in which the nonlinearity is encapsulated in a gearbox with a position-dependent gear ratio of g(z).

A New Nonlinear Model of Contact Normal Force

Morteza Azad and Roy Featherstone and introduced a nonlinear equation for the normal force between a sphere and the ground as

Abstract—This paper presents a new nonlinear model of the normal

F = kz n + λz p ˙z q (1)

force that arises during compliant contact between two spheres, or between a sphere and a flat plate. It differs from a well-known existing model by only

where z is the deformation variable, which is defined as the penetration

a single term. The advantage of the new model is that it accurately predicts the measured values of the coefficient of restitution between spheres and

distance of the undeformed sphere into the undeformed ground, ˙z is

plates of various materials, whereas other models do not.

the rate of deformation, k and λ are the coefficients of the spring and damper, respectively, and

n, p, and q are constant parameters. They chose the values of these parameters as n= linear damping, normal force. 3 (to get similar results to Hertz’s theory [6]), p= 3 2 2 , and q = 1 (to be able to determine the value of λ with respect to k conveniently), resulting in the equation

Index Terms—Animation and simulation, compliant contact model, non-

NTRODUCTION I. I

Since robots usually make contact with their environment during the ˙z. (2) execution of their tasks (e.g., grasping, walking, rolling, etc.), modeling

3 F = kz 3 2 + λz 2

This model has since appeared in [8] and [10], and has become the contact is an unavoidable part of most studies in this field. Two major well known in the robotics community. We shall refer to it as the forces appear during contact: the normal force and the friction force. In Hunt/Crossley model. Following a different line of reasoning, Lee and this paper, we focus on the normal force and introduce a new model for Wang [9] obtained a model identical to (2), but with p = 1 instead of calculating it. This new model agrees well with existing measurements

2 . Our new model also differs from (2) only in the value of of the coefficient of restitution between spheres and plates of various p, which

we show has to be p= 1 2 .

materials. In the remainder of this paper, we first derive the new model and Contact models can be classified into two types: rigid and com- then test it against experimentally measured values of the coefficient of pliant [3]. The model presented in this paper is compliant. The key

difference is that compliant models assume a small amount of local restitution between spheres and plates of various materials. It is shown deformation at the contact, which allows the contact forces to be ex-

that the new model accurately fits the experimental data, whereas the Hunt/Crossley model does not. The new model is based on an idea that

pressed as functions of local position and velocity variables. This fea- appeared originally in [1], which studied the contact between a rigid ture makes it relatively easy to incorporate a compliant contact model

sphere and a compliant plate.

into a dynamics simulator. Most previous studies of compliant contact models have considered the contact between two spheres, or between a sphere and a flat plate [2],

ONTACT II. C F ORCE M ODEL [5], [8], [10]. For example, Hunt and Crossley [5] modeled the ground

The contact force in (1) is the sum of an elastic component and a as a nonlinear spring–damper pair at the contact point with the sphere,

dissipative component. The expression 3 kz 2 in (2) is the elastic force predicted by Hertz’s theory for contact between a sphere and a plate, and

is known to be correct. As Hertz’s theory is based on an assumption of Manuscript received August 1, 2013; accepted November 21, 2013. Date of

linear elasticity in the contacting solids, it follows that the nonlinearity publication December 19, 2013; date of current version June 3, 2014. This paper

was recommended for publication by Associate Editor J. Dai and Editor B. J. in the elastic force expression is due to the nonlinear geometry at the Nelson upon evaluation of the reviewers’ comments.

contact (i.e., the curved surface of the sphere). M. Azad is with the School of Engineering, The Australian National

We can model the elastic contact force with a linear spring and a University, Canberra, Acton, A.C.T. 0200, Australia (e-mail: morteza.azad@

nonlinear gearbox, as shown in Fig. 1(a). The spring represents the anu.edu.au).

K depends Robotics, Istituto Italiano di Tecnologia, Genova 16163, Italy (e-mail:

lumped elasticity of the contacting bodies, and its stiffness R. Featherstone is a Visiting Professor with the Department Advanced

on the material properties of the contacting bodies. The gearbox has a roy.featherstone@ieee.org).

deformation-dependent gear ratio of g(z) and represents the nonlinear Digital Object Identifier 10.1109/TRO.2013.2293833

variation of contact area and strain distribution with z.