The modelling of end-effector's pose errors for the equivalent kinematical chain of the upper human limb.

Vacarescu, Valeria ; Lovasz, Erwin Christian ; Vacarescu, Cella Flavia 等

1. INTRODUCTION

A study upon the structure of the upper human limb is also used for acknowledging the morph-functional characteristics of the body (the interdependency between anatomy and biomechanics) and for obtaining important data in the "humanoid" modelling process for functional prosthesis. (Moeslung et al., 2005) proposed the modelling of the 3D position of a human arm only with two parameters to express the motion coursed by the anatomical joints. (Reinbolt et al., 2005) used the multi-joint kinematical model for the development of dynamics patient's specific model for the clinical problems in rehabilitation. A model of the human arm as a 7 DOF system is used by (Rosen et al., 2005) to resolve the equations of motion. (Bronzino et al., 2000) defines the geometry of joint's surfaces of upper human limb and the reference system attached to each joint, for studying upper human limb's biomechanics. (Vacarescu et al., 2005) models the upper human limb like an opened kinematical chain with 7 DOF and establishes the Jacobian's matrix for its analyses.

In this paper, it is proposed the differential modelling of 3D pose errors of the arm's kinematical chain with 7 DOF. Based on the above mentioned, the authors of this paper developed a model of human arm prosthesis, following the improvement of its performances through geometrical calibration using the proposed mathematical model.

2. THE MODELLING OF THE UPPER ARM

Structural, the upper human limb can be associated with an open kinematical chain with 7 DOF and eight links. In figure 1 is presented the equivalent kinematical chain. Because the shoulder's blades and the clavicles moves are too hard to model, in prosthesis fabrication, these links together with the afferent joints are included in the fixed link. So, the first kinematical joint of the equivalent kinematical chain is the shoulder's joint, respectively the spherical scapula-humeral joint. In kinematical studying, this joint can be replaced with three rotation joints, their axes having one common point. The three moves in this joint are maid along the axes perpendicular on the characteristic plans (sagittal, frontal, transversal). In the elbow joint, the main move is the flexion-extension move that is associated with a rotation kinematical joint. The wrists moves can be done, each, by a rotation kinematical joint. So, the equivalent kinematical chain is continued with three lower kinematical joints with perpendicular axes.

[FIGURE 1 OMITTED]

The directions of the three axes of the kinematical joints form a dextral trihedral and they are oriented upon the frontal, transversal and sagittal direction of the shoulder's joint. Properly choosing the other joints axes, the transform matrixes can be written, between the system's axes, for determining the position vector [bar.[r.sub.M]], of the characteristic point situated on the arm's end-effector. To do this, are used the Hartenberg-Denavit parameters. For fig.1, the values of the kinematical chains parameters Hartenberg Denavit are shown in the table 1.

3. THE DIFFERENTIAL MODELLING OF ERRORS

The major problem is the difference between the nominal geometry of the arm's kinematical chain, obtained by designing, in concordance with its functions and its real geometry, affected by manufacturing tolerances, assembling errors occurred during arm's assembling. Generally, the nominal geometrical models are simple, based on some presumptions like the parallelism and perpendicularity of axis.

Are used Hartenberg-Denavit parameters ([[theta].sub.i], [d.sub.i], [a.sub.i], [[alpha].sub.i]) in modelling of geometrical errors of the relative position parameters between the chains elements. It is known the fact that the small geometrical errors of these four parameters can lead to important variations of the pose parameters. It is necessary an external parameter, the [[beta].sub.i] angle, which defines a rotation system (i) as a new system (i'), around the [y.sub.i] axis (fig. 2). In the absence of [[beta].sub.i] angle (called twist angle), the non-parallelism of the axis must be set off by an artificial correction of lenghts [a.sub.i] and [d.sub.i], even if these parameters were initially correct. The [[beta].sub.i] rotation angle is used only for the parallel axis of the successive rotation joints. For the nominal geometrical model, [[beta].sub.i] = 0. The geometrical model allows to establish the pose vector of the arm's end-effector, as function of Hartenberg-Denavit geometrical parameters:

x = f([theta]([alpha], a, d, [beta]) (1)

where: [theta],[alpha],a,d,[beta] are [R.sup.n] vectors, for the arm's "n" joints. This relation between geometrical parameters is, generally, nonlinear. In their identification order, the geometrical model is linearised in around an initial estimation, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (nominal parameters) of the real parameters [theta],[alpha],a,d,[beta] . Since two axis of some successive rotation joints are (in the nominal geometrical model, [[beta].sub.sub.i] initialy supposed to be parallel, it is chosen [[??].sub.i] = 0 , to each element "i" of the arm. A differential model of the errors is obtaines such as:

[DELTA]x = [J.sub.[theta]] x [DELTA][theta] + J[alpha] x [DELTA][alpha] + [J.sub.a] x [[DELTA].sub.a] + [J.sub.d] x [[DELTA].sub.d] + [J.sub.[beta]] x [[DELTA].sub.[beta]] (2)

where: [DELTA][theta] is the error vector of the angle [[theta].sub.i] ( measured in the rotation joints with transducers); [DELTA][alpha], [DELTA]a, [DELTA]d, [DELTA][beta] are the error vectors of the torsion angle [[alpha].sub.i], the length [[alpha].sub.i], [d.sub.i] and angle [[beta].sub.i] (in the case of translation joints, [d.sub.i] is measured with transducers), and:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

[FIGURE 2 OMITTED]

Each one of these jacobian matrix is a sensitive matrix of end-effector's position and orientation, considering the variation of the Hartenberg-Denavit geometrical parameters (moderating coefficient). These matrix are calculated using the [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] geometrical parameters nominal values. Generally, the vector [J.sub.[theta]i] can be written such as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

where: "[]", "{}", and "x" mean: matrix, vector and vector product; [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the rotation matrix of the coordinate axis system related to the base (0) coordinate system; [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the end-effector's position vector related to system (i-1) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. A similar expression can be used for each jacobian matrix. The differential model of the errors, expressed in the relation (2) can be used for a geometrical calibration of the arm, in biomechanical applications. In this situation, is estabilshed the end-effector's pose error, expressed by [DELTA]x in the relation (2), by measuring with any system (with ultrasound measurement module or optical system, in our research project). Also are measured, with transducers, [DELTA][[theta].sub.i] or [DELTA][d.sub.i] in joints and using as initial dates in the relation (2), [DELTA]x, [DELTA][[theta].sub.i] (or [DELTA][d.sub.i]), such as the nominal values [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of the Hartenberg-Denavit geometrical parameters, are calculeted the errors [DELTA][theta], [DELTA][alpha], [DELTA]a, [DELTA]d, [DELTA][beta], used for obtaining real, corrected Hartenberg-Denavit geometrical parameters, [Q.sup.*], [[alpha].sup.*],[a.sup.*],[d.sup.*],[[beta].sup.*] :

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

The computation is made iterative until it riches the minimum of errors, by introducing in the relation (2) the new estimated values of the Hartenberg-Denavit geometrical parameters, [[theta].sup.*],[[alpha].sup.*],[a.sup.*],[d.sup.*],[[beta].sup.*].

4. CONCLUSION

The method presented in this paper it is developed in the CEEX 88 research project, for biometrical measurements. The present algorithm is applied in programming the control system of the human arm prosthesis, for the improvement of position and orientation accuracy of the end-effector. In the future work, the authors of the paper propose the developing of the human arm model, for the improvement of the prosthesis as well as for the use of a robot arm on a wheelchair.

5. REFERENCES

Bronzino, J. (2000). The biomedical engineering handbook, CRC Press, IEEE Press, IEEE Order No: PC5788, USA

Moeslund, T.; Granum, E.; (2005). Modeling the 3D pose of a human arm and shoulder complex using only two parameters, Integrated Computer Aided Engineering, vol.12, issue 2, 159-175, ISSN 1069-2509

Reinbolt, J; Schutte, J et al. (2005). Determination of the patient-specific multi-joint kinematic models through two-level optimization, Journal of biomechanics, vol.38, issue 3, 622-626, Elsevier, Florida

Rosen,J; Perry, J.C.; Manning, N. et al. (2005). The human arm kinematics and dynamics during daily activities--toward a 7DOF upper limb powered exoskeleton, Advanced Robotics ICAR'05 Proc., ISBN:0-7803-9178-0, Seattle

Vacarescu, V.; Vacarescu, C.F.(2005). The kinematics analysis of the equivalent kinematical chain of the upper human limb, 5th International Conference RaDMI 2005 Proceedings, Serbia & Montenegro, section E25, ISBN 86-83803-20-1. Tab. 1. The values of Hartenberg-Denavit parameters Joint i [q.sub.i] [[alpha].sub.i] [a.sub.i] [d.sub.i] 0 [[theta].sub.1] 0[grados] 0 0 1 [[theta].sub.2] 90[grados] 0 0 2 [[theta].sub.3] 90[grados] 0 0 3 [[theta].sub.4] -90[grados] [a.sub.3] 0 4 [[theta].sub.5] 90[grados] 0 0 5 [[theta].sub.6] -90[grados] 0 [d.sub.3] 6 [[theta].sub.7] 90[grados] [a.sub.7] 0