Geometric Design of 3R Robot Manipulators for Reaching Four End-Effector Spatial Poses

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1 Eric Lee Constantinos Mavroidis Robotics and Mechatronics Laboratory Department of Mechanical and Aerospace Engineering Rutgers University, The State University of New Jersey 98 Brett Road Piscataway, NJ 08854, USA Geometric Design of 3R Robot Manipulators for Reaching Four End-Effector Spatial Poses Abstract In this paper, the four-precision-point geometric design problem of serial-link robot manipulators with three revolute joints is solved using a polynomial continuation method. At each precision point, the end-effector spatial locations are defined. The dimensions of the geometric parameters of the 3R manipulator are computed so that the manipulator s end-effector will be able to reach these four pre-specified locations. Denavit and Hartenberg parameters and 4 4 homogeneous matrices are used to formulate the problem and obtain the design equations. Three of the design parameters are set as free choices and their values are selected arbitrarily. Two different cases for selecting the free choices are considered and their design equations are solved using polynomial homotopy continuation. In both cases for free choice selection, 36 distinct manipulators are found, the end-effectors of which can reach the four specified spatial positions and orientations. KEY WORDS geometric design, robot manipulators, polynomial homotopy continuation 1. Introduction The calculation of the geometric parameters of a multiarticulated mechanical or robotic system so that it guides a rigid body in a number of specified spatial locations or precision points is known as the Rigid Body Guidance Problem.It is also called the Geometric Design Problem (Lee and Mavroidis 2002a). This problem has been studied extensively for planar mechanisms and robotic systems and has recently drawn much attention to researchers for spatial multi-articulated systems (Bodduluri et al. 1993). The number of precision points that may be prescribed for a given mechanism or manipulator, so that it guides a rigid body exactly through the specified pre- The International Journal of Robotics Research Vol. 23, No. 3, March 2004, pp , DOI: / Sage Publications cision points, is limited by the system type and the number of design parameters that are selected to be free choices (Suh and Radcliffe 1978). This number can be calculated using the formula of Tsai (1972) and Roth (1986a). The design equations for the geometric design problem of mechanisms and manipulators are a set of non-linear, highly coupled multivariate polynomial equations. The solutions of these equations can be obtained by either numerical continuation methods or algebraic methods (Raghavan and Roth 1995). Using algebraic methods, the synthesis of planar mechanisms for rigid body guidance can be found in most textbooks on mechanism synthesis (Erdman and Sandor 1997; Sandor and Erdman 1984). For spatial mechanisms and manipulators, a few of them had been solved using algebraic methods (Tsai and Roth 1973; Roth 1986b; Perez and Mc- Carthy 2000; Mavroidis, Lee, and Alam 2001; Huang and Chang 2000; Murray and McCarthy 1999; Neilsen and Roth 1995; Innocenti 1994; McCarthy 2000; Lee and Mavroidis 2002b). Even though algebraic methods have been shown to be very efficient in solving several geometric design problems for spatial mechanical systems, the complexity of the design equations has limited their usage and there exist many types of robotic and mechanical systems that are used frequently, such as the 3R, 4R and 5R manipulators, for which the algebraic solutions of the geometric design problem have not yet been discovered. Polynomial continuation methods have been used extensively in the kinematic analysis and design of mechanisms and robotic systems (Wampler, Morgan, and Sommese 1990). They have been proven to be very effective methods in solving very difficult problems in the geometric design and analysis of robot manipulators (Wampler and Morgan 1991; Raghavan 1993; Roth and Freudenstein 1963; Morgan and Wampler 1990; Wampler, Morgan, and Sommese 1992; Dhingra, Cheng, and Kohli, 1994). For the spatial design problem, the seven-point sphere-sphere dyad design has been solved in Wampler, Morgan, and Sommese (1990). Recently, Lee and Mavroidis (2002a) applied the polynomial 247

2 248 THE INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH / March 2004 continuation method and solved the spatial 3R threeprecision-point problem. This paper extends the previous work done by Lee and Mavroidis to solve the spatial 3R four-precision-point geometric design problem. Tsai (1972) and Tsai and Roth (1971), using screw theory, were the first to obtain the design equations for the geometric design problem of 3R manipulators. However, they did not solve them. Lee and Mavroidis (2002a) solved the three-precision-point design problem with two different types of free choice selection and found eight distinct solutions for each. They were able to reduce the two problems to tracing 448 and paths, respectively, using polynomial continuation. Lee, Mavroidis, and Merlet (2002) studied the 3R five-precision-point design problem using interval analysis and found all the real solutions in a predefined bounded domain. For the five-precision-point case, the number of design solution remains to be determined. In the problem studied in this paper, four spatial positions and orientations are defined and the dimensions of the geometric parameters of the 3R manipulator are computed so that the manipulator will be able to place its end-effector at these four pre-specified locations. Denavit and Hartenberg (DH) parameters and 4 4 homogeneous matrices are used to formulate the problem and to obtain 24 design equations in 27 design unknowns. Three of the design parameters are set as free choices and their values are selected arbitrarily. Two different cases for selecting the free choices are considered and their design equations are solved using polynomial continuation. In both cases for free choice selection, 36 distinct manipulators are found that will be able to place their end-effector at the four specified spatial positions and orientations. The polynomial homotopy continuation method is implemented using the software package PHC developed by Verschelde and Cools (1996) and Verschelde (1999). 2. Polynomial Continuation Polynomial continuation is a numerical method that computes all the solutions of a system of polynomial equations by tracing a finite number of solution paths from a polynomial start system to the target system of interest. There are two main steps in using polynomial continuation method: (a) generate the start system; (b) trace a finite number of solution paths to obtain all the solutions of the polynomial system of interest. A start system is always generated based on an upper bound on the number of solutions of the target system. There are several known upper bounds; the only one relevant to us is the multi-homogeneous Bezout bound (Raghavan and Roth 1995). The computation of this bound is based on a partition of the variables. Detail of the procedures can be found in Lee and Mavroidis (2002a). In general, a different partition will result in a different multi-homogeneous Bezout number. Once a partition is specified, a start system with the number of solutions equal to the m-homogeneous Bezout number can be computed. The general procedure for start system generation can be found in Wampler (1994). After generating the start system, the second step is to follow a finite number of solution paths to obtain the solutions of the target system. Suppose that the target system is F(x) and that the start system is G(x). Then the homotopy H(x,t) is defined as H(x,t) = c(1 t) k G(x) + t k F(x) (1) where x = (x 1,...,x n ) is a complex n-tuple, t [0, 1], c is a randomly chosen complex number and k is a positive number (usually 1 or 2). It is obvious that H(x,0) = cg(x) and H(x,1) = F(x). The basic premise of the continuation method is that, if x(t) is a solution of H(x,t) = 0, then for a small increment t > 0, x(t + t) is near x(t).to compute x(t + t) from x(t), a predictor-corrector method can be used (Verschelde and Cools 1996; Verschelde 1999). Each solution of the start system (t = 0) represents a solution path of H(x,t) = 0 from t = 0tot = 1 and each path can be traced independently by successive small increments of t until t equals 1. The solutions of our target system are those that converge finitely in the continuation method as t approaches 1. More details of polynomial continuation methods can be found in Wampler, Morgan, and Sommese (1990), Verschelde and Cools (1996), Verschelde (1999), and Wampler (1994). 3. Problem Formulation In this paper, we describe the relative positions of links and joints in mechanisms and manipulators using the variant of DH notation introduced by Pieper and Roth (1969). In this section, only the essential elements of the formulation are described; the details can be found in our earlier paper (Lee and Mavroidis 2002a). Consider Figure 1, which denotes a generic 3R manipulator. A reference frame R i is attached at each link i(i = 0 4), frame R 0 is the fixed reference frame, and links 1, 2 and 3 are the moving links. An endeffector reference frame R e is also attached at the end-effector of the manipulator. (Note that frames R 4 and R e have the same z-axis.) The homogeneous transformation matrices A i, with i = 0, 1, 2, 3 describe frame R i+1 relative to R i. The homogeneous transformation matrix A c, which represents a screw displacement with a rotation φ around the z 4 -axis and a translation d along the same axis, relates R e to R 4. Homogeneous transformation matrix A h relates directly the end-effector reference frame R e to the frame R 0. Matrices A i and A c are given by c i s i c αi s i s αi a i c i A i = s i c i c αi c i s αi a i s i 0 s αi c αi d i (2)

3 Lee and Mavroidis / Geometric Design of 3R Robot Manipulators 249 Fig. 1. Schematic diagram of a 3R open-loop spatial manipulator. c φ s φ 0 0 A c = s φ c φ d where c i = cos(θ i ), s i = sin(θ i ), c αi = cos(α i ), s αi = sin(α i ), c φ = cos(φ) and s φ = sin(φ). The loop closure equation of the manipulator is used to obtain the design equations: A 0 A 1 A 2 A 3 A c = A h. (3) Equation (3) is a 4 4 matrix equation that results in six independent scalar equations. The right-hand side of eq. (3), i.e., the elements of matrix A h, is known since it represents the position and orientation of frame R e at each precision point. The left-hand side of eq. (3) contains all the unknown geometric parameters of the manipulator which are the DH parameters a i, α i, d i and θ i for i = 0, 1, 2, 3, and parameters φ and d of matrix A c. Joint angles θ 1, θ 2 and θ 3 have a different value for each precision point while all other 15 geometric parameters are constant. Thus, for n precision points there are n unknown parameters in total, and there are 6n scalar equations that are obtained. Therefore, the maximum number of precision points for exact synthesis is five. For four-precision-point synthesis, which is studied in this paper, there are 27 unknowns (15 structural parameters and 12 joint variables) and 24 scalar equations, thus we can select three structural parameters arbitrarily as free choices. 4. Design Equations at Each Precision Point Using the loop closure equation of the manipulator (eq. (3)), six scalar design equations are obtained at each precision point. The unknowns in these equations are the manipulator constant structural parameters and the joint variables θ 1, θ 2 and θ 3, which vary from precision point to precision point. To simplify the solution process, we eliminate the joint variables from the design equations at each precision point. Once the joint variables are eliminated, the new set of equations contains only unknowns that do not change from precision point to precision point. In this way, for each new precision point that is defined, new equations are added that have exactly the same structure as for the first precision point. In this section we present the method to obtain design equations devoid of the joint variables. By rearranging the loop closure eq. (3), we can rewrite it as follows A L = A R (4) where A L = A 1 A 2 and A R = A 1 0 A h A 1 c A 1 3. The third and fourth columns of the above equation are free of θ 3 and are easier to solve. We denote the third column vector of A L and A R as U L and U R, respectively, and the fourth column vector of A L and A R as V L and V R, respectively. (Note that we regard U L, U R, V L and V R as 3 1 matrices, i.e., we neglect the fourth component that is the homogeneous coordinate.) Then, we form the following three equations:

4 250 THE INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH / March 2004 U L V L = U R V R (5) V L V L = V R V R (6) U L (1)V L (2) U L (2)V L (1) = U R (1)V R (2) U R (2)V R (1). (7) Equations (5), (6) and (7) give a total of three scalar equations. Note that eq. (7) is the third component of the cross product equation U L V L = U R V R. It was found that eqs. (5), (6), and (7) are naturally devoid of θ 1. Joint angle θ 2 can also be eliminated by using the third and fourth elements of the third row of eq. (4) (Lee and Mavroidis 2002a). Thus, the three equations are free of θ 1, θ 2 and θ 3 and have, respectively, the following forms: f Xj,X k (α 0,θ 0,α 1 )X j X k = 0 (8) X j,x k W X j,x k W X j,x k W g Xj,X k (α 0,θ 0,α 1 )X j X k = 0 (9) h Xj,X k (α 0,θ 0,α 1 )X j X k = 0. (10) Here, W = {F,G,H,S,P,Q,R,d 2,a 0,a 1,d 0,d 1, 1} and F = λa, G = λb, H = λc, S = λcα 2 and λ = a 2 /sα 2. Note that f, g and h are polynomial functions of c 0, s 0, c α0, s α0, c α1 and s α1. In particular, f and g are first-order in each of the sine and cosine of the three angles while h is first-order in c 0, s 0, c α0, and s α0, and second-order in c α1 and s α1. The total of degrees is therefore five for both eqs. (8) and (9) and six for eq. (10). 5. Solution Procedure Using Polynomial Continuation In this paper we solve the geometric design problem of 3R manipulators with four precision points. In this case, there are 12 scalar equations in 15 unknowns. This means that we can select three design parameters as free choices so that a well-determined system of 12 equations in 12 unknowns is obtained. In this paper, two different ways for selecting free choices have been considered. Both types of free choice selection involve assigning parameters to the base of the manipulator. The design equations for both types of selections are obtained by substituting the free choices made into eqs. (8), (9), and (10) Type 1 of Free Choice Selection In this type, the free choices made are parameters α 0, a 0 and θ 0. By arbitrarily selecting the values for these parameters the designer selects the direction and partially locates the first joint of the manipulator with respect to a fixed reference frame. After substituting the values of the free choices into eqs. (8), (9), and (10), they become X j,x k T 1 f Xj,X k (α 1 )X j X k = 0 (11) X j,x k T 1 g Xj,X k (α 1 )X j X k = 0 (12) X j,x k T 1 h Xj,X k (α 1 )X j X k = 0 (13) where T 1 ={F,G,H,S,P,Q,R,d 0,d 1,a 1,d 2, 1}. After the substitution of the free choices, f and g become linear functions of c α1 and s α1, and h become quadratic functions of c α1 and s α1. By incorporating the constraint equations cα 2 1 +sα2 1 1 = 0, the new system is a multivariate polynomial system with 13 equations in 13 unknowns from T 1 {cα 1,sα 1 }. After linear reduction by subtracting eqs. (11), (12), and (13) at the first, second, and third precision points by the corresponding equations at the fourth precision point, the quadratic terms of c α1 and s α1 of eq. (13) are canceled and the total degree bound of the system is 1,417,176. However, using a two-partition G 1 ={F,G,H,S,P,Q,R,d 0,d 1,a 1,d 2 } and G 2 ={cα 1,sα 1 }, the multi-homogeneous Bezout bound is found to be 53248, and the number of paths needed to be traced in polynomial continuation is significantly reduced. Using PHC, a continuation method based on this twohomogeneous number is employed and the numerical values of the variables in G 1 G 2 are computed. The DH parameters of the design solutions are computed using the backsubstitution procedure outlined in Lee and Mavroidis (2002a). It is found that out of paths, only 144 paths converge to true solutions of the design problem; the remaining paths are solutions at infinity and extraneous solutions. These 144 solutions are all numerically different but contain only 36 geometrically distinct solutions, where each geometrically distinct solution has four equivalent different representations in terms of DH parameters. Therefore, at the most there are 36 distinct manipulators that can place their end-effectors in the four precision points specified by the designer in this case Type 2 of Free Choice Selection In this type, the free choices made are d 0, α 0 and θ 0. The design equations (8), (9), and (10) become X j,x k T 2 f Xj,X k (α 1 )X j X k = 0 (14) X j,x k T 2 g Xj,X k (α 1 )X j X k = 0 (15)

5 Lee and Mavroidis / Geometric Design of 3R Robot Manipulators 251 Table 1. DH Parameters of the 3R Manipulators for Real Solutions 1 8 Found Using Type 2 Free Choices a d a α d a α d a θ φ d X j,x k T 2 h Xj,X k (α 1 )X j X k = 0 (16) where T 2 ={F,G,H,S,P,Q,R,a 0,d 1,a 1,d 2, 1}. Together with the constraint equation cα sα2 1 1 = 0, the new system is again a multivariate polynomial system of 13 equations in 13 unknowns T 2 {cα 1,sα 1 }. Using a two-partition G 1 ={F,G,H,S,P,Q,R,a 0,d 1, a 1,d 2 } and G 2 ={cα 1,sα 1 }, after linear reduction, the twohomogeneous number is again As in type 1, a continuation method using PHC based on this two-homogeneous number gives 144 numerically distinct solutions and only 36 geometrically distinct solutions. 6. Numerical Example In this section, because of the similarity between the two types of free choices made, only one numerical example for the second type of free choices is shown. The computation is carried out using the software PHC run on the Sun Microsystems Enterprise system of the Rutgers University, Center for Advanced Information Processing (CAIP). PHC is a general-purpose polynomial equations solver with continuation method developed by Prof. Verschelde and which is publicly available (Verschelde and Cools 1996; Verschelde 1999). The start system is generated by a random linear product based on the multi-homogeneous Bezout number and the path tracking is carried out with a quadratic homotopy (i.e., k = 2 in eq. (1)). Four precision points are arbitrarily selected. These precision points are defined by the position coordinates of the origin of the end-effector frame with respect to the fixed reference frame and the direction cosines of the end-effector frame with respect to the fixed reference frame. These four precision points that are selected give the following A hi matrices where i = 1, 2, 3 and 4: A h1 = , A h2 = , A h3 = , A h4 = The numerical values of the free choices made are d 0 = 5,α 0 = and θ 0 = The computed values of the DH parameters of the 36 geometrically distinct manipulators solutions are given in Tables 1 5. Only the structural parameters are reported in the tables, and for each pair of conjugate complex solutions, only one of the two is reported. The computation is carried out using 15 digits but only eight digits are shown here. The average time required to complete the computation is 33 days. In Tables 1 5, the units for angular parameters are given in radians. Note that I is the square root of 1. In this example, there are only eight real manipulators that can place their endeffectors at the four specified precision points while the other solutions are complex. 7. Conclusions In this paper, the geometric design problem of serial-link spatial robot manipulators with three revolute (R) joints when four precision points are specified is solved using a

6 252 THE INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH / March 2004 Table 2. DH Parameters of the 3R Manipulators for Complex Solutions 9 12 Found Using Type 2 Free Choices a *I *I *I *I d *I *I *I e 1*I a *I *I *I *I α *I *I *I *I d *I *I *I *I a *I *I *I *I α e 1*I e 1*I *I *I d *I *I *I *I a *I *I *I *I θ e *I *I *I e 1*I φ *i *I e 1*I e 1*I d *I *I *I *I Table 3. DH Parameters of the 3R Manipulators for Complex Solutions Found Using Type 2 Free Choices a *I e 1*I e 1*I *I d e 1*I *I *I *I a *I *I e 1*I *I α *I *I e 2*I e 2*I d *I *I e 2*I *I a *I *I e 1*I *I α *I *I e 1*I e 1*I d *I *I e 1*I *I a *I *I e 1*I *I θ *I *I e 1*I e 2*I φ e 1*I *I e 3*I *I d *I *I e 1*I *I Table 4. DH Parameters of the 3R Manipulators for Solutions Found Using Type 2 Free Choices a *I *I *I *I d *I *I *I *I a *I *I *I *I α e 1*I *I *I *I d *I *I *I *I a *I *I *I *I α *I *I *I *I d *I *I *I *I a *I *I *I *I θ *I *I *I *I φ e 1*I *I *I e *I d *I *I *I *I

7 Lee and Mavroidis / Geometric Design of 3R Robot Manipulators 253 Table 5. DH Parameters of the 3R Manipulators for Complex Solutions 21 and 22 Found Using Type 2 Free Choices a *I *I d *I *I a *I *I α e *I e 1*I d *I *I a *I *I α *I e 1*I d *I *I a e *I *I θ *I *I φ e 1*I e 1*I d *I *I polynomial homotopy continuation method. Four spatial positions and orientations are defined and the DH parameters of the 3R manipulator are computed so that the manipulator will be able to place its end-effector at these four pre-specified locations. Two types of free choice selections are considered. It is shown that for both types of free choice selection, 36 manipulators can be found at most that can place their end-effectors at the four specified precision points. Acknowledgments This work was supported by a National Science Foundation CAREER Award to Professor Mavroidis under the grant DMI Mr. Eric Lee was supported by a Computational Sciences Graduate Fellowship from the Department of Energy. The authors would like to thank Dr. Charles Wampler of General Motors and Dr. Jan Verschelde of the University of Illinois at Chicago, for providing helpful suggestions in using polynomial continuation methods and assisting with the use of the software PHC. Any opinions, findings, conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the National Science Foundation. References Bodduluri, M., Ge, J., McCarthy, M. J., and Roth, B The synthesis of spatial linkages. In A. Erdman, editor, Modern Kinematics: Developments in the Last Forty Years, Wiley, New York. Dhingra, A. K., Cheng, J. C., and Kohli, D Synthesis of six-link, slider-crank and four-link mechanisms for function, path and motion generation using homotopy with m-homegenization. Transactions of the ASME, Journal of Mechanical Design 116: Erdman, A. G., and Sandor, G. N Mechanism Design: Analysis and Synthesis, Vol. 1. 3rd edition, Prentice-Hall, Englewood Cliffs, NJ. Huang, C., and Chang, Y.-J Polynomial solution to the five-position synthesis of spatial CC dyads via dialytic elimination. In Proceedings of the ASME Design Technical Conferences, September 10 13, Baltimore, MD, Paper Number DETC2000/MECH Innocenti, C Polynomial solution of the spatial Burmester problem. Mechanism Synthesis and Analysis ASME DE 70: Lee, E., and Mavroidis, C. 2002a. Solving the geometric design problem of spatial 3R robot manipulators using polynomial continuation. Transactions of the ASME, Journal of Mechanical Design 124(4): Lee, E., and Mavroidis, C. 2002b. Geometric design of spatial PRR manipulators using polynomial elimination techniques. In Proceedings of the ASME Design Technical Conferences, September 30-October 2, Montreal, Canada, Paper Number DETC2002/MECH Lee, E., Mavroidis, C., and Merlet, J. P Five precision point synthesis of spatial RRR manipulators using interval analysis. In Proceedings of the ASME Design Technical Conferences, September 30-October 2, Montreal, Canada, Paper Number DETC2002/MECH Mavroidis, C., Lee, E., and Alam, M A new polynomial solution to the geometric design problem of spatial R-R robot manipulators using the Denavit and Hartenberg parameters. Transactions of the ASME, Journal of Mechanical Design 123(1): McCarthy, M Algebraic synthesis of spatial chains. In The Geometric Design of Linkages, Chapter 11, McGraw- Hill, New York. Morgan, A. P., and Wampler, C. W Solving a planar four-bar design problem using continuation. Transactions of the ASME, Journal of Mechanical Design 112: Murray, A. P., and McCarthy, J. M Burmester lines of a spatial five position synthesis from the analysis of a 3-CPC platform. Transactions of the ASME, Journal of Mechanical Design 121: Neilsen, J., and Roth, B Elimination methods for

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