Inverse Kinematics of the General 6R Manipulator and Related Linkages

Raghavan, Madhusudan; Roth, Bernard

doi:10.1115/1.2919218

Cited by 190 publications

(131 citation statements)

References 4 publications

Supporting

Mentioning

129

Contrasting

Order By: Relevance

“…Moreover, the way this result is obtained can be used to construct poses of T in this region. This outcome is not completely surprising, since the conditions under which the IK of a 6-dof serial linkage has less than 16 solutions have been established in [15,19] and a 6-dof fragment of a protein backbone does not satisfy any of them.…”

Section: Introductionmentioning

confidence: 98%

See 1 more Smart Citation

On the structure of the inverse kinematics map of a fragment of protein backbone

2007

View full text Add to dashboard Cite

Loop closure in proteins requires computing the values of the inverse kinematics (IK) map for a backbone fragment with 2n ≥ 6 torsional degrees of freedom (dofs). It occurs in a variety of contexts, e.g., structure determination from electron-density maps, loop insertion in homology-based structure prediction, backbone tweaking for protein energy minimization, and study of protein mobility in folded states. The first part of this paper analyzes the global structure of the IK map for a fragment of protein backbone with 6 torsional dofs and a slightly idealized kinematic model, called the canonical model. This model, which assumes that every two consecutive torsional bonds C α -C and N-C α are exactly parallel, makes it possible to separately compute the inverse orientation map and the inverse position map. The singularities of both maps and their images, the critical sets, respectively decompose SO(3) and R 3 into open regions where the number of IK solutions is constant. This decomposition leads to a constructive proof of the existence of a region in R 3 × SO(3) where the IK of the 6-dof fragment has 16 solutions. The second part of this paper extends this analysis to study fragments with more than 6 torsional dofs. It describes an efficient recursive algorithm to sample IK solutions for such fragments, by identifying the feasible range of each successive torsional dof. A numerical homotopy algorithm is then used to deform the IK solutions for a canonical fragment into solutions for a non-canonical fragment. Computational results for fragments ranging from 8 to 30 dofs are presented.

show abstract

Section: Introductionmentioning

confidence: 98%

“…, and s t = 2ξ/1 + ξ 2 in Equation (19). The result is an 8 th -order polynomial of ξ, whose roots can be derived using the Sturm method [5,24].…”

mentioning

confidence: 99%

On the structure of the inverse kinematics map of a fragment of protein backbone

2007

View full text Add to dashboard Cite

show abstract

“…In contrast with FK, inverse kinematics (IK) problems are usually very difficult to solve in joint parameters: typically the dependence of the end effector configurations on joint parameters (particularly for rotational joints) is highly nonlinear. Yet past research has led to a large body of impressive results; a small set of representative work appears in [1], [2], [3], [4], [5], [6], [7], [8], [9], including recent formulations of IK problems using distance constraints and vector equations. We recently introduced a new set of linkage parameters tailored to the study of inverse kinematics.…”

Section: Overviewmentioning

confidence: 99%

A Unified Geometric Approach for Inverse Kinematics of a Spatial Chain with Spherical Joints

Han

Rudolph

2007

Proceedings 2007 IEEE International Conference on Robotics and Automation

View full text Add to dashboard Cite

Abstract-Conventionally, joint angles are used as parameters for a spatial chain with spherical joints, where they serve very well for the study of forward kinematics (FK). However, the inverse kinematics (IK) problem is very difficult to solve directly using these angular parameters, on which complex nonlinear loop closure constraints are imposed by required end effector configurations. In a recent paper, our newly developed anchored triangle parameters were presented and shown to be well suited for the study of IK problems in many broad classes of linkages. The focus of that paper was the parameterization of non-singular solutions; among many specific types of IK problems, only one, that of a spatial chain with spherical joints imposing 5 dimensional constraints, was developed in detail.Here we present a unified approach to the solutions of that and two other types of IK problems. The critical concepts in our approach-the geometric formulation in anchored triangle parameters, and the application of loop deformation spacesare general for all IK problems, and especially useful for redundant systems. For the three IK problems addressed in this paper, we demonstrate convexity properties of the set of IK solutions. We also give detailed descriptions of the parameterization of singular deformations. Similar ideas apply readily to linkages involving multiple loops. I. OVERVIEWKinematics is fundamental in the study of linkage systems. This has long been known in robotics, where kinematics analysis, robot design, motion planning and trajectory control all involve kinematical issues. More recently, kinematics has been generalized and applied to the study of protein conformations. A key, yet somewhat un-emphasized, issue in the study of kinematics is the choice of parameters. By far the most commonly used kinematic parameters are joint parameters [1], including angles for rotational joints, translational displacements for prismatic joints, and related twists [2]. For linkages, such joint parameters are a natural default since they correspond directly to the actuation of the joints and are well suited for forward kinematics (FK) computation (e.g., by taking the product of link transformation matrices). In contrast with FK, inverse kinematics (IK) problems are usually very difficult to solve in joint parameters: typically the dependence of the end effector configurations on joint parameters (particularly for rotational joints) is highly nonlinear. Yet past research has led to a large body of impressive results; a small set of representative work appears in We recently introduced a new set of linkage parameters tailored to the study of inverse kinematics. These anchored triangle parameters consist of certain inter-joint distances (called diagonal lengths) and certain triangle orientation parameters (signs or dihedral angles); they were described in some detail for serial chains with spherical joints in space or revolute joints in the plane in [10], and for closed planar chains with revolute joints in [11]. It was shown that...

show abstract

“…A while loop is then executed until P gets empty (lines [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18], by iterating the following steps. Line 4 SOLVE-LINKAGE(B, L, P, H, σ, ρ) 1: S ← ∅ 2: P ← {B} 3: while P = ∅ do 4: B c ← EXTRACT(P ) 5: repeat 6: V p ← VOLUME(B c ) 7: SHRINK-BOX(B c , L, P, H)…”

Section: Pseudocodementioning

confidence: 99%

“…The roots of this polynomial, once backsubstituted into other equations, yield all solutions of the original system. These methods have proved quite efficient in fairly non-trivial problems such as the inverse kinematics of general 6R manipulators [12], [1], distance computations of twodimensional objects [13], or the generation of configurationspace obstacles [14]. Recent progress on the theory of sparse resultants, moreover, qualifies them as a very promising set of techniques [15].…”

Section: Introductionmentioning

confidence: 99%

Multi-loop Position Analysis via Iterated Linear Programming

Porta

Ros²,

Thomas³

2006

Robotics: Science and Systems II

View full text Add to dashboard Cite

Abstract-This paper presents a numerical method able to isolate all configurations that an arbitrary loop linkage can adopt, within given ranges for its degrees of freedom. The procedure is general, in the sense that it can be applied to single or multiple intermingled loops of arbitrary topology. It is also complete, meaning that all possible solutions get accurately bounded, irrespectively of whether the analyzed linkage is rigid or mobile. The problem is tackled by formulating a system of linear, parabolic, and hyperbolic equations, which is here solved by a new strategy exploiting its structure. The method is conceptually simple, geometric in nature, and easy to implement, yet it provides solutions at the desired accuracy in short computation times.

show abstract

Inverse Kinematics of the General 6R Manipulator and Related Linkages

Cited by 190 publications

References 4 publications

On the structure of the inverse kinematics map of a fragment of protein backbone

On the structure of the inverse kinematics map of a fragment of protein backbone

A Unified Geometric Approach for Inverse Kinematics of a Spatial Chain with Spherical Joints

Multi-loop Position Analysis via Iterated Linear Programming

Contact Info

Product

Resources

About