Positional Solution of a Novel Knee Rehabilitation Mechanism Jian-Feng LI 1,a,*, Xiang-Qiang HUANG 2,b, Chun-Jing TAO 3,c, Run JI 4,d

Transcription

1 016 International Conference on Mechanics Design, Manufacturing and Automation (MDM 016) ISBN: Positional Solution of a Novel Knee Rehabilitation Mechanism Jian-Feng LI 1,a,*, Xiang-Qiang HUANG,b, Chun-Jing AO,c, Run JI 4,d 1, College of Mechanical Engineering and Applied Electronics echnology, Beijing University of echnology, Beijing, 10014, China.,4 National Research Center for Rehabilitation echnical Aids, Beijing, , China a lijianfeng@bjut.edu.cn, b @qq.com, c taochj@gmail.com, d jirun@ gmail.com *Corresponding author Keywords: Knee rehabilitation exoskeleton, Screw axis analysis, Kinematical compatibility, Human-machine closed chain, Positional solution. Abstract. In this paper, the design of knee assistive exoskeleton mechanism was discussed, which could enable human knee axis to align the exoskeleton active joint axis ideally. Based on the data of the knee joint flexion experiment, the knee joint screw motion parameters could be extracted. According to the knee joint physiological structure and the instantaneous characteristics of screw axis, the knee joint kinematic model was established. hen, according to the DOFs analysis of human-machine closed chain and the human-machine compatibility, a novel knee exoskeleton mechanism configuration was proposed. And the positional solution of human-machine closed kinematic chain was solved. It can be found that the motion between knee assistive exoskeleton mechanism s joint and human knee joint had certainly quantificational relationship, and the knee assistive mechanism had a fine motion synchronicity with the human knee joint. In addition, while the method of screw axis and the exoskeleton design were proposed for human knee joint, they are also applied to other human limbs. Instruction Robotic exoskeletons are mechatronic devices worn by persons and work cooperatively in parallel with human bodies by the interaction loads. By retrievalling literatures about the design characteristics of knee rehabilitation exoskeleton, designing the exoskeleton was mainly based on the kinematics of human knee joint. According to the proposed knee kinematic model, they can be roughly classified into three categories: (1) knee kinematic model can be simplified to a revolute pair whose rotation axis is fixed and perpendicular to the sagittal plane [1]; () knee kinematic model is considered as a revolute pair whose rotation axis is not fixed []; () knee kinematic model is a laxity joint whose axis position and posture change instantaneously [-5]. In the existing knee exoskeleton configurations, knee kinematic model was usually simplified as a revolute pair which axis was fixed/or not fixed and was perpendicular to the sagittal plane [6-10]. For example, the exoskeleton SERKA [6] and Lokomat [7] all regarded the knee joint as a rotation pair with fixed rotation axis. Ergin. et al [8] proposed an knee rehabilitation exoskeleton with planar degrees of freedom, and could enable the exoskeleton joint axis to align the knee joint axis in the sagittal plane. Also, Amigo [9] and Celebi [10] presented respectively a virtual joint consisted of three active rotation pair and an active knee rehabilitation exoskeleton which could guarantee the alignment between the exoskeleton joint s axis and the knee joint axis. he advantage of

2 existing mechanism design is that obtaining the simple mechanism configurations. But these rehabilitation institutions only can guarantee the exoskeleton active joint s axis aligns with the human knee joint s axis in the sagittal plane without considering other planes. Few researchers discuss the ways about satisfying the axis alignment between human and machine from the point of the design of exoskeleton mechanism. Combining the instantaneous characteristics of the knee joint axis, the establishment of the human knee joint kinematics model and the DOF analysis of the human-machine closed chain, the problem about designing the rehabilitation exoskeleton with good human-machine compatibility was studied in this article. A knee exoskeleton mechanism was put forward for the rotation assistance of the human knee complex. When it was connected to the human knee joint, a one-dof human-machine closed chain was formed, and the kinematics of the human-machine closed chain was investigated. Proposing the Rehabilitation Mechanism Collecting the Experimental Data he experimental data were acquired by the NDI OPORAK optical motion capture system. Landmark s location is shown in the Fig.1. he NDI capture system continuously tracked the D location of the markers which were expressed in global coordinate system, which concluded marker s XYZ coordinates of fifteen flexion angles. Figure 1. he location of the landmarkers. Calculation Method of Screw Parameters Once the coordinates of these landmarkers of the moving object were collected, the screw parameters which described the screw motion of rigid body could be obtained. By applying point cluster technology to the landmarkers coordinates, the homogeneous transformation matrices were extracted [11]. hen, screw parameters of the knee joint instantaneous motion were extracted from the homogeneous transformation matrix [1]. Establishing Kinematic Model and Mechanism Configuration he location and orientation change of these screw axes describes the kinematic characteristic of the knee joint s rotation axis, shown in the Fig.. Figure. Screw axes.

3 he figure shows that the location and orientation of knee joint screw axis change with the change of joint angle. According to the physiological structure of knee joint and the transient properties of screw axis, knee kinematics model can be equivalent to a generalized 1 DOF revolute pair. he human knee joint kinematic model can be considered as a generalized 1 DOF revolute pair. Basis on the bionic design of knee joint movement, the active joint of the knee exoskeleton is set as a 1 DOF revolute joint. Considering structural simplicity, interaction force between human and machine and the choice of the DOF to be freed, a human-machine movement compatibility knee rehabilitation exoskeleton mechanism configuration was proposed, as shown in Fig.. Figure. Exoskeleton mechanism configuration R1CS. Positional Solution of the Human-Machine Closed Chain he human-machine closed chain is illustrated in Fig.4. In the serial kinematic chain R1CS of knee assistive mechanism. he human knee joint can be equivalent to a generalized 1 DOF rotation model which axis changes instantaneously. Figure 4. Human-machine closed chain. As showed in the Fig. 4, the axes of human knee joint move in the double quasi-conic frustum with an elliptical cross section. herefore, the center point O of the ellipse section is supposed as the rotation center of all the screw axes. o carry out the kinematic analysis, the frames O X YZ 0 0 0, P x y z p p p, C x y z 1 1 1, D x y z, F x y z, O x1 y z 1 1 and F x y z are defined and showed in Fig.5. In the initial state, the posture of the knee joint rotation axis is same with the posture of the knee assistive mechanism active rotation axis. However, because of the position invisibility of human knee joint axis, the wearing position errors between the

4 assistive mechanism and human knee complex are defined. a b c O= P denotes the wearing position error vector of point P described in frame O X YZ Position Solution Method of Human-Machine Closed Chain Supposing the human-machine closed chain is disparted at the S joint, and two open serial sub-chains OPABCDEF and OGF are obtained. (1) he homogeneous transformation matrix 0 of sub-chain OPABCDEF a b rans( Z, c) rans( Y, b) rans( X, a)= c O P O O O (1) Where, a b c O= denotes the position vector of point P described in frame O X YZ P l1 sin cos rans( z, l l ) rans( x, l ) Rot(y,90 ) Rot( z, )= cos sin 0 l l P 0 1 P 0 P 1 P 1 1 () Where, 1 denotes the counterclockwise rotation angles about the z 1 axis; l 0, l 1 andl denote the length of PA, AB and BC, respectively l d cos sin 0 0 rans( x, l ) Rot (y, 90 ) Rot ( z,90 ) Rot( z, ) rans( z, d ) = sin cos () Where, denotes the counter-clockwise rotation angles about the z axis; d denotes the displacement of joint C; l denotes the length of CD; l 5 rans( z, l ) rans(y, l ) Rot( z, 90 )= l (4) Where, l4 and l5 denote the length of DE and EF respectively. herefore, the transformation matrix 0 which transfers the coordinates frame F x y z frame O X YZ 0 0 0can be written as: to

5 n o a X x x x n o a Y n o a Z z z z P 1 y y y = P 1 (5) Where, n n n, x y z o o o and x y z a a a denote the orientation vectors of the x x y z, y and z axes respectively, vector F ( X Y Z ) denotes the position vector of the center point F of the S joint, and all of them are described in frame O X YZ () he homogeneous transformation matrix 0 ' of sub-chain OGF M M M M M M 0 1 Rot( Z, ) Rot ( Y, ) Rot( X, ) Rot( Y,90) Rot ( z, )= M M M ' O O O 0 1' k (6) Where,, and denote the counterclockwise rotation angles about the X 0, Y0 and Z 0 axes; k denotes the counterclockwise rotation angles about the axis of the knee joint; M ij are the polynomial about,, and k ; M cos cos 11, the rest of elements are omitted here l rans( x, l ) rans( z, l ) Rot( y, 90 )= l ' 0 ' 1' 7 1' 6 1' (7) Where, l6 andl 7 denote the length of FG and OG respectively. herefore, the transformation matrix 0 which transfers the coordinates frame F x y z ' ' ' ' frame O X YZ 0 0 0can be written as: to n o a X ' x ' x ' x ' n o a Y n o a Z 'z 'z 'z ' ' 'y 'y 'y ' = ' 1' ' (8) Where, n n n, ' x ' y ' z o o o and ' x ' y ' z a a a denote the orientation vectors of the ' x ' y ' z x ', y' and z ' axes respectively, vector F ( X Y Z ) denotes the position vector of the ' ' ' center point F of the S joint, and all of them are described in frame O X YZ According to the concept of kinematics consistency, restriction equations can be acquired:

6 X X ', Y Y ', Z Z ' (9) From Eqs (5) and (8), the variables X, Y, Z, X, Y and Z ' ' ' in Eq. (9) can be obtained as: X = Acos A sin cos d cos A Y = B cos B sin B cos cos d sin B Z C cos C sin C sin sin d cos C (10) X lm lm Y l M l M Z lm lm ' ' ' (11) In Eqs (10) and (11), A l 1 1, the rest of coefficients A i, B i, C i are omitted here. Substitute Eqs (10) and (11) into Eq. (9) and eliminate the variable d, we can obtain: N cos N sin N cos cos N sin cos N cos N sin N cos cos N sin N sin cos (1) In Eq. (1), N M l M l b ; the rest of elements N ij are omitted. Let x tan( ) 1 and y tan( ), we can obtain: sin 1 cos 1 1 sin 1 cos x x x x y y y y (1) Substitute Eq. (1) into Eq. (1), we get: N N y N N x N N y N N x N N 11 1 y N N N N 5 y N N 5 x N N y N y N N N N y N y N N x (14) Using Eqs (14) eliminate the interim variable x, we get: Fy Fy Fy Fy Fy Fy+ Fy Fy F 0 (15)

7 Where, F 4 1 N N N N N N N N ; he rest of coefficients Fi are omitted. When the values of variable y are obtained, the value of variables x, 1 and can be solved through Eq. (14) and Eq. (1) respectively; d can be obtained through Eq. (11) and Eq. (10); the angular displacements of the revolute joints, 4 and 5 of the S joint should be solved, and they can be obtained as: arctan( P P ) arcsin( P ) 4 1 arctan( P P ) (16) Where, P 11, P 1, P 1, P and P are the components of the following matrix S. [ ] P P P P P P P P s ' P P P P 1 4 P P P P (17) In Eq. (17), P 11 M 1 cos M cos 1 sin M sin 1 sin, the rest of elements P ij are omitted here. Position Analysis of the Knee Assistive Mechanism In this section, a position solution example of the knee assistive mechanism was proposed, in which the angular displacement of human knee joint are adopted as the inputs of human-machine closed chain. he angle displacements of the knee joint are shown in able 1. able 1. he displacement trajectories of the knee joint [ o ] k In this example, the dimensional parameters of the knee assistive mechanism were proposed in able. In addition, the position offsets between the frame P x y z p p pand the frame O X YZ were supposed as a b c mm. he orientation offsets (,, ) between the frame P x y z p p pand the frame O X YZ are shown in able. able. Parameters of human-machine closed chain [mm]. l l l l l l l l able. he orientation offsets [o]

8 Furthermore, the joint displacements of the knee mechanism were calculated and the corresponding joint displacements were instructed in Fig.5 a)-f). a) Angle displacement of revolute joint R1 b) Angle displacement of cylindrical pair C c) Displacement trajectory of cylindrical pair C d) Angle displacement of the first joint of spherical joint S e) Angle displacement of the second joint of spherical joint S f) Angle displacement of the third joint of spherical joint S Figure 5. Joint displacement trajectories of the knee mechanism. Summary (1) he screw motion parameters of knee joint were extracted on the experiment data; According to the instantaneous characteristics of the knee screw axis, the kinematic model of knee joint was established; In terms of the DOFs of human-machine closed chain and the human-machine compatibility, a knee rehabilitation exoskeleton mechanism was optimized. () On the basis of the concept of the kinematic consistency, the kinematic equations of the human-machine closed chain were established and predigested as a 8 th polynomial equation of one variable. And the position solution of human-machine closed chain can be obtained. Acknowledgement his research was supported by Natural Science Foundation of China (No. 6174).

9 References [1] M. Dollar Aaron, Herr Hugh, Lower Extremity exoskeletons and active orthoses: challenges and state-of-the-art, J. IEEE ransactions on Robotics, 008, 4(1): [] J.J. O Connor,.L. Sherclif, E. Biden, et al. he geometry of the knee in the sagittal plane, J. Part H: Journal of Engineering in Medicine, 1989, Vol. 0: -. [] A.J. Va den Bogert, C. Reinschmidt, A. Lundberg, Helical axes of skeletal knee joint motion during running, J. Journal of Biomechanics, 008, Vol.41: [4] Frances. Sheehan. he finite helical axis of the knee joint (a non-invasive in vivo study using fast-pc MRI), J. Journal of Riomechanics, 007, Vol. 40: [5] Wolf Alon. Instantaneous screws of weight-bearing knee: What can the screws tell us about the knee Motion, J. Journal of Biomechanical Engineering, 014, Vol. 16: [6] F. Veneman Jan, Kruidhof Rik, E.G. Hekman Edsko, et al. Design and evaluation of the LOPES exoskeleton robot for interactive gait rehabilitation, J. IEEE ransactions on Neural Systems and Rehabilitation engineering, 007, Vol. 15, No. : [7] Riener Robert, Lünenburger, Colombo Gery. Human-centered robotics applied to gait training and assessment, J. Journal of Rehabilitation Research and Development, 006, Vol. 4, No. 5: [8] Ergin Mehmet Alper, Patoglu Volkan. A self-adjusting knee exoskeleton for robotassisted treatment of knee injuries, J. 011 IEEE/RSJ international conference on intelligent robots and systems, Sep 5-0, 011: [9] L.E. Amigo, A. Casals, J. Amat, Design of a -DOF joint system with Dynamic servo-adaptation in orthotic applications, J. Shanghai international conference center, May 7-1, 011: [10] Celebi Besir, Yalcin Mustafa, Patoglu Volkan, Assiston-knee: a self-aligning knee exoskeleton, J. 01 IEEE/RSJ international conference on intelligent robots and systems (IROS), Nov -7, 01: [11].P. Andriacchi, E.J. Alexander, M.K. oney, et al. A point cluster method for in vivo motion analysis: Applied to a study of knee kinematics, J. Journal of Biomechanical Engineering, 1998, Vol. 10: [1] L.K. Davidson, K.H, Hunt. Robots and screw theory, applications of kinematics and statics to robotics. Oxford, Oxford University Press, 004: