Working Space Representation for the Human Upper Limb in Motion

Transcription

1 7th WSEAS Int. Conf. on APPLIED COMPUTER & APPLIED COMPUTATIONAL SCIENCE (ACACOS '8), Hangzhou, China, April 6-8, 28 Working Space Representation for the Human Upper Limb in Motion ANTOANELA NAAJI Faculty of Computer Science Vasile Goldis Western University Arad 3125, Bd. Revolutiei nr ROMANIA Abstract: - Theoretically speaking, the modelling process is based on developing a model representing all properties of an object or phenomenon corresponding to the purpose of the analysis. To this effect, carrying out a virtual model of the human body consists of representing its main characteristics, as close to reality as possible. Motive activities of the human upper limb result from the motion of kinematical chains made by skeletal elements connected by joints. The aim of this paper is to describe the human upper limb as a kinematical structure similar to that of a robot with five degrees of freedom for which we apply the conventions of Robotics, obtaining the kinematical model of the human upper limb. In this case, the arm has three degrees of freedom in the shoulder joint, the forearm one degree of freedom in the elbow joint and the hand one degree of freedom in the radiocarpal joint. The kinematical equations of motion obtained are parametric and express the position and orientation of the hand in relation to a fixed reference frame. One of the most important applications of parametric representation of the systems is the working space modelling described by the end of the kinematical chain during the motion. The approach to hyper-surfaces modelling is based on the direct geometric model considering the human upper limb as a kinematical structure similar to that of a robot. For generating the surfaces in which the end-effector of the human upper limb moves, we applied certain specific functions of the Matlab program. Key-Words: - Kinematical chain, kinematical model, parametric representation, working space, hyper-surfaces modelling 1 Introduction The human upper limb represents not only a concatenation of body segments in a functional unit, very important for labor and creation activities, but also a human body prolongation which, considered as a whole, is characterized by a specific pathology. The belt of the human upper limb, the shoulder, the arm, the forearm and the hand function together during different motions either as an open kinematical chain or a close kinematical chain. The degrees of freedom of the human upper limb segments are summing up, which leads to the amplification of motion possibilities. In this paper we considered the human upper limb as a kinematical structure similar to that of a robot with five degrees of freedom, based on Denavit- Hartenberg convention [3, 4, 5]. The degrees of freedom are associated to the three joints: the arm joint, the forearm joint and the hand joint. Based on the kinematical model of human upper limb, respectively on the kinematical equations of motion, in this paper we modelled the hypersurfaces inside which the origin trajectories of the end-effector are to be found, starting from simple motions to complex ones. 2 The Human Upper Limb as Kinematical Chain Regarding the motions mentioned, for human upper limb as an open kinematical chain [1, 5, 8], the elevation and the descending motions and also the circumduction motions of arms depend in the first place on the kinematical chain: belt-shoulder-arm. Gripping depends on the chain: forearm - arm neck - arm. Pushing, griping, striking, body maintain, climbing and falling in hands train compulsorily all the upper limb segments. During the abduction motion, the two extremities of the humerus move in opposite courses: at the same time when the lower extremity ascends, the upper extremity descends. The motion takes part around a biomechanical ante-posterior axis, which ISBN: ISSN:

2 7th WSEAS Int. Conf. on APPLIED COMPUTER & APPLIED COMPUTATIONAL SCIENCE (ACACOS '8), Hangzhou, China, April 6-8, 28 passed the lower external part of the humeral head, little inside the anatomical neck. The adduction motion takes part in an opposite course. In orthostatic position, the important role in executing it occurs to the upper limb weight and gravitational forces. The forward and backward projection is produced around a transversal axis, which passes through the center of the greater tuberosity and through the center of the glenoidal cavity. The amplitude of forward projection is 95 while that of backward projection 2. The inside and outside rotation motions are produced around a vertical axis, which passes through the humeral head. In the outside rotation motion, the humeral head glides from backwards through the glenoidal cavity. Their amplitude is about 8 for rotation and about 95 for internal rotation. The circumduction motion is summing up the previous motions and executes around the three axes. The humeral head describes a small circle, while the lower extremity of the humerus describes a large circle, but in the opposite course. Flexion-extension is executed in a sagital plane around a transversal axis, which practically superpose the bi-epicondial line, through the flexion motion, the forearm approaching the arm. Its active normal amplitude is approximately 15. The forearm is structured so that it allows the pronation and supination motions. The pronation and supination motions are rotation motions that are executed around a vertical axis, which passes through the middle radial couples. The average normal amplitude of an active pronation-supination motionis is 18. The neck hand s motions take place around a center, which can be considered to be is the big bone, the central pivot around which the other carpals bones move. Flexion-extension takes place in the sagital plane around a transversal axis, which passes through the head of the big bone. The active and passive flexion motions have 9 amplitude. The radial and cubital bending motions take place around an ante-posterior axis, which passes through the center of the big bone. The motions of the active bending have amplitude of 4, but the passive ones have 45. The circumduction motion results from the passing through the flexion, abduction extension, adduction positions or invert. The motion does not represent a circle, but an ellipse, because the flexion and extension are more ample than the lateral bending [2]. 3 Kinematical Model of Human Upper Limb It is considered that the human upper limb has three joints (figure 1): The arm joint is represented through three revolute joints corresponding to the three degrees of freedom resulting from the simple flexion-extension motions (around a transversal axis), abductionadduction motions (around a sagital axis), inside and outside rotation (around a vertical axis); The forearm joint is represented by a single revolute joint corresponding to a single degree of freedom resulting from simple flexion-extension motion (around a transversal axis); The hand joint is represented by a revolute joint corresponding to a degree of freedom resulted from simple flexion-extension motion (around transversal axis). Fig. 1 Simplified kinematical chain of human upper limb We represented the fixed reference frame with the origin in the center of the scapular belt. The origins of the first three systems (notated with, 1 and 2) can be consider superpose for modelling through three simple revolute joints, a joint with three degrees of freedom. The distance between the fix reference system and the origins of these systems is notated with c, being equal with the half of distance to scapular belt. The distance between the origins of the second and third systems can be considered equal to the length of humerus and it is notated with h. The origin of the third system is modelled through a cylindrical joint in the elbow ISBN: ISSN:

3 7th WSEAS Int. Conf. on APPLIED COMPUTER & APPLIED COMPUTATIONAL SCIENCE (ACACOS '8), Hangzhou, China, April 6-8, 28 joint, with one degree of freedom. The distance between the origin of the third and fourth systems can be considered equal to the length of the radius and ulna, and it is notated with r and the distance between the origin of the fourth and fifth systems is equal to the length of the hand and it is notated with p+d. Thus, the hand joint is modelled as a joint with one degree of freedom. For the determination of the kinematical equations we used the Denavit-Hartenberg convention [3]. We considered that the position and the direction of the last element with respect to the fixed reference system, at one moment, is o expressed through the matrices G5 which resulted from the multiplication of transfer matrices expressing the position of each element from the kinematical chain, with respect to the anterior one [6]. For the determination of the transfer matrices i 1 T we used the data from table 1. i Nr Articular cos sin l variable i θ I d i α i α i α i 1 q 1 =θ 1 θ 1 c q 2 =θ 2 θ q 3 =θ 3 θ 3 h q 4 =θ 4 r θ q 5 =θ 5 p+d θ 5 1 Table1 The articular variables corresponding to the three joints of the human upper limb model with five degrees of freedom o The transfer matrix G5 which represents the position and the direction of the reference system attached to the last element front to the fixed reference frame can be determined by multiplying the transfer matrices: G 5 = [ T 1 ] [ 1 T 2 ] [ 2 T 3 ] [ 3 T 4 ] [ 4 T 5 ] (1) The transfer matrices corresponding to the kinematical chains of the human upper limb - the simplified model are: T 1 cosθ1 = sinθ1 1 sinθ1 cosθ1 c 1 (2) cosθ 2 sinθ 2 1 = sinθ 2 cosθ 2 T T 3 cosθ3 = sinθ3 1 sinθ3 cosθ3 1 h 1 cosθ 4 sinθ 4 r cosϑ4 3 = sinθ 4 cosθ 4 r sinθ 4 T cosθ 5 sinθ 5 ( p+ d) cosθ5 4 sinθ 5 cosθ 5 ( p+ d) sinθ 5 T 5 = 1 1 (3) (4) (5) (6) Replacing in relation (1), the transfer matrices obtained and using the notation m for (p+d), through o the identification of the matrix G5, we obtained the kinematical equation of the system: n x =cos(θ5)*cos(θ4)*cos(θ1)*cos(θ2)*cos(θ3)- cos(θ5)*cos(θ4)*sin(θ1)*sin(θ3)+cos(θ5)*cos(θ1)* sin(θ2)*sin(θ4)-sin(θ5)*sin(θ4)*cos(θ1)*cos(θ2)* cos(θ3)+sin(θ5)*sin(θ4)*sin(θ1)*sin(θ3)+sin(θ5)* cos(θ1)*sin(θ2)*cos(θ4) (7) n y =cos(θ5)*cos(θ4)*sin(θ1)*cos(θ2)*cos(θ3)+ cos(θ5)*cos(θ4)*cos(θ1)*sin(θ3)+cos(θ5)*sin(θ1)* sin(θ2)*sin(θ4)-sin(θ5)*sin(θ4)*sin(θ1)*cos(θ2)* cos(θ3)-sin(θ5)*sin(θ4)*cos(θ1)*sin(θ3)+sin(θ5)* sin(θ1)* sin(θ2)*cos(θ4) (8) n z =cos(θ5)*sin(θ2)*cos(θ3)*cos(θ4)-cos(θ5)* cos(θ2)*sin(θ4)-sin(θ5)*sin(θ2)*cos(θ3)*sin(θ4)- sin(θ5)* cos(θ2)*cos(θ4) (9) o x =-sin(θ5)*cos(θ4)*cos(θ1)*cos(θ2)*cos(θ3)+ sin(θ5)*cos(θ4)*sin(θ1)*sin(θ3)-sin(θ5)*cos(θ1)* sin(θ2)* sin(θ4)-cos(θ5)*sin(θ4)*cos(θ1)*cos(θ2)* cos(θ3)+cos(θ5)*sin(θ4)*sin(θ1)*sin(θ3)+cos(θ5)* cos(θ1)* sin(θ2)*cos(θ4) (1) o y =-sin(θ5)*cos(θ4)*sin(θ1)*cos(θ2)*cos(θ3)- ISBN: ISSN:

4 7th WSEAS Int. Conf. on APPLIED COMPUTER & APPLIED COMPUTATIONAL SCIENCE (ACACOS '8), Hangzhou, China, April 6-8, 28 sin(θ5)*cos(θ4)*cos(θ1)*sin(θ3)-sin(θ5)*sin(θ1)* sin(θ2)* sin(θ4)-cos(θ5)*sin(θ4)*sin(θ1)*cos(θ2)* cos(θ3)-cos(θ5)*sin(θ4)*cos(θ1)*sin(θ3)+cos(θ5)* sin(θ1)*sin(θ2)*cos(θ4) (11) o z =-sin(θ5)*sin(θ2)*cos(θ3)*cos(θ4)+sin(θ5)* cos(θ2)* sin(θ4)-cos(θ5)*sin(θ2)*cos(θ3)*sin(θ4)- cos(θ5)* cos(θ2)*cos(θ4) (12) a x =-cos(θ1)*cos(θ2)*sin(θ3)-sin(θ1)*cos(θ3) (13) a y =-sin(θ1)*cos(θ2)*sin(θ3)+cos(θ1)*cos(θ3) (14) a z =-sin(θ2)*sin(θ3) (15) p x =m*cos(θ5)*cos(θ4)*cos(θ1)*cos(θ2)*cos(θ3)- m*cos(θ5)*cos(θ4)*sin(θ1)*sin(θ3)+m*cos(θ5)* cos(θ1)*sin(θ2)*sin(θ4)-m*sin(θ5)*sin(θ4)* cos(θ1)*cos(θ2)*cos(θ3)+m*sin(θ5)*sin(θ4)* sin(θ1)*sin(θ3)+m*sin(θ5)*cos(θ1)*sin(θ2)* cos(θ4)+r*cos(θ4)*cos(θ1)*cos(θ2)*cos(θ3)- r*cos(θ4)*sin(θ1)*sin(θ3)+cos(θ1)*sin(θ2)*r* sin(θ4)-cos(θ1)* sin(θ2)*h (16) p y =m*cos(θ5)*cos(θ4)*sin(θ1)*cos(θ2)*cos(θ3)+ m*cos(θ5)*cos(θ4)*cos(θ1)*sin(θ3)+m*cos(θ5)* sin(θ1)*sin(θ2)*sin(θ4)-m*sin(θ5)*sin(θ4)* sin(θ1)*cos(θ2)*cos(θ3)-m*sin(θ5)*sin(θ4)* cos(θ1)*sin(θ3)+m*sin(θ5)*sin(θ1)*sin(θ2)* cos(θ4)+r*cos(θ4)*sin(θ1)*cos(θ2)*cos(θ3)+r* cos(θ4)*cos(θ1)*sin(θ3)+sin(θ1)*sin(θ2)*r* sin(θ4)-sin(θ1)* sin(θ2)*h (17) p z =m*cos(θ5)*sin(θ2)*cos(θ3)*cos(θ4)-m*cos(θ5)* cos(θ2)*sin(θ4)-m*sin(θ5)*sin(θ2)*cos(θ3)* sin(θ4)-m*sin(θ5)*cos(θ2)*cos(θ4)+sin(θ2)* cos(θ3)*r*cos(θ4)-cos(θ2)*r*sin(θ4)+ cos(θ2)*h+c (18) 4 Parametric representation of human upper limb working space One of the most important applications of systems parametric representation is the working space modelling described by the end of the kinematical chain while doing its working task [7]. The kinematical equations of motion [5] are parametric equations which describe the position and orientation of the hand with respect to the fixed reference frame. Their representation in 3D space leads to modelling the space inside which the endeffector moves. For generating the graphic representations, we used some functions of Matlab program [9]. Initially, we declared the variables used in program. In our case, these variables were articular variables q i, respectively θ i angles of revolute joints. Matlab is a programming language that uses expressions. Expressions contain operators or other special characters, functions and names of variables. Considering that Matlab executes the operations with vectors and matrices faster with one order than compilation/interpretation operations, we ll obtain a higher speed for data processing if the algorithms contains in.m files are vectorized (for example: t=:.1:1). For simplifying the representation, certain articular variables are vectorized, for example θ 1 =:pi/2:.638*pi), and other are cycled using for instruction. This instruction allows one to repeat a group of instructions a certain number of times. The graphics are represented by designing 3D curves which contain all the points generating the triplet p x, p y, p z obtained from the solution of the human upper limb s direct geometric model. We specified these considerations in order to justify why sometimes the use of professional software, such as Matlab, is preferred instead of creating some original programs. An important feature of known professional programs is that beside the general character of application, they ensure the possibility of exchanging results, absolutely necessary in any kind of interdisciplinary research. Using the human upper limb kinematical model, the last column of matrices that express the position and orientation of the reference frame attached to the hand with respect to the fixed one, represents the equations of motion p x = p x (t), p y = p y (t), p z = p z (t) used for the parametric representation of hypersurfaces inside end-effector moves. Using the facilities of Matlab, we shall present in the following part of the paper the representation of trajectories and working space for the simplified model of human upper limb. Considering h=29cm, r=23.5cm, c=2cm, m=16cm and the possible values of articular variables: θ 1 =º-115º, θ 2 =º-18º, θ 3 =º-18º, θ 4 =º- 15º, θ 5 =º-9º, we represented a few variants of working space modelling, from simple motions to more complex ones (figures 2-1). ISBN: ISSN:

5 7th WSEAS Int. Conf. on APPLIED COMPUTER & APPLIED COMPUTATIONAL SCIENCE (ACACOS '8), Hangzhou, China, April 6-8, 28 Variant 1: θ 1 =º-115º, θ 2 =º, θ 3 =º, θ 4 =º, θ 5 =º Variant 4: θ 1 =º-115º, θ 2 =º-18º, θ 3 =º, θ 4 =º, θ 5 =º Fig. 2 Motion trajectory representation for variant 1 Variant 2: θ 1 =º,θ 2 =º-18º,θ 3 =º,θ 4 =º,θ 5 =º Fig. 5 Working surface representation for variant 4 Variant 5: θ 1 =º, θ 2 =º-18º, θ 3 =º-18º, θ 4 =º, θ 5 =º Fig. 3 Motion trajectory representation for variant 2 Variant 3: θ 1 =º, θ 2 =º, θ 3 =º-18º, θ 4 =º, θ 5 =º Fig. 6 Working surface representation for variant 5 Variant 6: θ 1 =º-115º, θ 2 =º-18º, θ 3 =º-18º, θ 4 =º, θ 5 =º Fig. 4 Motion trajectory representation for variant 3 Fig. 7 Working surface representation for variant 6 ISBN: ISSN:

6 7th WSEAS Int. Conf. on APPLIED COMPUTER & APPLIED COMPUTATIONAL SCIENCE (ACACOS '8), Hangzhou, China, April 6-8, 28 Variant 7: θ 1 =º, θ 2 =º, θ 3 =º, θ 4 =º-15º, θ 5 =º. 5 Conclusion The kinematical equations obtained are the parametrical equations which express the position and direction of the hand with respect to the fixed reference frame. Their representations in the 3D space leads to the modelling of the surface inside which the final effector of the human upper limb moves while doing its working task. The use of programs for the parametric representation of hyper-surfaces described by the distal end of the human upper limb during its motion is a method of checking the accuracy of the resulted model. Fig. 8 Motion trajectory representation for variant 7 Variant 8: θ 1 =º, θ 2 =º, θ 3 =º, θ 4 =º, θ 5 =º-9º. Fig. 9 Motion trajectory representation for variant 8 Variant 9: θ 1 =º, θ 2 =º-18º, θ 3 =º, θ 4 =º-15º, θ 5 =º-9º. References: [1] Baciu C., The Locomotor Apparatus, Editura medicala, Bucuresti, 198 [2] Denischi A., Marin Gh., Antonescu D., Biomechanics, Editura Academiei Bucuresti, 1989 [3] Dragulescu D., Dynamics of Robots, Editura Didactica si Pedagogica Bucuresti, 1997 [4] Dragulescu D., Toth-Tascau M., Couturier C., Human upper and lower limbs modeling using Denavit-Hartenberg s convention, Proceedings Situation and Perspective of Research and Development in Chemical and Mechanical Industry, Krusevac, 21 [5] Naaji A., Kinematical Modeling of the Human Upper Limb, Scientific Bulletin of the Politehnica University of Timişoara, Romania Transactions of Mechanics, Vol. Proceedings of the XI-th Conference on Mechanical Vibrations, 25, pp [6] Naaji A., Modeling Techniques. System for Diagnosis of the Human Upper Limb, Editura Orizonturi Universitare, Timisoara, 25 [7] Panjabi M., White A. III, Biomechanics in the muskuloskeletal system, Churchill Livingstone, New York, 21 [8] Papilian V., Human Anatomy, Vol. I The Locomotor Apparatus, Editura ALL, Bucuresti, 1998 [9] *** MatLab 7.1 User Guide, The MathWorks, Inc., 24 Fig. 1 Working surface representation for variant 9 ISBN: ISSN: