DynamicsHumanArm_En_25_Chapter_Author

See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/272179862 Dynamics Modeling of 3D Human Arm Using Switched Linear Systems Conference Paper · March 2015 DOI: 10.1007/978-3-319-15705-4_25 CITATIONS READS 8 102 4 authors: Artur Babiarz Silesian University of Technology Michał Niezabitowski 92 PUBLICATIONS   504 CITATIONS    Silesian University of Technology 107 PUBLICATIONS   834 CITATIONS    SEE PROFILE SEE PROFILE Adam Czornik Silesian University of Technology Jerzy Klamka 124 PUBLICATIONS   937 CITATIONS    Institute of Theoretical and Applied Informaticsw Polish Academy of Science 177 PUBLICATIONS   2,988 CITATIONS    SEE PROFILE SEE PROFILE Some of the authors of this publication are also working on these related projects: Controllability of dynamical systems View project All content following this page was uploaded by Artur Babiarz on 15 April 2015. The user has requested enhancement of the downloaded file.

Metadata of the chapter that will be visualized in SpringerLink Book Title Intelligent Information and Database Systems Series Title Chapter Title Dynamics Modeling of 3D Human Arm Using Switched Linear Systems Copyright Year Copyright HolderName 2015 Corresponding Author Springer International Publishing Switzerland Author Family Name Babiarz Author Particle Artur Author Given Name Prefix Suffix Division Faculty of Automatic Control, Electronics and Computer Science, Institute of Automatic Control Organization Silesian University of Technology Address Akademicka 16 Street, 44-101, Gliwice, Poland Email [email protected] Family Name Czornik Particle Adam Given Name Prefix Suffix Division Faculty of Automatic Control, Electronics and Computer Science, Institute of Automatic Control Organization Silesian University of Technology Address Akademicka 16 Street, 44-101, Gliwice, Poland Email [email protected] Family Name Klamka Particle Jerzy Given Name Prefix Suffix Division Faculty of Automatic Control, Electronics and Computer Science, Institute of Automatic Control Organization Silesian University of Technology Address Akademicka 16 Street, 44-101, Gliwice, Poland Email [email protected] Family Name Niezabitowski Particle Michał Given Name Prefix Suffix

Division Faculty of Automatic Control, Electronics and Computer Science, Institute of Automatic Control Organization Address Silesian University of Technology Email Akademicka 16 Street, 44-101, Gliwice, Poland [email protected] Abstract A novel approach to modeling human arm with hybrid systems theory is presented. The 3D human arm Keywords (separated by '-') mathematical model is described. The arm model is built using three rotational link. Each of them is represented as a truncated cone prism. The shape of each link is changing during any motion. Based on the analysis of the arm motion a mathematical model of a switched linear system is proposed. The design process of state-dependent switching function and division of the state-space are shown. At the end, a few simulation results are presented. Dynamics of human arm - Switched system - Switching rule

Author Proof Dynamics Modeling of 3D Human Arm Using Switched Linear Systems Artur Babiarz(B), Adam Czornik, Jerzy Klamka, and Michal Niezabitowski Faculty of Automatic Control, Electronics and Computer Science, Institute of Automatic Control, Silesian University of Technology, Akademicka 16 Street, 44-101 Gliwice, Poland {artur.babiarz,adam.czornik,jerzy.klamka,michal.niezabitowski}@polsl.pl http://www.ia.polsl.pl Abstract. A novel approach to modeling human arm with hybrid sys- tems theory is presented. The 3D human arm mathematical model is described. The arm model is built using three rotational link. Each of them is represented as a truncated cone prism. The shape of each link is changing during any motion. Based on the analysis of the arm motion a mathematical model of a switched linear system is proposed. The design process of state-dependent switching function and division of the state- space are shown. At the end, a few simulation results are presented. Keywords: Dynamics of human arm · Switched system · Switching rule 1 Background and Significant Over the past decade, greater and more specific attention undoubtedly focused on the modeling of mechanical systems such as exoskeletons. The exoskeletons are used primarily to support the movement of people with disabilities or to increase the strength of the human body [1], [2]. The models mentioned above objects mainly use knowledge about the structure of the human musculoskeletal system (also known as the locomotor system) [3]. The starting point for most of the work is the model described by the Euler-Lagrange formalism. In the literature, we can find simple models exoskeletons [4], [5], as well as models with kinematic chain which is the equivalent of the human locomotor system. Obviously, these objects are very complicated and simplification of dynamic models are acceptable [6], [7], [8]. A model of the exoskeleton with five degrees of freedom that was attached to human arm using an arm orthosis is described in [9], [10]. In addition, the articles dealt with the problem of measuring the orientation of the human arm by an orientation sensor and a rotary encoder. In [11], it is studied the trajectory planning of an anthropomorphic human arm and a concept of movement primitive. Besides, the authors present human arm triangle space as an intermediate space between joint space and task space. The primary motivation for using switched linear systems comes partly from reaserch results published in [12], [13], [14], [15]. In this article, we focus on the c Springer International Publishing Switzerland 2015 N.T. Nguyen et al. (Eds.): ACIIDS 2015, Part II, LNAI 9012, pp. 258–267, 2015. DOI: 10.1007/978-3-319-15705-4 25

Dynamics Modeling of 3D Human Arm Using Switched Linear Systems 259 Author Proof switched linear systems that can be described as follows: x˙ = Aσ(·)x + Bσ(·)u, (1) y = Cσ(·)x + Dσ(·)u. where: x ∈ Rn is the state, u ∈ Rm is control signal, y ∈ Rq is output, σ(·) : P → {1, 2, . . . , N } is the switching rule, and Ai, Bi, Ci, Di, i = 1, 2, . . . , N are constant matrices. In the literature, there are no known cases of the above-mentioned systems for modeling the human arm. The initial work focused on a model of human upper limb, which the state space is divided into two regions only [16]. Subsequent research concentrated on modeling of lower or upper human limbs [17], [18]. In these studies, the authors show mathematical models of human arm or leg as switched linear systems. The state space of presented switched linear systems was divided for more regions than in [16]. 1.1 Motivation The research results presented in [14], [15] confirm that the shape of the human lower and upper limbs deformed during the execution of any movement. More- over, the dimensions and shape of the human limbs do not depend on a single muscle, but group of muscles (a phenomenon so-called the muscle synergism). Remark 1. Under the above conclusions, we can assume that the matrix of iner- tia and the distance from the center of gravity of each joint, are changed. In addition, changes of these parameters are dependent on the angular displace- ment of the arm. Besides, it is mentioned the muscles effect were omitted. They have influence on the shape of the each link only. Furthermore, research results published in [19] justify the application of hybrid systems for modeling objects with complex biomechanical structure. On the other hand, results presented in the work [12], [13] indicate that the human arm is unstable in the considered range of motion. Due to the biomechanical limita- tions of motion of each link, a operating space of human limbs can be naturally divided. As a result, we can obtain a set of subsystems. The conclusion from the above analysis is the basis for modeling of the human arm using a switched system. Such a system has a switching function depending on the state vec- tor. Consequently, we deduce the using theory of hybrid systems is a very novel approach to modeling and analysis of dynamic properties of human limbs. The structure of this paper is as follows. At the beginning, the mathematical model of three-dimensional human arm is presented. The next section focuses on the description of the human arm dynamics using switched linear systems. In this section, the partition of the state-space and switching function are shown. The simulation results are presented in Section 3. Finally, we conclude our approach to mathematical modeling of human arm dynamics.

260 A. Babiarz et al. Author Proof 2 Mathematical Model of Human Arm 2.1 State-Space Model The mathematical model of the arm can be obtained using the standard Euler- Lagrange description. The equation of motion is as follows: M (q)q¨ + C(q, q˙)q˙ + G(q) + Bq˙ = u (2) where: M (q) ∈ R3×3 - is a positive definite symmetric inertia matrix, C(q, q˙) ∈ R3×3 - is Coriolis and centrifugal forces matrix, G(q) ∈ R3 - is gravity forces vector, B ∈ R3×3 - is the joint friction matrix, u ∈ R3 - is forces and moments acting on the system, q ∈ R3 - is angular displacement. Fig. 1. Kinematics scheme of 3D human arm The human arm model is presented in Fig. 1. The dynamic equation of a three-dimensional human arm is described by nonlinear state equation (2) where: ⎡⎤ ⎡⎤ m11 m12 m13 c11 c12 c13 M (q) = ⎣ m21 m22 m23 ⎦ C(q, q˙) = ⎣ c21 c22 c23 ⎦ m31 m⎡32 m3⎤3 ⎡ c3⎤1 c32 c33 g11 b11 b12 b13 G(q) = ⎣ g21 ⎦ B = ⎣ b21 b22 b23 ⎦ g31 b31 b32 b33 m11 =m1d21 cos2(q2) + Iz1 + m2l12 cos2(q2) + m2l12 cos2(q2) sin(q3) + m2l12 sin(q2) cos(q2) cos(q3) + m2l12 cos2(q2) cos(q3) + m2l12 sin(q2) cos(q2) sin(q3) + m2d22 cos(q2 + q3),

Dynamics Modeling of 3D Human Arm Using Switched Linear Systems 261 Author Proof m12 = 0, m13 = 0, m21 = 0, m22 = 2m2l1d2 cos(q3) + m2d22 + m2l12 + m1d21 + Ix1, m23 = m2l1d2 cos(q3) + m2d22, m31 = 0, m32 = m2l1d2 cos(q3) + m2d22 m33 = m2d22 + Iz2 c11 = 0, c12 = − 2m1d21 cos(q2) sin(q2)q˙1 − 2m2l12 cos(q2) sin(q2)q˙1 − 2m2l12 sin(q2) cos(q2) sin(q3)q˙1 + m2l12 cos2(q2) cos(q3)q˙1 − m2l12 sin2(q2) cos(q3)q˙1 − 2m2l12 sin(q2) cos(q2) cos(q3)q˙1 + m2l12 cos2(q2) sin(q3)q˙1 − m2l12 sin2(q2) sin(q3)q˙1 − m2d22 sin(q2) cos(q2)q˙1 − m2d22 cos(q2) sin(q3)q˙1 c13 = m2l12 cos2(q2) cos(q3)q˙1 − m2l12 sin(q2) cos(q2) sin(q3)q˙1 −m2l12 sin(q2) cos(q2) cos(q3)q˙1 − m2d22 sin(q2 + q3)q˙1, c21 =0.5m2l1d2 cos(q2)(cos(q2 + q3) − sin(q2 + q3))q˙1 − 0.5m2l1d2(sin(q2 + q3) + cos(q2 + q3))q˙1 − m1d12 cos(q2) sin(q2)q˙1 − m2l12 cos(q2) sin(q2)q˙1 − m2d22 cos(q2 + q3) sin(q2 + q3), c22 = 0 c23 = −m2l1d2 sin(q3)q˙3 c31 = −(0.5m2l1d2c2(cos(q2 +q3)−sin(q2 +q3))+m2d22 cos(q2 +q3) sin(q2 +q3))q˙1 c32 = m2l1d2 sin(q3)q˙2 c33 = 0, g11 = m1gd1 sin(q1) cos(q2) − m2g sin(q1)(l1 cos(q2) + d2 cos(q2 + q3)) g21 = m1gd1 cos(q1) sin(q2) − m2g cos(q1)(l1 sin(q2) + d2 sin(q2 + q3)) g31 = m2gd2 cos(q1) sin(q2 + q3) b11 = b22 = b33 = 0.1, b12 = b21 = b31 = 0.2, b13 = b23 = b32 = 0.05. where: m - is the mass, li - is the link length, di - is the distance from the joint to the center of mass, Izi, Ixi - is the moment of inertia around the z and x axis, respectively. The dynamics of model in terms of the state vector qT , q˙T T can be expressed as: d dt q = q˙ , (3) q˙ M (q)−1[u − C(q, q˙)q˙ − G(q)]

262 A. Babiarz et al. Author Proof where it is assumed that the inertia matrix M is invertible. Positive definiteness of M is seen directly by the fact that the kinetic energy is always nonnegative and is equal to zero if and only if all the joint velocities are zero. Thus, M is invertible and equation (3) is valid. Now, a new set of variables can be assigned to each of the derivatives. The new set of state variables and their equivalences can be expressed as follows: x1 =q1, x2 = q2, x3 = q3, (4) x4 =x˙ 1 = q˙1, x5 = x˙ 2 = q˙2, x6 = x˙ 3 = q˙3. We can write the general output and state equations: x˙ = Ax + Bu, (5) y = Cx + Du, (6) where: x˙ = q˙1 q˙2 q˙3 q¨1 q¨2 q¨3 T , x = q1 q2 q3q˙1q˙2q˙3 T , (7) y = q¨1 q¨2 q¨3 T , u = u1 u2 u3 T . Matrices A, B, C and D are computed using the series expansion linearization method. 2.2 Switched Linear System Model of the human arm has limitations of movement. The limitations arise from the physics of motion in the joint. We assume that the ranges of each joint movement are constrained: −3.14 [rad] x1 0 [rad], −1.22 [rad] x2 3.14 [rad], 0 [rad] x3 2.96 [rad], According to section 1.1, we can design switched linear system which based on (3) and (7). It should be pointed out, that the switching function is state-dependent [17], [20]. Then, the mathematical model can be described by equations: x˙ (t) = Aσ(x)x(t) + Bσ(x)u(t), (8) y(t) = Cσ(x)x(t) + Dσ(x)u(t). (9) Generally speaking, we consider the switched linear systems with state-dependent switching. The choice of active dynamics is determined strictly by the current state x(t). So Rn is divided into a collection of disjoint regions Φ1, ..., Φi, . . . , ΦN with Φ1 · · · ΦN = Rn, and then ⎧ ⎪⎪⎨⎪⎪⎪⎪ A1x + B1 u if x ∈ Φ1 ... x˙ = ⎩⎪⎪⎪⎪⎪⎪ Aix + Biu if x ∈ Φi ... if x ∈ ΦN u AN x + BN

Dynamics Modeling of 3D Human Arm Using Switched Linear Systems 263 Author Proof Furthermore, it is assumed that only the angular displacement (the first three elements of the state vector: x1, x2 and x3) influences on the change of the shape parameters of the each link during the movement. Consequently, the angular velocity (the last three elements of the state vector: x4, x5 and x6) may be any for each subsystem of model described by equations (10) and (11). ⎧ A1x + B1u if −1.57 ≤ x1 < 0, x2 = 0, x3 ≥ 0 ⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪ A2x + B2u if −1.57 ≤ x1 < 0, x2 > 0, x3 > 0 x˙ = ⎪⎪⎪⎪⎪⎪⎩⎪⎪⎪⎪ A3x + B3u if −1.57 ≤ x1 < 0, x2 < 0, x3 = 0 (10) A4x + B4u if −1.57 ≤ x1 < 0, x2 < 0, x3 > 0 A5x + B5u if −3.14 ≤ x1 ≤ −1.57, x2 = 0, x3 ≥ 0 A6x + B6u if −3.14 ≤ x1 ≤ −1.57, x2 > 0, x3 > 0 A7x + B7u if −3.14 ≤ x1 ≤ −1.57, x2 < 0, x3 = 0 A8x + B8u if −3.14 ≤ x1 ≤ −1.57, x2 < 0, x3 > 0 ⎧ C1x + D1u if −1.57 ≤ x1 < 0, x2 = 0, x3 ≥ 0 ⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪ C2x + D2u if −1.57 ≤ x1 < 0, x2 > 0, x3 > 0 y = ⎪⎪⎩⎪⎪⎪⎪⎪⎪⎪⎪ C3x + D3u if −1.57 ≤ x1 < 0, x2 < 0, x3 = 0 (11) C4x + D4u if −1.57 ≤ x1 < 0, x2 < 0, x3 > 0 C5x + D5u if −3.14 ≤ x1 ≤ −1.57, x2 = 0, x3 ≥ 0 C6x + D6u if −3.14 ≤ x1 ≤ −1.57, x2 > 0, x3 > 0 C7x + D7u if −3.14 ≤ x1 ≤ −1.57, x2 < 0, x3 = 0 C8x + D8u if −3.14 ≤ x1 ≤ −1.57, x2 < 0, x3 > 0 The switching rule has been designed in accordance with mechanical movement limitations. 3 Experimental Results In order to obtain the switched linear model, the operating points have been arbitrarily selected from each subspaces Ωi, i = 1, 2, . . . , 8 of a state space. Fixed parameters, used for linearization of the arm, are shown in Table 1. Whereas, Table 2 contains the variable model parameters: the distance from the joint to the center of mass and the moment of inertia for each Ωi. The shape of each arm link is approximated a truncated cone, the dimensions of which depend on the configuration of the arm. Strictly speaking, the moment of inertia of each link depends on the value of the state vector elements x1, x2 and x3. Table 1. The parameters of human arm Link 1 Link 2 m [kg] 1.4 1.1 l [m] 0.3 0.33

264 A. Babiarz et al. Author Proof Table 2. The parameters of switched model d1 [m] d2 [m] Ix1 [kgm2] Iz1 [kgm2] Iz2 [kgm2] The case I 0.11 0.16 0.027 0.025 0.045 The case II 0.1 0.16 0.025 0.029 0.045 The case III 0.12 0.14 0.021 0.024 0.043 The case IV 0.1 0.15 0.024 0.023 0.046 The case V 0.14 0.16 0.024 0.022 0.043 The case VI 0.13 0.14 0.022 0.023 0.044 The case VII 0.13 0.15 0.024 0.023 0.044 The case VIII 0.15 0.14 0.024 0.023 0.042 The simulation results were obtained using Matlab Simulink package and S- Function [21]. The control signal is modeled as u1 = sin(5t), u2 = cos(2t), u3 = 0.1 sin(5t) and the initial condition is equal x0 = −0.52 [rad]; 0.52 [rad]; 0.17 [rad]; rad rad rad T s s s 0 [ ]; 0 [ ]; 0 [ ] . Figures 2-7 present time history of six elements of state vector. angle [rad] State variable − x1 0 −0.1 −0.2 −0.3 −0.4 −0.5 −0.6 −0.7 0 2 4 6 8 10 12 14 16 18 20 time [s] Fig. 2. A scope of x1 state variables angle [rad] State variable − x2 1.5 1 0.5 0 −0.5 −1 −1.5 0 2 4 6 8 10 12 14 16 18 20 time [s] Fig. 3. A scope of x2 state variables

Author Proof angle [rad]Dynamics Modeling of 3D Human Arm Using Switched Linear Systems 265 angular velocity [rad/s] State variable − x3 1.2 angular velocity [rad/s] 1 angular velocity [rad/s] 0.8 0.6 0.4 0.2 0 0 2 4 6 8 10 12 14 16 18 20 time [s] Fig. 4. A scope of x3 state variables State variable − x4 3 2 1 0 −1 −2 −3 0 2 4 6 8 10 12 14 16 18 20 time [s] Fig. 5. A scope of x4 state variables State variable −x5 5 4 3 2 1 0 −1 −2 −3 −4 −5 0 2 4 6 8 10 12 14 16 18 20 time [s] Fig. 6. A scope of x5 state variables State variable − x6 5 4 3 2 1 0 −1 −2 −3 −4 −5 0 2 4 6 8 10 12 14 16 18 20 time [s] Fig. 7. A scope of x6 state variables

Author Proof 266 A. Babiarz et al. 4 Conclusions In the article, we showed a way of obtaining a three degrees-of-freedom mathe- matical model of a human arm. Switched linear system model was used. Addi- tionally, it was shown that the switching function should be state-dependent. The consideration on the use other elements of the state vector for the construction of a switching rule will be the next step in our research. In consequence, the research tasks should include also the analysis of switching dependent of the elements of state vector. A fortiori, referring to the results of [19], [22] subsystems of the model of human arm might be unstable or on the stability boundary and may be unobservable. For this reason, we plan to perform an analysis of such properties as stability, control- lability and observability [23], [24]. The subsequent research will address the appli- cation of the fractional order switched systems theory and discrete-time switched systems to modeling of human arm [25], [26]. Acknowledgments. The research presented here were funded by the Silesian Univer- sity of Technology grant BK-265/RAu1/2014/2 (A.B.), the National Science Centre according to decision the National Science Centre granted according to decisions DEC- 2012/05/B/ST7/00065 (A.C.), DEC-2012/07/N/ST7/03236 (M.N.) and DEC-2012/07/ B/ST7/01404 (J.K.). References 1. Kong, K., Tomizuka, M.: Control of exoskeletons inspired by fictitious gain in human model. IEEE/ASME Transactions on Mechatronics 14, 689–698 (2009) 2. Pons, J.L., Moreno, J.C., Brunetti, F.J., Rocon, E.: Lower-Limb Wearable Exoskele- ton In Rehabilitation Robotics, pp. 471–498. I-Tech Education and Publishing (2007) 3. Sekine, M., Sugimori, K., Gonzalez, J., Yu, W.: Optimization-Based Design of a Small Pneumatic-Actuator-Driven Parallel Mechanism for a Shoulder Prosthetic Arm with Statics and Spatial Accessibility. Evaluation. International Journal of Advanced Robotic Systems 286 (2013) 4. Csercsik, D.: Analysis and control of a simple nonlinear limb model. PhD Thesis, University of Technology (2005) 5. Choudhury, T.T., Rahman, M.M., Khorshidtalab, A., Khan, M.R.: Modeling of Human Arm Movement: A Study on Daily Movement. In: Fifth International Con- ference on Computational Intelligence, Modelling and Simulation (CIMSim), pp. 63–68 (2013) 6. Zawiski, R., Blachuta, M.: Model development and optimal control of quadrotor aerial robot. In: 17th International Conference on Methods and Models in Automa- tion and Robotics (MMAR), pp. 475–480 (2012) 7. Blachuta, M., Czyba, R., Janusz, W., Szafran´ski, G.: Data Fusion Algorithm for the Altitude and Vertical Speed Estimation of the VTOL Platform. Journal of Intelli- gent and Robotic Systems 74, 413–420 (2014) 8. Garrido, A.J., Garrido, I., Amundarain, M., Alberdi, M.: Sliding-mode control of wave power generation plants. IEEE Transactions on Industry Applications 48, 2372–2381 (2012)

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