In robotics the group of proper rigid transformations of 3-dimensional space is of central import... more In robotics the group of proper rigid transformations of 3-dimensional space is of central importance. The relevant Clifford algebra in this case is a degenerate one with three generators that square to — 1 and a single generator that squares to 0. The algebra contains a copy of the group’s double cover.
The Cayley map for the rotation group SO(3) is extended to a map from the Lie algebra of the grou... more The Cayley map for the rotation group SO(3) is extended to a map from the Lie algebra of the group of rigid body motions SE(3) to the group itself. This is done in several inequivalent ways. A close connection between these maps and linear line complexes associated with a finite screw motions is found.
Molien’s theorem concerns the invariants of finite groups. So we will introduce a small finite gr... more Molien’s theorem concerns the invariants of finite groups. So we will introduce a small finite group to use for our examples. The permutation group or symmetric group on 3 letters has the presentation, S3 =< σ, τ |σ2 = τ = e, στ = τσ > that is,σ andτ are generators of the group, e being the identity element. These generators satisfy the relations, σ = τ = e, that isσ has order 2 andτ order 3, in factσ can be thought of as a swap and τ as a cycle. The final relation is στ = τσ, or στσ = τ which shows that the subgroup generated by τ on its own is a normal subgroup. In cycle notation we could write, σ = (12), τ = (123).
Further the groupGL(n) is a Lie group, this just means that the multiplication and the map which ... more Further the groupGL(n) is a Lie group, this just means that the multiplication and the map which sends every element to its inverse are both differentiable maps. This is easy to see since the maps can be defined in terms of addition, multiplication and division by non-zero numbers. We will usually take the ground field, the set containing the matrix entries, to be complex numbers. This is just for simplicity, it is sometimes more useful to know the case where the ground field is the real numbers or some finite field. It is possible to say something about these cases but as usual, when we use complex numbers the theory is more complete. We can write GL(n,R) to distinguish the case where the ground field is the real numbers. As a manifold the groupGL(n) will be an open set in the space C 2 , the complement of the closed set consisting of the singular matrices. So the group has complex dimensionn, real dimension2n. Notice that the groupGL(n,R), real dimension n, has 2 disjoint componen...
The model of a ball rolling on an inclined plane is derived using screw theory and Lie algebra. T... more The model of a ball rolling on an inclined plane is derived using screw theory and Lie algebra. The plane is supported by a gimbal and the inclination can be controlled in two orthogonal directions. The model includes effects due gravity, Coriolis and centripetal torques.
Fifth World Congress on Intelligent Control and Automation (IEEE Cat. No.04EX788)
ABSTRACT Holzer method is a very efficient lumped parameter dynamic modeling method for flexible ... more ABSTRACT Holzer method is a very efficient lumped parameter dynamic modeling method for flexible manipulators. However, it does have some faults. An improved efficiency dynamic modeling method based on the Holzer method is proposed: This new method is called Ding-Holzer method, it is applied to the dynamic modeling of planar single arm, multiple link manipulators and the arm with spatial compliance, respectively. A sequence comparative study and theoretical analysis convince the improved method.
This work argue that directional statistics are important in Robotics. That is, statistics on gen... more This work argue that directional statistics are important in Robotics. That is, statistics on general manifolds. Historically the subject began with statistics on circles and spheres, hence the title. It is still a relatively new discipline, see [4] for an overview. In order to make the case a specific example is studied: Finding the rigid transformation undergone by a camera from a knowledge of the images of a number of points. This very simple, perhaps naive, example allows us to study different models for error in the observations and estimators for the rigid transformation.
A line symmetric motion is the motion obtained by reflecting a rigid body in the successive gener... more A line symmetric motion is the motion obtained by reflecting a rigid body in the successive generator lines of a ruled surface. In this work we review the dual quaternion approach to rigid body displacements, in particular the representation of the group SE(3) by the Study quadric. Then some classical work on reflections in lines or half-turns is reviewed. Next two new characterisations of line symmetric motions are presented. These are used to study a number of examples one of which is a novel line symmetric motion given by a rational degree five curve in the Study quadric. The rest of the paper investigates the connection between sets of half-turns and linear subspaces of the Study quadric. Line symmetric motions produced by some degenerate ruled surfaces are shown to be restricted to certain 2-planes in the Study quadric. Reflections in the lines of a linear line complex lie in the intersection of a the Study-quadric with a 4-plane.
We discuss a unified approach to a class of geometric combinatorics incidence problems in two dim... more We discuss a unified approach to a class of geometric combinatorics incidence problems in two dimensions, of the Erdős distance type. The goal is obtaining the second moment estimate. That is, given a finite point set S in 2D, and a function f on S × S, find the upper bound for the number of solutions of the equation (1) f (p, p ′) = f (q, q ′) = 0, (p, p ′ , q, q ′) ∈ S × S × S × S. E.g., f is the Euclidean distance in the plane, sphere, or a sheet of the two-sheeted hyperboloid. Our ultimate tool is the Guth-Katz incidence theorem for lines in RP 3 , but we focus on how the original problem in 2D gets reduced to its application. The corresponding procedure was initiated by Elekes and Sharir, based on symmetry considerations. The point we make here is that symmetry considerations, not necessarily straightforward and potentially requiring group representation machinery, can be bypassed or made implicit. The classical Plücker-Klein formalism for line geometry enables one to directly interpret a solution of (1) as intersection of two lines in RP 3. This, e.g., allows for a very brief argument as to how the Euclidean plane distance argument extends to the spherical and hyperbolic distances. The space of lines in the projective three-space, the Klein quadric K, is four-dimensional. Thus, we start out with an injective map F : S × S → K, that is from a pair of points (p, q) to a line lpq, and seek a corresponding combinatorial problem in the form (1) in two dimensions, which can be solved by applying the Guth-Katz theorem to the set of lines {lpq} in RP 3. We identify a few arguably new such problems, and hence applications of the Guth-Katz theorem and make generalisations of the existing ones. It is the direct approach in question that is the main purpose of this paper.
Proceedings of IEEE International Conference on Robotics and Automation
ABSTRACT We give a geometrical description of Raibert and Craig&#39;s hybrid force/position c... more ABSTRACT We give a geometrical description of Raibert and Craig&#39;s hybrid force/position control method (1981). Our description is coordinate free, hence answering the criticism of the original work that it was not transformation invariant. However, our approach avoids the complications introduced in the work of Lipkin and Duffy (1988). This simplification is achieved by recognising that velocity screws and wrenches are different geometrical objects and then keeping them separate throughout the discussion. So we do not use any metric properties of the screw space of infinitesimal rigid body motions. Rather, we employ the duality between the vector space of screws and the linear functionals on them. We give several examples and show how changes of coordinates should be handled
In robotics the group of proper rigid transformations of 3-dimensional space is of central import... more In robotics the group of proper rigid transformations of 3-dimensional space is of central importance. The relevant Clifford algebra in this case is a degenerate one with three generators that square to — 1 and a single generator that squares to 0. The algebra contains a copy of the group’s double cover.
The Cayley map for the rotation group SO(3) is extended to a map from the Lie algebra of the grou... more The Cayley map for the rotation group SO(3) is extended to a map from the Lie algebra of the group of rigid body motions SE(3) to the group itself. This is done in several inequivalent ways. A close connection between these maps and linear line complexes associated with a finite screw motions is found.
Molien’s theorem concerns the invariants of finite groups. So we will introduce a small finite gr... more Molien’s theorem concerns the invariants of finite groups. So we will introduce a small finite group to use for our examples. The permutation group or symmetric group on 3 letters has the presentation, S3 =< σ, τ |σ2 = τ = e, στ = τσ > that is,σ andτ are generators of the group, e being the identity element. These generators satisfy the relations, σ = τ = e, that isσ has order 2 andτ order 3, in factσ can be thought of as a swap and τ as a cycle. The final relation is στ = τσ, or στσ = τ which shows that the subgroup generated by τ on its own is a normal subgroup. In cycle notation we could write, σ = (12), τ = (123).
Further the groupGL(n) is a Lie group, this just means that the multiplication and the map which ... more Further the groupGL(n) is a Lie group, this just means that the multiplication and the map which sends every element to its inverse are both differentiable maps. This is easy to see since the maps can be defined in terms of addition, multiplication and division by non-zero numbers. We will usually take the ground field, the set containing the matrix entries, to be complex numbers. This is just for simplicity, it is sometimes more useful to know the case where the ground field is the real numbers or some finite field. It is possible to say something about these cases but as usual, when we use complex numbers the theory is more complete. We can write GL(n,R) to distinguish the case where the ground field is the real numbers. As a manifold the groupGL(n) will be an open set in the space C 2 , the complement of the closed set consisting of the singular matrices. So the group has complex dimensionn, real dimension2n. Notice that the groupGL(n,R), real dimension n, has 2 disjoint componen...
The model of a ball rolling on an inclined plane is derived using screw theory and Lie algebra. T... more The model of a ball rolling on an inclined plane is derived using screw theory and Lie algebra. The plane is supported by a gimbal and the inclination can be controlled in two orthogonal directions. The model includes effects due gravity, Coriolis and centripetal torques.
Fifth World Congress on Intelligent Control and Automation (IEEE Cat. No.04EX788)
ABSTRACT Holzer method is a very efficient lumped parameter dynamic modeling method for flexible ... more ABSTRACT Holzer method is a very efficient lumped parameter dynamic modeling method for flexible manipulators. However, it does have some faults. An improved efficiency dynamic modeling method based on the Holzer method is proposed: This new method is called Ding-Holzer method, it is applied to the dynamic modeling of planar single arm, multiple link manipulators and the arm with spatial compliance, respectively. A sequence comparative study and theoretical analysis convince the improved method.
This work argue that directional statistics are important in Robotics. That is, statistics on gen... more This work argue that directional statistics are important in Robotics. That is, statistics on general manifolds. Historically the subject began with statistics on circles and spheres, hence the title. It is still a relatively new discipline, see [4] for an overview. In order to make the case a specific example is studied: Finding the rigid transformation undergone by a camera from a knowledge of the images of a number of points. This very simple, perhaps naive, example allows us to study different models for error in the observations and estimators for the rigid transformation.
A line symmetric motion is the motion obtained by reflecting a rigid body in the successive gener... more A line symmetric motion is the motion obtained by reflecting a rigid body in the successive generator lines of a ruled surface. In this work we review the dual quaternion approach to rigid body displacements, in particular the representation of the group SE(3) by the Study quadric. Then some classical work on reflections in lines or half-turns is reviewed. Next two new characterisations of line symmetric motions are presented. These are used to study a number of examples one of which is a novel line symmetric motion given by a rational degree five curve in the Study quadric. The rest of the paper investigates the connection between sets of half-turns and linear subspaces of the Study quadric. Line symmetric motions produced by some degenerate ruled surfaces are shown to be restricted to certain 2-planes in the Study quadric. Reflections in the lines of a linear line complex lie in the intersection of a the Study-quadric with a 4-plane.
We discuss a unified approach to a class of geometric combinatorics incidence problems in two dim... more We discuss a unified approach to a class of geometric combinatorics incidence problems in two dimensions, of the Erdős distance type. The goal is obtaining the second moment estimate. That is, given a finite point set S in 2D, and a function f on S × S, find the upper bound for the number of solutions of the equation (1) f (p, p ′) = f (q, q ′) = 0, (p, p ′ , q, q ′) ∈ S × S × S × S. E.g., f is the Euclidean distance in the plane, sphere, or a sheet of the two-sheeted hyperboloid. Our ultimate tool is the Guth-Katz incidence theorem for lines in RP 3 , but we focus on how the original problem in 2D gets reduced to its application. The corresponding procedure was initiated by Elekes and Sharir, based on symmetry considerations. The point we make here is that symmetry considerations, not necessarily straightforward and potentially requiring group representation machinery, can be bypassed or made implicit. The classical Plücker-Klein formalism for line geometry enables one to directly interpret a solution of (1) as intersection of two lines in RP 3. This, e.g., allows for a very brief argument as to how the Euclidean plane distance argument extends to the spherical and hyperbolic distances. The space of lines in the projective three-space, the Klein quadric K, is four-dimensional. Thus, we start out with an injective map F : S × S → K, that is from a pair of points (p, q) to a line lpq, and seek a corresponding combinatorial problem in the form (1) in two dimensions, which can be solved by applying the Guth-Katz theorem to the set of lines {lpq} in RP 3. We identify a few arguably new such problems, and hence applications of the Guth-Katz theorem and make generalisations of the existing ones. It is the direct approach in question that is the main purpose of this paper.
Proceedings of IEEE International Conference on Robotics and Automation
ABSTRACT We give a geometrical description of Raibert and Craig&#39;s hybrid force/position c... more ABSTRACT We give a geometrical description of Raibert and Craig&#39;s hybrid force/position control method (1981). Our description is coordinate free, hence answering the criticism of the original work that it was not transformation invariant. However, our approach avoids the complications introduced in the work of Lipkin and Duffy (1988). This simplification is achieved by recognising that velocity screws and wrenches are different geometrical objects and then keeping them separate throughout the discussion. So we do not use any metric properties of the screw space of infinitesimal rigid body motions. Rather, we employ the duality between the vector space of screws and the linear functionals on them. We give several examples and show how changes of coordinates should be handled
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Papers by J. Selig