Motion Planning for a Class of Planar Closed-Chain Manipulators

Transcription

1 Motion Planning for a Class of Planar Closed-Chain Manipulators Guanfeng Liu Department of Computer Science Stanford University Stanford, CA liugf@cs.stanford.edu J.C. Trinkle Department of Computer Science Rensselaer Polytechnic Institute 8th St., Troy, NY trink@cs.rpi.edu

2 Motion Planning for a Class of Planar Closed-Chain Manipulators Abstract We study the motion problem for planar star-shaped manipulators. These manipulators are formed by joining k legs to a common point (like the thorax of an insect) and then fixing the feet to the ground. The result is a planar parallel manipulator with k independent closed loops. A topological analysis is used to understand the global structure the configuration space so that planning problem can be solved exactly. The worst-case complexity of our algorithm is O(k 3 + k N 3 ), where N is the maximum number of links in a leg. A simple example illustrating our method is given. I. INTRODUCTION Due to the computational complexity and difficulty of implementing general exact motion planning algorithms, such as Canny s [], today sample-base algorithms, such as Kavraki s [3] dominate motion planning research. However, there are important classes of problems for which these algorithms do not perform well and for which specialized exact methods have not been developed. This provides motivation to try to develop effective exact algorithms. The problem class of interest here arises in systems whose configuration space (C-space) cannot effectively be represented as a set of parameters with simple bounds, but rather is most naturally represented as a sub-space of co-dimension one or greater embedded in a higher-dimensional ambient space [9]. In particular, we are interested in planar star-shaped manipulators. These manipulators are formed by joining k planar legs to a common point (like the thorax of an insect) and then fixing the feet to the ground. The result is a planar parallel manipulator with k independent closed loops. Each independent loop imposes an algebraic kinematic constraint equation on the system, and so the C-space of star-shaped manipulators is an algebraic variety embedded in the joint space. Walking robots whose legs are SCARA robots with axes perpendicular to the ground can be modeled as a planar star-shaped manipulators. The most relevant previous work is that of Milgram, Trinkle, and Liu [6], [7], [8]. In those papers, the C-space of manipulators forming a single closed loop are analysed from a global topological perspective, and the topological properties are used to guide the design of complete algorithms. The range of problem covered includes three-dimensional single-loop closed chains with spherical joints without obstacles and twodimensional single-loop closed chains with point obstacles. In the former case, the complexity of the planning algorithm given was O(n 3 ), where n is the number of links in the loop. The worst-case complexity of the algorithm obtained for the latter was Ω(m n 3 ) and O(m n 7 ) with m being the number of point obstacles in the workspace, so unfortunately its practicality is currently limited to closed chains with about 6 links and obstacles. Here, previous topological methods for C-space connectivity analysis are extended to the case of planar star-shaped manipulators without obstacles. We derive important global properties of C-space and use these results to derive a necessary and sufficient condition for path existence problem. A complete algorithm based on the properties is implemented and examples are presented. II. NOTATION Manipulator Notation M - Manipulator A - Root junction or thorax of M o i - Grounding point of foot i of M n j - Number of links in M j l j,i - Length of link i of M j ; i =,..., n j θ j,i - Angle of link i relative to link i M j - Leg j of M with foot fixed at o j and other end free, j =,..., k M j (p) - Leg j of M with foot fixed at o j and other end fixed at p M(p) - Manipulator with A fixed at p L j - Sum of lengths of links of M j L j (p) - A set of long links of Mj (p) L j (p) - Number of long links of Mj (p) Workspace Notation W A - Workspace of A d U i - Cell of dimension d of W A p - Point in the plane of M γ = p(t) - Curve in the plane of M f - Kinematic map of A f j - Kinematic map of endpoint of M j Σ A - Singular set of f in W A Σ j - Singular set of f j Configuration Space (C-space) Notation C - C-space of M C(p) - C-space of M(p) C j - C-space of M j C j (p) - C-space of Mj (p) c - Point in C-space III. PRELIMINARIES The class of planar manipulators studied here are refered to as planar star-shaped manipulators (see Fig. ). A star-shaped manipulator is composed of k serial chains with all revolute joints. Leg M j is composed of n j links of lengths l j,i, i =,..., n j with angles θ j,i, i =,..., n j. At one end (the foot), M j is connected to ground by a revolute joint fixed at the point o j. At the other end, it is connected by another revolute

3 o o l l 6 l l M 9 M Fig.. Star-shaped manipulator with k = 4. l l 3 3 A 3 l 33 l 43 8 l l 3 l 3 joint to a junction point denoted by A. Note that when k is one, a star-shaped manipulator is an open serial chain. When k is two, it is a single-loop closed chain. Assuming that the foot of M j is fixed at o j, let f j (Θ j ) = p denote the kinematic map of M j, where Θ j = (θ j,,, θ j,nj ) is the tuple of joint angles, and p is the location of the endpoint of the leg (the thorax end). When M j is detached from the junction A, the image of its joint space is the reachable set of positions of the free end of the leg, called the workspace W j. The workspace W j is an annulus if and only if there exists one link with length strictly greater than the sum of all the other link lengths. Otherwise it is a disk. Clearly, the workspace W A of A when all the legs are connected to A is given by: W A = M l o M k W j. () j= In our study of C, it will be convenient to refer to several other C-spaces. The C-space of leg M j when detached from the rest of the manipulator will be denoted by C j. When the endpoint is fixed at the point p, leg j will be denoted by M j (p), where the tilde is used to emphasize the fact that the endpoint has been fixed. Note that Mj (p) is a single-loop planar closed chain, about which much is known (see [6]), including global structural properties of its C-space, denoted by C j (p) = f j (p). When the junction A of a star-shaped manipulator is fixed at point p, its C-space will be denoted by C(p). Since collisions are ignored, the motions of the legs are independent, and therefore the C-space of the manipulator (with fixed junction) is the product of the C-spaces of the legs with all endpoints fixed at p: C(p) = C (p) C k (p) = f = f (p) (p) fk (p) where by analogy, f is a total kinematic map of the star-shaped manipulator. Loosely speaking, the union of the C-spaces C(p) at each point p in W A gives the C-space of a star-shaped o () A M j oj p p3 p Σ j γ - f j ( γ ) - - f p f j ( p ) j ( 3 ) - f j ( p ) Fig.. Left: The workspace W j of a three-link open chain M j based at o j. The singular set Σ j of the kinematic map f j is four concentric circles. The small circles, figure eights, and points at o clock show the topology of the C-space C j (p) of the leg when its endpoint is fixed at a point in one of the seven regions delineated by the singular circles (one of the four circles or one of the three open annular regions between them). Right: The inverse image of the curve γ - a pair of pants. manipulator: C = p W A C(p). (3) Several properties of the C-spaces C j and C j (p) are highly relevant and so are reviewed here before analyzing the C- space of star-shaped manipulators. It is well known that the C-space of M j is a product of circles (i.e., C j = (S ) n j ). The workspace W j contains a singular set Σ j which is composed of all points p in W j for which the Jacobian of the kinematic map Df j (Θ j ) drops rank for some Θ j f j (p). These points form concentric circles of radii l j, ±l j, ± ±l j,nj, as shown in Fig. When A coincides with a point in Σ j, the links can be arranged such that they are all colinear, in which case the number of instantaneous degrees of freedom of the endpoint of the leg is reduced from two to one. Now consider the case where the endpoint of leg j is fixed to the point p. In other words, we are interested in the C-space C j (p) of Mj (p). In the o clock position in Fig., points, circles, and figure eights are drawn to represent the global structures of C j (p) in the seven regions of W j. Specifically, when A is fixed to a point p on the outer-most singular circle, C j (p) is a single point. For p fixed to any point in the largest open annular region, C-space is a single circle. Continuing inward, the possible C-space types are a figure eight (on the second largest singular circle), two disconnected circles, a figure eight again, a single circle, and a single point (on the inner-most singular circle). A detailed analysis of C j (p) with an arbitrary number of links in M j (p) can be found in [6]. The results that will be particularly useful in the analysis of star-shaped manipulators follow. First, the connectivity of C j (p) is uniquely determined by the number of long links. Consider the augmented link set composed of the links of M j and o j p, which will be called the fixed base link with length denoted by l j,. Let L j be the sum of all the link lengths including the fixed base link (i.e., L j = n j i= l j,i). Further, let L j (p) be a subset of {,,..., n j } such that l j,α + l j,β > L j /; α, β L j (p), α β. Over all

4 such sets, let L j (p) be a set of maximal cardinality. Then the number of long links of Mj (p) is defined as L j (p), where denotes set cardinality. Lemma : Kapovich and Milson [], Trinkle and Milgram [6] The C-space C j (p) = f j (p) has two components if and only if L j (p) = 3, and is connected if and only if L j (p) = or. No other cardinality is possible. Let us return to the discussion of Fig.. Viewing W j as a base manifold and the C-space corresponding to each end point location as a fibre, it is apparent that the singular set Σ j partitions W j into regions over which the C-spaces C j (p) form a trivial fibration. The implications of this observation are useful in determining the C-space of more complicated mechanisms. Consider a modification to M j (p) that allows the endpoint to move along a one-dimensional curve segment γ within W j. Then as long as γ is entirely contained in one of the regions defined by the singular circles, Cj (γ) = C j (p) I, where I is the interval. If γ crosses a singular circle transversally, then C j (γ) = ( C j (p ) I) Cj (p 3 ) ( C j (p ) I), where p is a Σ j+ o j+ W A M j+ A M j oj Σ j Fig. 3. The workspace W A of A for a star-shaped manipulator with k = is the intersection of the workspaces of A for each leg considered separately. The singular set Σ is composed of the black circular arcs where they bound or intersect the gray area. point in one of the two open angular regions containing γ, p is a point in the other, and p 3 is a point on the singular circle crossed by γ, and denotes the standard gluing operation. In Fig., an example γ and the corresponding C-space C j (γ) are shown. o j+ o j IV. ANALYSIS OF STAR-SHAPED MANIPULATORS For star-shaped manipulators with one or two legs, the global topological properties of the C-space C are fully understood (for one, see [4]; for two, see [6], []). The goals of this section are to study the global properties of C when M has more than two legs and to derive necessary and sufficient conditions for solution existence to the motion planning problem. ) Local Analysis: As a direct generalization of the singular set of a single leg, we define the singular set of a star-shaped manipulator as a subset Σ of W A such that for every p Σ, there exists a configuration c such that at least one of the Jacobians {Df (c),, Df k (c)} drops rank. By definition we have: ( k ) Σ = Σ i WA. (4) i= An advantage of this definition is that Σ can be used to stratify W A such that each stratum is trivially fibred. Figure 3 shows a star-shaped manipulator with two legs. The singular set Σ is the boundary of the lune formed by the intersection of the outer singular circles of their individual workspaces. For every point interior to the lune, the fibre is two circles (the direct product of two points with one circle). The fibres associated to the vertices of the lune are single points, which correspond to simultaneous full extension of the two legs. Fig. 4 shows a possible workspace for a star-shaped manipulator with three legs. The singular set defines 6 distinct sets d U i of varying dimension d, where i is an arbitrarily assigned index that simply counts components. We will refer o j+ Fig. 4. Workspace (shaded gray) of a star-shaped manipulator with three legs. The singular set partitions W A into two-dimensional, 3 one-dimensional, and zero-dimensional chambers. to these sets as chambers. There are two-dimensional, 3 one-dimensional, and zero-dimensional chambers, each of which is trivially fibred. More generally, the intersections among the arcs composing Σ are zero-dimensional chambers, denoted U i, i =,, m. Removing the U i from Σ partitions it into open one-dimensional chambers U i, i =,, m. Removing U i and U i from W A yields open twodimensional sets U i, i =,, m, for which the following relationships hold: Σ = W A Σ = m i= U i m i= U i () m U i. (6) i= Proposition. : For all d =,, and i, f ( d U i ) = d U i f (p), where p is any point in d U i and the operator denotes the direct product. Gluing the f ( d U i ) for all i and d gives the total C-space C. Proof: When d =, U i contains a single point, the result follows. When d =, U i belongs to one singular circle of

5 one leg, say M j. Any two points p, p U i are related by a Euclidean rotation p o j = R(p o j ), indicating that C j (p ) and C j (p ) are homeomorphic. Thus C j (p) for all p U i have equivalent topological structure. For the other legs M l, l j, according to [] (Lemma 6. and Corollary 6.) C l (p) for all p U i have equivalent topological structures as U i is free of singular points of Ml (p). Thus f (p) = C (p) C k (p) for all p U i have equivalent topological structures. The case when d = can be proved by applying Lemma 6. and Corollary 6. of [] to all legs. Proposition and the fact that d U i is a simply connected set, reveal that each component of f ( d U i ) is a direct product of one component of C j (p), j =,, k, with a d-dimensional disk. Using L j (p), j =,, k and Lemma, one can show that the number of components of f ( d U i ) is k, where k k is the number of legs for which L j (p) = 3. ) Local Path Existence: Before considering the global path existence problem, consider motion planning between two valid configurations c init and c goal for which the junction A lies in the same chamber. Since the fibre over every point in d U i is equivalent, path existence amounts to checking the component memberships of the configurations c init and c goal. For a single leg M j (p), if the number of long links L j (p) is not three, then any two configurations of Mj (p) are in the same component. When L j (p) = 3, choose any two long links and test the sign of the angle between them (with full extension taken as zero). There are two possible signs, one corresponding to elbow-up and the other to elbow-down. If for two distinct configurations of M j, A lies in the same chamber, there is a continuous motion between them while keeping A in this chamber, if and only if the elbow sign is the same at both configurations (naturally, one must perform the sign test with the same two links and in the same order for both configurations). Considering all the legs together, a continuous motion of A in d U i exists if and only if a motion exists for each leg individually. The previous discussion serves to prove the following result. Proposition. : Restricted to f ( d U i ), two configurations c, c f ( d U i ) are path connected if and only if for each leg M j with L j = 3 in d U i, the elbow angle of Mj has the same sign at c and c. Proposition completely solves the path existence problem if W A consists of a single chamber. However, things become complex when W A has more than one chamber. 3) Singular Set and Global C-space Analysis: Recall that the C-space C is a union of f ( d U i ), d {,, }, i =,, d m and that f (p), p d U i for d and all i is a set containing at least a singularity of f. Combining the local C-space and singular set analysis yields the global structure of C-space. Proposition. 3: For all p Σ j, f j (p) is a singular set containing isolated singularities. If a singularity separates its neighborhood V in f j (p), then it is these singularities which glue the two separated components in f j (q) where q W A Σ j is a point sufficiently close to p. Proof: First it is obvious that f j (p) contains isolated singularities for there are finite ways to colinearize all the links of a close chain. Second, let γ : ( ε, ε) W A, γ() = p be a curve that is transverse to Σ j. According to Corollary 6.6 of [], the distance function s(γ(t)) = t γ dt defines a Morse function on f j (γ) s f j : f j (γ) R. Note that is a singular value of s f j and the isolated singularities of f j (p) are also singularities of s f j. The result of Morse theory applying to s f j yields that (s f j ) () = f j (p) is given by attaching a handle to (s f j ) (ε ) = f j (q) for a sufficiently small ε and q a point sufficiently close to p. The Proposition follows. Next, we establish necessary and sufficient conditions for the connectivity of C. Let J be the index set such that for all j J, L j = 3 for at least one chamber d U i. We prove the following theorem. Theorem : Suppose W A = d= f (W A ) is connected if and only if: ) W A is connected; ) Σ j WA for all j J. ( d m i= d U i ). Then C = Proof: (i) Necessity : Since C is a fibration of the base manifold W A, it can have one component only when W A has one component. Thus item of Theorem is required. Second, in order that C be connected, for each leg M j restricted to W A, the C-space C j (W A ) = f j (W A ) must be connected. By definition, for all j J, there exists a chamber d U i such that L j = 3. The result of Proposition 3 means that C j (W A ) is connected only if W A Σj. (ii) Sufficiency : Item and imply that C j (W A ) are path connected for all j. Moreover, C is a fibration over W A. The result follows. Fig. illustrates the global connectivity for an example W A corresponding to a star-shaped manipulator with two legs and a workspace for which there are two chambers U and U 3 where leg has three long links and another chamber U 4 where both legs have three long links. Among these chambers, U and U belong to Σ, and U 3 belongs to Σ. According to Theorem, the C-space is path connected. In this example, the C is the product of the two structures shown. Corollary : Two configurations c and c of a star-shaped manipulator are in the same component if and only if ) f(c ) and f(c ) are in the same component of W A ; ) For each leg j with L j = 3 for all chambers d U i in the component of W A which contains f(c ) and f(c ), the elbow sign is same at both c and c. Remark : As a matter of fact, Σ completely determines the connectivity of C-space. To plan a path between two given configurations, often motions of the junction to points on Σ are produced to adjust the signs of the leg angles. However, possible deviation of the junction from Σ caused by numerical

6 M f ( γ) + +! " # $ 9 8 M f ( γ) + % & % & 3 4 : U U U U U 3 U4 γ U 3 ( ) ' % & ' ' %, - λ. * λ / + Fig.. C-space of a star-shape manipulator with two legs. For purposes of simplicity, only the portion of f (γ) is shown, where γ is a continuous curve in W A that visits all chambers. Fig. 6. Logical flow and complexity of the major steps of PathExists. errors, somtimes makes it impossible to adjust the sign of the leg while fixing its end point. For these reasons, points in D chambers are preferred for sign adjustment. V. A POLYNOMIAL-TIME, EXACT, COMPLETE ALGORITHM Our algorithm consists of two main routines, PathExists and ConstructPath. The logical flow of PathExists is illustrated in Figure 6. Its input is the topology and link lengths of a star-shaped manipulator and two valid configurations, c init and c goal. The output is the answer to the path existence question. Below we will show that the complexity of PathExists is O(k 3 + kn), where N is the maximum number of links in a leg and k is the number of legs. The approach taken is to compute W A and then, for each leg with its end point constrained to lie in W A, to determine if its initial and goal configurations are path connected. Since the C-space of a leg is guaranteed to be connected if one of its singular circles Σ j intersects W A, the most straight forward way to test connectivity is to explicitly perform the intersections. However, since there are as many as n j singular circles, any algorithm based on this approach will have worst-case complexity that is at least exponential in N. The key idea of PathExists is a polynomial-time algorithm for checking the existence of an intersection between W A and a singular circle.. Construct W A Recall that W A is the intersection of the workspaces of the legs when they are disconnected from A. Each workspace is a disk or annulus, which can be determined by finding the length of the longest link and comparing it with the sum of all the other link lengths of that leg. For each leg, computation of the workspace is O(n j ), where recall that n j is the number of links in leg j. Since there are k legs, computing the workspaces of the legs is O(kN). The last step in the construction of W A is the pairwise intersection of the O(k) boundary circles of the legs. Thus the overall complexity of constructing W A is O(k + kn).. Are p init and p goal in same component of W A? One can compute p init and p goal from c init and c goal from the kinematic map of any leg, which is O(N). Extend rays from p init and p goal and count the number of intersections of each ray with the boundary of each component of W A. The component with an odd number of intersections (counting tangencies as two intersections) contains the point. Since there are at most O(k ) arcs defining W A, this step is O(k ). 3. Compute J This step is used to filter out easy solution existence checks, based on the cardinality and members of the sets L j (p init) and L j (p goal). For each leg M j (p init ), compute L j (see Section III) and find the three longest links of the set {l j,,..., l jnj }. Denote these links by (p init ; λ j,, λ j,, λ j,3 ). Do the same for (p goal ) and define (p goal ; λ j,, λ j,, λ j,3 ). This requires O(N) work. Finally, L j (p ( )) = 3 if and only if λ j, + λ j,3 > L j /. If L j (p init) = L j (p goal) and L j (p init) = 3, and if the signs of the long links are different at c init and c goal, then add j into J. Computing J is O(kN). 4. Compute extreme values of p o j ; p W A, j J If o j is interior to W A, then the minimum length is zero. This will be determined as a byproduct of the ray-shooting test above. Otherwise the minimum and maximum can be computed by finding the minimum and maximum distances from o j to each arc on the boundary of W A. In worst case this requires O(k ) intersections between a line segment and a circular arc, for each leg, so this step is O(k 3 ).. Does the set of long links vary for all j J? If and only

7 if p W A exists such that L j (p) L j (p init), then it is possible to make the long links colinear and thus change the signs of their relative angles. This can be done by computing a base length l j, satisfying λ j, (l j, )+λ j,3 (l j, ) = L j (l j, )/ (Note that L j varies linearly with l j,, and λ j, and λ j,3 ) vary discontinuously with l j,.) Finding such a base length or determinining nonexistence of such can be computed in constant time for each leg, so this step is O(k). The basic idea of ConstructPath is that when moving from c init to c goal, some of the legs with three long links when A is fixed at p init and p goal will require a change in the signs of relative angles between long links, which is always possible at a singular point of the corresponding leg. A natural approach then is to use two kinds of motion generation algorithms: accordion move and sign-adjust move. The former moves the thorax endpoint (at A) along a specified path segment without changing the sign of the relative angle between a pair of long links in legs. This can be realized using constrained accordion move. The latter keeps the endpoint fixed at a way point p Σ j while moving leg j into a colinear configuration and then to a nearby configuration with the sign of the relative angle between a pair of long links in this leg chosen to match those of c goal. The input of ConstructPath is W A, c init, c goal, the set of way points p j W A, j J computed during the execution of PathExist, and a few other special way points. ConstructPath constructs a path in W A connecting p init to p goal and visiting all of the way points. Since there are at most k way points, this can be done in O(k 3 ) time by connecting all these points with straight lines to the boundary of W A. Though crude, a path can be constructed by moving from c init to the boundary, around the boundary to the line segment connected to one of the way points, along the line to the way point, back to the boundary, and so on, until all way points have been visited and p goal has been reached. The path in C then is produced by using accordion moves along the path and sign-adjust moves at the way points. Once A is coincident with p goal, one is assured by the previous steps, that with A fixed at p goal, the configuration of each leg is in the same component of its current C-space C j (p goal ) as c goal. The final move can be accomplished using a special accordion move algorithm found in [6]. The complexity of the accordion move algorithms reported in [6] are O(N 3 ). Since there are O(k ) boundary segments, the number of path segments is also O(k ). If we assume that each boundary segment in the path is approximated by a constant number of line segments, then the complexity of ConstructPath is O(k N 3 ). Note that accordion move algorithms with the required behavior can be designed to be O(N ), so the complexity of ConstructPath could be reduced. Overall, our path planning algorithm is O(k 3 + k N 3 ). VI. EXAMPLE In this example, the manipulator has three three-link legs, one of which has three long links when A is fixed at p goal. Figure 7 shows the manipulator in its starting and goal initial config. goal config. base points initial junction goal junction Fig. 7. Manipulator s initial configuration (junction on the right, drawn red) and goal configuration (junction just below the top foot, drawn blue. Accordion move to one singular circle of leg Fig. 8. All legs use an accordion move to move the junction A to a point on a singular circle of leg. configurations. Since when A is fixed at p goal, the C-space of two of the legs have one component, our algorithm requires that A move from its initial location to one way point, and then to the goal. At the way point, the two lower legs can remain fixed while the other leg is moved to adjust the signs of its relative angles. In this particular example, we chose to move A to a point on the singular set of the leg in question. This leg was then moved to make all its links colinear. Before leaving the way point via the next accordion move, leg s angles were adjusted to match the relative signs in the goal configuration. Figures 8, 9,, and show the progress of the manipulation plan as the steps of the complete planning algorithm are carried out. This way point is one of the special ones not computed as a byproduct of PathExist.

8 Adjust sign of leg VII. CONCLUSION In this paper, we study the global structural properties of planar star-shaped manipulators. Via the analysis of the singular set Σ, we derive the global connectivity of the C-space, and necessary and sufficient conditions for path existence. Based on these results, we devise a complete algorithm for motion planning. Simulation examples are used to illustrate the key ideas behind the motion planning problem of planar star-shaped manipulators. Fig. 9. With the junction A at a point on a singular circle of leg, its join angles can be adjusted to achieve the signs required at the goal configuration. Accordion move to goal junction REFERENCES [] J.F. Canny, The Complexity of Robot Motion Planning. Cambridge, MA: MIT Press, 988. [] M. Kapovich and J. Millson, On the moduli spaces of polygons in the Euclidean plane. Journal of Differential Geometry, Vol. 4, PP , 99. [3] L.E. Kavraki, P. Švestka, J.C. Latombe, and M.H. Overmars, Probablistic Roadmaps for path planning in high-dimensional configuration space. IEEE Transactions on Robotics and Automation, (4):66-8, 996. [4] J.C. Latombe, Robot Motion PLanning. Kluwer Academic Publishers, 99. [] R.J. Milgram and J.C. Trinkle, The Geometry of Configuration Spaces for Closed Chains in Two and Three Dimensions. Homology Homotopy and Applications, 6():37-67, 4. [6] J.C. Trinkle and R.J. Milgram, Complete Path Planning for Closed Kinematic Chains with Spherical Joints. International Jounral of Robotics Research, (9): , December,. [7] G.F. Liu, J.C. Trinkle and R.J. Milgram, Toward Complete Path Planning for Planar 3R-Manipulators Among Point Obstacles. In Algorithmic Foundations of Robotics VI, STAR 7, Springer-Verlag, ,. [8] G.F. Liu and J.C. Trinkle, Complete Path Planning for Planar Closed Chains Among Point Obstacles. In Robotics: Science and Systems, MIT Press,. [9] J. Yakey, S. M. LaValle, and L. E. Kavraki, Randomized path planning for linkages with closed kinematic chains. IEEE Transactions on Robotics and Automation, 7(6):9 98, December. Fig.. All legs use an accordion move to move the junction A to its goal location. The signs of the joint angles are preserved guaranteeing that leg will be in the correct C-space component once A is fixed at the goal position. Trinkle milgram algorithm: overlap with goal configuration Fig.. All legs use the Trinkle-Milgram algorithm to achieve their goal configurations with the junction A fixed.