99 7. Applications and generalizations of the results
and ωi2 and ωn are also linearly independent. Calculate the spherical Jacobian of the fictitious manipulator with joint axes ωi1, ωi2 and ωn (withωr=ωn), then determineγ ∈Randr ∈R3for which the spherical Jacobian is regular, that can be done because of Theorem 12. Then calculate the spherical Jacobian for the redundant arm as
JS =ωnωn⊤JΩ+JΩ×ωn, (7.5) and the regularized spherical Jacobian as
JSreg =JS+γ
I−ωnωn⊤
ω1×r ω2×r . . . ωn−1×r 0
. (7.6) Since this is a3×nmatrix, and the columns with indicesi1,i2andnare linearly independent by the construction, the matrixJSregis full rank.
100 7. Applications and generalizations of the results
Now, the regularized end effector Jacobian can be defined as J˜e,reg =
JV reg JSreg
, (7.10)
withJV reg being defined as
JV reg= (Ad(I,−γVrV)Je)V (7.11) andJSregbeing defined as
JSreg =JS+γΩ
I−ωnω⊤n
ω1×rΩ ω2×rΩ . . . ωn−1×rΩ 0 . (7.12) Note that the regularization vectorsrV andrΩare independent of each other, and so are the constantsγV and γΩ. Using these definitions, the joint velocity needed for the desired end effector velocityx˙d can be cal-culated as
θ˙= ( ˜Je,reg)#( ˙gxd) (7.13) where#denotes the Moore–Penrose pseudoinverse of the matrix.
8
THESES
Thesis Group 1 (Chapter 3)
I provided new properties of theadandAdoperators related to the nested Lie brackets of motion generators and their exponential mappings, and the properties of the generators of adjacent robot joints. I provided the action point transformation to determine the velocity generator of an ar-bitrary point of the rigid body and the end effector Jacobian describing the velocity generators of the end effector frame.
Publications related to the theses are: [DH09], [DH12b], [DH11], [DH13], [DH14], [HD09].
Thesis 1.1
I proved that the multiple action of theadoperator inse(3)is i, adixy = 0, fori >1ifxΩ = 0.
ii,
adixy=
adxy ifimod4 = 1 ad2xy ifimod4 = 2
−adxy ifimod4 = 3
−ad2xy ifimod4 = 0 andad0xy =y, ifxΩ 6= 0.
I proved that theAdoperator has the following properties:
i, ∀ξ1∈se(3)and∀t∈R,Ad
eξˆ1tξ1 =ξ1.
101
102 8. Theses
ii, Ifξ1, ξ2∈se(3)are linearly independent, then∀t∈R,ξ1andAdeξˆ1tξ2 are also linearly independent.
iii, ∀ξ1, ξ2 ∈se(3)and∀t∈R,h ξ1,Ad
eξˆ1tξ2i
= Adeξˆ1t[ξ1, ξ2].
iv, Ifeξtˆ is a pure translation, i.e.ξΩ = 0, thenAdeξtˆ ξ0=ξ0+tadξξ0. I proved that if the transformation g ∈ SE(3) is not a pure trans-lation, then Ad can be calculated with a closed formula similar to the Rodrigues formula: if ξ1,Ω 6= 0, then Ad
eξˆ1θ1 ξ2 can be written in the following closed form:
Adeξˆ1θ1ξ2=ξ2+ sin (θ1) [ξ1, ξ2] + (1−cos (θ1)) [ξ1,[ξ1, ξ2]].
The Lie algebrase(3)has two linearly independent invariant (under coordinate transformations) bilinear forms: the Killing form and the reciprocal product. The Killing form of two motion generators is
κ(ξ1, ξ2) =−1
2hω1, ω2i,
while the reciprocal product of two motion generators is ξ1⊙ξ2 =hω1, v2i+hω2, v1i.
I proved that the Killing form and the reciprocal product of adjacent joints are invariant of the joint configuration.
Thesis 1.2
I introduced the action point transformation, with which one can specify the point whose linear velocity is of particular interest, and after the application of the action point transformation on the generatorξi, itsvi component will describe the infinitesimal translation of the new point instead of the origin of the base frame. Letp be the point of interest, I proved that the application of the action point transformation on ξi = (ωi, vi)can be calculated as follows:
vi′ ωi′
= Ad(I,−p)
vi ωi
=
vi+ωi×p ωi
.
Thesis 1.3
I defined the end effector Jacobian, whose columns are motion genera-tors that define the infinitesimal motion of the end effector frame de-scribed in the base frame. I proved, that the end effector Jacobian can be calculated as
Je(θ) = Ad(I,−p(θ))Js(θ) (8.1)
103 8. Theses
or
Je(θ) = Ad(R(θ),0)Jb(θ), (8.2) where p(θ) is the position of the origin of the end effector frame given in the base frame, whileR(θ) is the transformation between the base frame and the end effector frame (the orientation of the end effector frame), both in the joint configurationθ. I proved, that the ranks of the JacobiansJe, JsandJbare identical in every joint configuration.
Thesis Group 2 (Chapter 4)
I provided a method to regularize the differential inverse positioning problem using the action point transformation. I generalized the reg-ularization method for the spatial inverse positioning problem for ma-nipulators with arbitrary degrees-of-freedom. I provided sufficient and necessary condition for regularizability in the general case, and for three degrees-of-freedom manipulators I provided bounds on the singular val-ues and a closed formula for the determinant of the regularized Jacobian using the Lie brackets of the generators.
Publications related to the theses are: [DH13], [DH12b], [DH11].
Thesis 2.1
I proved, that the inverse positioning problem can always be regularized in the plane. I justified, that a suitable choice of the regularization vector is the linear velocity generator of the second joint. I proved, that the singular values of the regularized task Jacobian are bounded as follows:
σ(Jreg) ≤ 2p
(l1+l2)2+γ2 σ(Jreg) ≥ |detJregSV |
2p
(l1+l2)2+γ2
withl1 andl2 being the lengths of the segments of the manipulator and detJreg being the determinant of the regularized Jacobian. I justified, that motion in singular direction gets rescaled, when the regularized Jacobian is used, however, motion in regular direction is unaffected. I also analyzed the effect of regularization in the configuration space of the planar manipulator.
Thesis 2.2
I proved, that the inverse positioning problem of manipulators moving on a surface (i.e. two joint manipulators with joint axes not parallel)
104 8. Theses
can always be regularized. I justified, that the regularization vector can be chosen as the linear velocity generator of the last joint. I also proved, that the singular values of the regularized task Jacobian are bounded the same way as in the planar case.
Thesis 2.3
I proved, that if joint axes of the manipulator are not all parallel, or does not intersect each other in a point, or the end effector point is not on the last joint axis, then a sufficient and necessary condition for regu-larizability of the task Jacobian is that the end effector Jacobian is full rank.
I proved, that the regularization vector has to be in the range space of the task Jacobian, that is useful for its construction. I provided closed form expression on the determinant of the regularized Jacobian using the Lie brackets of the motion generators, and I proved, that the singu-lar values of the regusingu-larized Jacobian are bounded as follows:
σ(Jreg) ≤ 3p
L2+γ2 σ(Jreg) ≥ |detJreg| 9(L2+γ2),
whereLis the length of the manipulator in a fully extended state.
Thesis Group 3 (Chapter 5)
I provided a method to regularize the inverse orientation problem. I de-fined the spherical representation that defines infinitesimal rotation as a one-dimensional infinitesimal rotation and a two-dimensional infinites-imal translation. I provided the task Jacobian in the spherical repre-sentation, called the spherical Jacobian, and regularized the spherical Jacobian using the action point transformation. I provided bounds on the singular values of the regularized spherical Jacobian.
Publications related to the thesises: [DH14], [DH12b], [DH11], [DH13].
Thesis 3.1
I introduced the spherical representation, in which the orientation change is defined as infinitesimal movement on the tangent space of a sphere, and one-dimensional rotation around the normal of the sphere. I intro-duced the spherical Jacobian, that is the task Jacobian for the orienta-tion problem described in the spherical representaorienta-tion. I proved, that the rank of the task JacobianJΩand the spherical JacobianJSare iden-tical in every joint configuration. I also proved, that the rank deficiency
105 8. Theses
of these matrices is at most one. I proved, that if the spherical Jaco-bian is singular, then the singular direction is one of the translational generators.
Thesis 3.2
Since the singularity of the inverse orientation problem appears in the linear velocity generator that defines translation on the tangent space of a sphere, it can be regularized using the methods from Thesis Group 2. I introduced the regularized spherical Jacobian using the regulariza-tion method described for the inverse orientaregulariza-tion problem, I provided conditions for regularizability and I proved, that the singular values of the regularized Jacobian are bounded as follows:
σ(JSreg) ≤ 3p 2 +γ2 σ(JSreg) ≥ |detJSreg|
9(2 +γ2) .
withdetJSreg being the determinant of the regularized spherical Jaco-bian.
Thesis Group 4 (Chapter 6)
I provided a method to incorporate joint constraints into the differential inverse kinematics algorithm. I proved that the method can be used in conjunction with other algorithms based on the differential inverse kine-matics algorithm. I identified singular configurations arising when joint limits are reached, and provided a method to overcome these singulari-ties.
Publications related to the thesises: [DH12a], [DH12b], [DH11].
Thesis 4.1
I introduced a nonlinear transformation on the joint space of the ma-nipulator, that maps the real line onto the open interval of the admis-sible joint variable values (i.e. an interval with lower and upper limits being the lower and upper limits of the joint variables). I call this trans-formation the joint constraint function, while its inverse the inverse joint constraint function. Using the inverse joint constraint function, the joint variables can be mapped onto the real line, where the differen-tial inverse kinematics algorithm can be carried out. I introduced the constrained Jacobian, that is the Jacobian matrix after the nonlinear transformation. I proved, that the constrained Jacobian does not nec-essarily have to be applied, it is enough to calculate the joint velocities in the usual way, and transform them only before the integration takes place.
106 8. Theses
Thesis 4.2
The nonlinear joint transformation causes singularities if some of the joint variables reach their limits. I call this situation a constrained sin-gularity. I analyzed constrained singularities, and provided a solution to overcome these singularities. The solution is based on the Singular Value Decomposition of the matrix containing the differentials of the in-verse constrained functions, for which I provided a closed-form solution, and tends to drive the joints away from the joint limits if this motion is advantageous for the desired end effector motion, and leaves the joint variables at their limits otherwise.
9
CONCLUSION
The dissertation was about the solution of the differential inverse kine-matics algorithm. It focused on the operations needed to be done in one iteration, and did not concern the consequences of using the proposed methods throughout more iterations of the differential inverse kinemat-ics algorithm.
The mathematical background of the thesis is the Lie group and Lie algebra describing rigid body movements in three dimensions. Some properties of the operators and bilinear forms on this Lie group and Lie algebra were explored in Chapter 3. The end effector Jacobian was in-troduced, that is the analytical Jacobian of the forward kinematics map, based on motion generators. The action point transformation was de-fined, that transforms the action point of the linear velocity generators to arbitrary points in the space.
The regularization of the inverse positioning problem was discussed in Chapter 4. The idea was introduced on a planar manipulator with two joints; the existence of the regularization vector was proved. Bounds on the singular values of the regularized task Jacobian were given, and it was shown, that motion is scaled down in singular directions, but re-mains unaffected in regular directions due to the regularization. The methodology was generalized for spatial manipulators as well.
The regularization of the inverse orientation problem was discussed in Chapter 5. The inverse orientation problem is transformed to the verse position on a plane and inverse orientation around an axis by in-troducing the spherical representation and the spherical Jacobian. The spherical Jacobian is then regularized using the ideas in Chapter 4.
The joint constraints were incorporated into the differential inverse kinematics algorithm in Chapter 6. The joint variables are transformed
107
108 9. Conclusion
to a fictitious joint space, and then transformed back after the integra-tion in the differential inverse kinematics algorithm, with a transfor-mation ensuring that the joint variables always stay between the limit.
The effect of the transformation was analyzed, and the singularities arising because of the transformation were handled.
The regularization of the task Jacobians was generalized for the end effector Jacobian of manipulators with n joints in Chapter 7. It was shown, that the regularization and joint constraint incorporation has no effect on each other, thus they can be applied simultaneously.
The automatic calculation of the regularization vectors is straight-forward for planar manipulators and in the case of the inverse orienta-tion problem, however it is not in the case of spatial manipulators. De-termining theγ parameter as a function of time or the function of the condition number or determinant of the task Jacobian is still a problem to be solved, along with the analysis of the effect of the regularization algorithm during more iterations of the differential inverse kinematics algorithm. Note that since lower bounds can be imposed on the smallest singular values of the task Jacobians according to Theorems 2, 9 and 13, thus stable closed-loop inverse kinematics algorithm can be developed using the results in [72].
INDEX
action point transformation, 28 ad operator, 16
Adjoint operator, 18 automorphism, 18
closed-loop inverse kinematics, 23 configuration space, 44
constrained end effector Jacobian, 88
constrained Jacobian, 88 constrained singularity, 92 constraint function, 86
coordinate transformation, 18 Damped Least Squares, 92 discrete time, 22
discretization, 22 DLS pseudoinverse, 92 fictitious joint space, 87 fictitious joint variable, 86 forward kinematics, 20 generator, 13
linear velocity, 13 rotational, 13 translational, 13 group, 11
Special Euclidean, 11 Special Orthogonal, 12 homogeneous coordinates, 10
invariant bilinear form, 18 inverse constraint function, 86 isomorphism, 14
Jacobian
analytical, 29 constrained, 88
constrained end effector, 88 end effector, 29
spherical, 76 Killing form, 18 Lie algebra, 13
special Euclidean, 13 special orthogonal, 13 Lie group, 12
Lie subalgebra, 17
manipulator Jacobian, 28 body, 21
spatial, 21 nilpotency index, 17 nilpotent
Lie algebra, 17 operator, 17
one-parameter subgroup, 15 P-type control, 23
planar manipulator, 33 109
110 INDEX
product of exponentials, 20 reciprocal product, 19 regularizable, 39, 52, 54 regularization, 37
regularization vector, 39, 52, 54, 81 regularized spherical Jacobian, 80 singular line, 45
Singular Value Decomposition, 92 spatial manipulator, 53
spherical joint
two degrees-of-freedom, 48 spherical representation, 75 SVD, 92
task Jacobian, 30 universal joint, 48 velocity
body, 21 spatial, 21
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