Cable-Suspended Planar Parallel Robots with Redundant Cables: Controllers with Positive Cable Tensions

Transcription

1 Proceedings of the 2003 IEEE International Conference on Robotics & Automation Taipei, Taiwan, September 4-9, 2003 Cable-Suspended Planar Parallel Robots with Redundant Cables: Controllers with Positive Cable Tensions So-Ryeok Oh, Ph.D. Student Sunil K. Agrawal, Ph.D., Professor Mechanical Systems Laboratory, Department of Mechanical Engineering University of Delaware, Newark, DE 976, U.S.A. oh, Abstract Cable-suspended robots are structurally similar to parallel actuated robots but with the fundamental difference that cables can only pull the end-effector but not push it. From a scientific point of view, this feature makes feedback control of cable-suspended robots lot more challenging than their counterpart parallel-actuated robots. In the case with redundant cables, feedback control laws can be designed to make all tensions positive while attaining desired control performance. This paper describes these approaches and their effectiveness is demostrated through simulations of a three degree-offreedom cable suspended robots with four, five, and six cables. quadratic programming. Section 5 shows a method to find the globally continuous tensions by trajectory planning. The results of simulation are presented in Section 6. We have fabricated an experiment test-bed of a planar cable suspended robot with four cables as shown in Fig.. In this cable robot, the suspension points of the cables in the ground frame and attachment points of the cable in the end-effector frame can be changed. Due to presence of four cables, it is a redundantly actuated system and will provide a setup for validation of the ideas described in this paper. Introduction In the last decades, robots have made tremendous in-roads into industries for manufacturing and assembly. However, for long reach robotics such as inspection and repair in shipyards and airplane hangars, application of robotics is still in its infancy([],[2]). Conventional robots with serial or parallel structures are impractical for these applications since the workspace requirements are higher by three to four orders of magnitude than what the conventional robots can provide. However, cables have the unique property - they carry loads in tension but not in compression[3]. This paper presents approaches to handle input tension constraints using the robot s redundancy([4],[5]). We use the tension in the redundant cables to satisfy the positive input constraints. At a point in the state space, positive tension constraints in the input can be represented by a set of linear inequalities in the input tensions. The feasible space enclosed by these linear inequalities can be characterized mathematically and sketched graphically for up to six cables. However, selection of input tensions point-wise in the state space may not always provide continuity of the tension trajectories. Due to this reason, we also propose a second method of trajectory planning which uses trajectory parameterization in conjunction with collocation to ensure smooth input tensions during the path [6]. The organization of this paper is as follows: Section 2 describes the kinematic and dynamic equations of the robot. Section 3 outlines a method for control based on feedback linearization. Section 4 provides the details of obtaining the feasible region for the cases with 4, 5, and 6 cables. In addition, the constraint problem is reformulated for linear and Figure : A prototype of a planar cable robot with four cables fabricated at University of Delaware. The fourth cable and the motor are located at the back and are not visible in the front view. 2 System Dynamic Model Our model of a planar cable robot consists of a moving platform (MP) that is connected by n cables to a base platform showninfig.2. Thecablei is connected to MP as shown in Fig. 2. M is the center of the MP. The cable separation angles on MP is denoted by α i. An inertial reference frame F 0(XY ) is located at 0 and a moving reference frame F M (xy) is located on MP at its center of mass M. The orientation of MP is specified by θ e. The origin of F M is given by a vector from 0 to M with x e and y e as its components. 2. Cable Kinematics and Statics The position vector of point a i with respect to F M is written as [ bicα i b isα i ] T. () /03/$ IEEE 3023

2 where c and s stand for cos and sin, respectively and b i is the distance between points M and a i. The transformation matrix of frame F M with respect to frame F 0 can be written as cθe sθ e x e 0 T M = sθ e cθ e y e. (2) 0 0 Therefore, the position vector of points a i with respect to F 0 is [ 0 ] [ r M ] i = 0 r T i M,i= n. (3) Figure 3: A sketch of parameter s i. where cθ i = lix l i,sθi = liy l i,i= n and s i is the normal distance between M and the cable axis i and can be expressed using Fig. 3 as s i = b i s(θ e + α i θ i). Eqs.(7) can be written in matrix form as Au = F (8) Figure 2: A sketch of the cable system along with geometric parameters for the robot with n cable. Upon substitution of 0 T M from Eq.(2) into Eq.(3), one leads to 0 xe + b i cθ ecα i b i sθ esα i r i =,i= n. (4) y e + b i sθ ecα i + b i cθ esα i Moreover, the position vector of suspension point A i of cable i with respect to reference point 0 is written as 0 di p i =,i= n. (5) h i Hence, the vector a ia i for cable i is l i = 0 p i 0 lix r i = l iy di x e b i cθ ecα i + b i sθ esα i = h i y e b i sθ ecα i b i cθ esα i i = n. (6) The static equilibrium equation of MP can be used to obtain the forces in the cables. n Fx =0 T icθ i =0 n Fy =0 T isθ i + mg =0 (7) n Mz =0 T is i =0 where l x l l nx l n A = l y l l ny (9) l n s s n u = [ T T 2 T n ] 2.2 System Dynamics F = [ F x F y M z ]. (0) During the motion, mẍ F = m(ÿ e g), () I z θe where m is the mass and I z is the moment of inertia of the end-effector about its center of mass along Ẑ. The equations of motion can be written alternatively in the following general form Dẍ + G = A(x)u (2) where D is the inertia matrix for the system and g is the vector of gravity terms. Their expressions are [ m 0 ] 0 [ 0 ] D = 0 m 0, G = mg 0 0 I Z 0 and x =[x e,y e,θ e] T. The above dynamic model is valid only for u 0, i.e., the cables are in tension. A positive tension implies that the cable is pulling the attachment point of the end-effector. 3024

3 3 Feedback Controller In this section, we consider controllers based on feedback linearization theory to asymptotically stabilize the system to x d (t), while satisfying the property u(t) 0, t>0. We define x(t) =x(t) x d (t). For the system given in Eq. (2), with a feedback law of the form A(x)u = Dv + G, (3) the system dynamics becomes ẍ = v. (4) With the choice v = ẍ d K p x K d x, where K p and K d are positive diagonal matrices, we can ensure that x(t) will asymptotically track x d (t). The solution of Eq.(3) depends on the number of cables. With three cables, Eq.(3) has 3 equations in 3 unknowns. If the three equations are linearly independent, there is a unique solution for the problem. For more than three cables, Eq.(3) is an underdetermined system of equations and has many solutions if AA T is invertible. In this case, the general solution for Eq.(3) can be written as u = u + N(A)m. (5) Here, u is the minimum norm solution of Eq.(3) derived using the pseudo inverse of matrix A and is given by u = A T (AA T ) [ D(ẍ d K p x K d x)+g ]. (6) Here, N(A) is the null space or kernel of matrix A and m is a(n r) dimensional underdetermined vector, where r is the rank of matrix A and n is the number of cables. 4 Pointwise Feasible Region On using the input constraint u(t) 0 for all the time t>0, the resulting condition is u + N(A)m 0. (7) Since A(x) is nonlinear in x, it is hard to get a solution that is globally valid in the state space. However, it is possible to get the solution at a specific point x in the state space. It is clear from Eq.(7) that a feasible solution at a specific x is characterized by a convex region bounded by n linear inequalities on the elements of m i,i=,,k, where k is the number of linearly independent columns or rank of N(A). In the following discussions, we assume that matrix A has full row rank of 3 and the null space has a dimension (n 3). 4. One extra cable With one extra cable, the four linear inequalities in m become u n u 2 n 2 u + 3 n m 0. (8) 3 u 4 n 4 So, the feasible region F A of m is described by the common interval bounded by four linear inequalities shown in Fig. 4(a). Here p i is the solution point when each component of Eq.(8) is made to be an equality. If F A is empty, the tension constraints can not be met. Figure 4: A sketch of feasible region for m with (a) 4 cables and (b) 5 cables. 4.2 Two extra cable Two extra cables result in 2-dimensional vector m and five linear inequalities given by u n n 2 u 2 n 2 n 22 m u 3 + n 3 n (9) u 4 n 4 n 42 m 2 u 5 n 5 n 52 To determine the feasible area, the main computation steps are: (i) find the intersection points p i of every pair of two equations formed by converting the inequalities to equalities, (ii) check if the solution satisfies all the remaining inequalities. The computation of p i requires solving sets of n C 2 linear equations in two variables, followed by inequality checks to determine feasibility. A typical feasible region is shown in Fig. 4(b). 4.3 Three extra cable Three extra cables result in 3-dimensional vector m and six linear inequalities given by u n n 2 n 3 u 2 n 2 n 22 n 23 u 3 n 3 n 32 n 33 m u 4 u 5 u 6 + n 4 n 42 n 43 n 5 n 52 n 53 n 6 n 62 n 63 m 2 m 3 0. (20) It s feasible region is determined using a procedure similar to the case with two extra cables. One needs to find the intersection points p i of every set of three equations formed by converting the inequalities to corresponding equalities and check if the solution satisfies all the remaining inequalities. The feasible domain is typically a volume in 3-dimensionsl space of (m,m 2,m 3). One can generalize this method for more cables. As expected, the computations will increase with the number of extra cables. If all components of the minimum norm solution in Eq.(6) are greater than zero, one may not need to compute the feasible region. In case, the minimum norm solution results in some negative tensions, a criteria is needed to choose a 3025

4 point inside the feasible region. Several criteria are discussed in the upcoming sections based on linear programming and quadratic programming. In addition, we take the criteria of selecting a point m in the feasible area with minimum norm, i.e., minimum tensions in the extra cables that make all the tensions positive. We label this selection criteria FA. 4.4 Linear Programming A Linear Programming(LP) can be described as follows: minimize cx (2) subject to Ax b x l x x u (22) where x is the vector of decision variables, A is a matrix with constant coefficients, c and b are constant vectors. It is well known that the solution of a LP problem is one of the vertices of the feasible area. Geometrically, this solution can be obtained by shifting the level set cx = k parallel to itself along the direction of decreasing k until the solution becomes infeasible. Essentially, this results in a vertex solution. The LP cost to be minimized can be taken as m + m m k. 4.5 Quadratic Programming The structure of constraints in a quadratic programming(qp) is similar to that of LP. The objective function is allowed to contain a quadratic term such as minimize x T Hx (23) coefficients. Here, φ i(t) are chosen such that together with Φ 0(t), they span a complete set. When the form of x(t) from Eq.(24) is substituted in Eq.(7), the inequalities becomes u = u(t, P i)+n(a(t, P i))m 0. (25) Eq.(25) must be valid at all time during [0,t f ], i.e., it represents an infinite number of constraints on the coefficients P,,P M,m,,m k. A number of schemes may be used to transform these infinite constraints into a finite number of constraints. We use a collocation grid in time to form a finite number of nonlinear inequality constraints in the coefficients P,,P M,m,,m k. A finite collocation grid is selected within [0,t f ]. At each collocation point, the n constraints of Eq.(25) are satisfied. If needed, one can ensure satisfaction of the constraints between the collocation points by bounding a finite number of derivatives of the constraints at the collocation points. On choosing N +2 collocation points at t 0,t,,t N,t f such that t 0 <t < <t N <t f we get a total of (N +2)n inequalities onthemodecoefficientsp,,p M,m,,m k which have the following form u j = u(t j,p m)+n(a(t j,p m))m l 0 (26) j =,,N +2,m=,,M,l =,,k. The parameters P,,P M,m,,m k canbesolvedusing Matlab 6 routines such as fmincon, designed to solve nonlinear programming problems. This method allows computation of continuous trajectories for control. 6 Simulation where H is a positive definite matrix. With the given structure of the QP problem, the optimal solution is obtained as the point where the level set, an ellipsoid x T Hx = k centered about the origin touches the feasible region for the smallest value of k. The feasible solution may not lie on a vertex. 5 Globally Continuous Solution The solution of Eq.(7), through characterization of the feasible space in Section 4, may result in tensions which are pointwise feasible but discontinuous in time. From an experimentation point of view, it is important for the solution to be continuous to avoid instability. The objective is to develope trajectories x(t) over[0,t f ]to steer the system between given boundary conditions and satisfy the positive input constraints. One can choose x(t) to have the following form: x(t) =Φ 0(t)+ M P iφ i(t). (24) i= The form of x(t) is made admissible in the following way: (i) Φ 0(t) are 3-dimension vector functions that satisfy boundary conditions of x(t) and its derivatives, (ii) φ i(t) are scalar functions that satisfy the boundary conditions in their homogeneous form, and (iii) P i are 3-dimension vectors of constant Figure 5: A Simulink block diagram for the cable system. A simulation of the cable robot was developed in Matlab Simulink to verify the feedback controller concepts outlined in the previous sections. Fig. 5 shows the Simulink model for the controllers. The user specifies a desired position and orientation for the end-effector plate. The controller block implements the control law given in Eq.(5). The ouputs of the controller block are the cable tensions. If each tension is positive, it is fed to the system dynamic model. If at least one is negative, the inputs are modified to become positive using the null space contribution by computing the feasible solution using FA, LP, QP. 6. Four cable robot We consider a specific move of the end-effector from x 0 = [0.48, 0.24, 0 o ] T. Fig. 6 show the graphs for four control meth- 3026

5 Figure 6: Plots of position, orientation, feasible area, and cable tensions for the 4-cable robot without and with null space components using FA, LP, and QP. Figure 7: Plots of position, orientation, and cable tensions for the 5-cable robot without and with null space contributions using FA, LP, and QP. ods: (i) No null space contribution, (ii) null space contribution using FA, i.e., minimum norm of the tension in extra cable, (iii) Linear Pragramming(LP), and (iv) Quadratic Programming(QP). The first three plots of Fig. 6 show that the motions of the cable robot in position and orientation are exactly the same in all four cases. This is due to the fact that the null space term does not affect the control trajectories and thereby the state trajectories. The fourth plot is the feasible area for m. The dotted line in this plot shows the feasible point selected by the algorithm. If Eq.(6) results in positive tension, the null space contribution is not invoked. Otherwise, the algorithm is invoked to make all tensions positive. The last four plots show the cable tensions. When not using the null space term, the tension in the fourth cable is negative after 0.8 seconds. However, FA, LP, and QP use the available feasible region of the null space to prevent the tension from becoming negative. The tension histories from the algorithm FA, LP, and QP turn out to be the same. 6.2 Five cable robot For 5 cable robot, the position, orientation, and correspoding tension graphs are shown in Fig. 7. The results are similar to the case with 4 cables. Fig. 8 shows the shape of the feasible region, which has the shape of a triangle at any time instant. The figure also shows the points selected by FA method during the simulation. 6.3 Six cable robot A 6 cable model was developed to check the effect of redundancy on the feasible region in the null space. As one will expect, the feasible region is a volume at each time instance. We can observe that last two tension graphs have negative values when the null space contribution were not used. Utilizing the null space term, we can satisfy positive input constraints. The plots 4-7 in Fig. 9 show that the tensions of QP are slightly different from those of FA and LP. Figure 8: Plots of the feasible region for the null space vector of the 5-cable robot during the trajectory and selected point using FA. 6.4 Continuous Trajectory The trajectory planning method is to prevent discontinuity in tensions illustrated by the example of a four cable robot. We select the feasible solution of x(t) to have the following form. ] ] ] x(t) = [ Φ0 Φ 20 Φ 30 + [ p p 2 p 3 φ + [ p2 p 22 p 23 φ 2 (27) where Φ 0 = a 0 + a t + a 2t 2 + a 3t 3 + a 4t 3 (t f t)+a 5t 3 (t f t) 2 Φ 20 = b 0 + b t + b 2t 2 + b 3t 3 + b 4t 3 (t f t)+b 5t 3 (t f t) 2 Φ 30 = c 0 + c t + c 2t 2 + c 3t 3 + c 4t 3 (t f t)+c 5t 3 (t f t) 2 φ = t 3 (t f t) 3 φ 2 = t 4 (t f t) 3 (28). The coefficients a i,b i,c i are determined from eighteen conditions, (x, ẋ, ẍ, y, ẏ, ÿ, θ, θ, θ) 0 =(0.48, 0, 0, 0.24, 0, 0, 0 o, 0, 0) and (x, ẋ, ẍ, y, ẏ, ÿ, θ, θ, θ) 6 = (0.6, 0, 0, 0.4, 0, 0, 20 o, 0, 0) with t [0, 6]. 3027

6 Figure 9: Plots of position, orientation, and cable tensions for the 6-cable robot without and with null space contribution using FA, LP, and QP. Equally spaced five collocation points were used in the simulation and the objective function was defined by the sum of norms for x(t) and its derivatives. The solutions of p ij,m are p =0.0,p 2 = 0.0,p 3 =0.02,p 2 =0.0,p 22 =0,p 23 = 0.0, and m =2.78. The solid lines in plots -3 of Fig. 0 are the planned trajectories which are calculated by the proposed scheme mentioned and are used as desired trajectories of the planar robot. Dotted lines are the actual state trajectories while tracking the desired trajectories starting out from an initial error. Plot 4-8 in Fig. 0 are the tension histories, which show that this method satisfies not only the tension constraints but also results in smooth tensions. Figure 0: Plots of planned trajectory(solid line), actual states(dotted line) of the positions and orientation, and four tension. 7 Conclusions This paper presented approaches for control of cable suspended robots with redundant cables to follow prescribed trajectories, while keeping all tensions positive during motion. This was achieved by adjusting the tensions in the extra cables by identifying a feasible space for these tensions. It was shown that this feasible space is a polytope at any given point in the state space and is described by a set of linear inequalities. This polytope was sketched for planar cable robots with one, two, and three extra cables. The feasible space is a line for one extra cable and increases in dimension by one, i.e., becomes an area and a volume as extra cables are added. The performance of the controllers was compared when null space contribution was added from the cables using simple heuristics, linear programming, and quadratic programming. These methods were classified as point-wise since they do not ensure continuity of the cable tensions as the trajectory evolves in time. An alternative method was proposed which uses a global continuous description of the trajectory while satisfying the constraints. Dynamic simulations were presented to show the effectiveness of all these methods with satisfactory results. 8 Acknowledgments The authors appreciate financial supports of NSF award No. IIS-07733, NIST MEL award No. 60NANB2D037, and PTI/NIST award No. AGR References [] Aghili, F., Buehler, and Hollerbach, J. M., Dynamics and Control of Direct-Drive Robots with Positive Joint Torque Feedback, Proceedings of International Conference on Robotics and Automation, Albuquerque, New Mexico, 56-6, 997. [2] Albus, J., Bostelman, R., and Dagalakis, N., The NIST Robocrane, Journal of Research of National Institute of Science and Technology, Vol. 97, No. 3, , 992. [3] Abdullah B. Alp, Sunil K. Agrawal, Cable Suspended Robots: Design, Planning and Control, Proceedings of International Conference on Robotics and Automation, Washington, DC, , [4] A.A. Maciejewski, C.A. Klein, Obstacle avoidance for kinematically redundant manipulators in dynamically varying environments, International Journal of Robotics Research, Vol 5, No. 4, 09-7, 985. [5] Y. Nakamura, H. Hanafusa, Inverse Kinematics Solutions with Singularity Robustness for Robot Manipulator Control, Transaction of the ASME Journal of Dynamic System, Measurement, and Control, Vol. 08, 63-7, 986. [6] Faiz, N. and Agrawal, S. K., Murray, R. M., Trajectory Planning of Differentially Flat Systems with Dynamics and Inequalities, AIAA Journal of Guidance, Control, ad Dynamics, Vol. 24, No. 2, ,