October 2018, Vol. 18, No. 5 MANUFACTURING TECHNOLOGY ISSN 1213–2489 704 indexed on: http://www.scopus.com Mathematical Model of the RRR Anthropomorphic Mechanism for 2D Biomechanical Analy- sis of a Deep Squat and Related Forms of Movement Václav Bittner 1,2 , Radim Štryncl 2 , Karel Jelen 2 , Martin Svoboda 3 1 Faculty of Science, Humanities and Education, Technical University of Liberec, Studentská 2, Liberec, Czech Republic, E-mail: vaclav.bittner@tul.cz 2 Faculty of Physical Education and Sport, Charles University in Prague, José Martího 31, Praha 6, Czech Republic, E-mail: jelen@ftvs.cuni.cz, radimstryncl@seznam.cz 3 Faculty of Mechanical Engineering, Jan Evangelista Purkyně University in Ústí nad Labem, Pasteurova 1, Ústí nad Labem, Czech Republic E-mail: martin.svoboda@ujep.cz The aim of this study was to create a mathematical model of the RRR anthropomorphic mechanism for a 2D biomechanical analysis of a deep squat and related forms of movement. The segment stick model is designed to diagnose the movement with sagittal plan symmetry. Based on the input data from kinematic and dynamometric analysis, and from the anthropometric data of the monitored person, it is possible to estimate the resulting mo- mentum of the forces acting on the main joints of the lower body. The technology may be applied in analysing deep squats, studying the dynamics of vertical reflection as well as in the biomechanical analysis of related forms of movement (e.g. standing-up, squatting with a dumbbell, skiing in downhill posture, etc.). The derived motion equa- tions may be used to analyse the dynamics of the movement of anthropomorphic or mechatronic systems with the same geometry. Keywords: Mathematical model, RRR mechatronic system, anthropomorphic mechanism, biomechanical analysis of movement Introduction One of the most topical biomechanical issues of today is the deep squat. It represents the basic movement model of primates and ranks among their natural postural posi- tions. It is a posture where the flexion in the knee joint enables the back of the thighs to touch the calves, the heels stay on the ground and the spine is upright in a neu- tral position. This posture may be seen in young children. Based on an innate movement model, they instinctively use a deep squat if they want to reach the ground with their hand. They also play in this posture. Practical experience shows that the majority of the Euro-Atlantic population in developed countries is losing the ability to reach the bottom position of a deep squat, or they are not using this movement pattern at all. However, studies proving the positive effect of the deep squat on the production of muscle power production and the perfor- mance of lower limbs may be found in world literature [1], [2], [3], [4]. Doubts regarding the overstraining of the knee joints are then disproved by Bryanton [2]. He found out that together with the engagement of the glutaei, it is mainly the load of the hip joints that increases with the depth of a squat, not the knee joints. Most studies on the deep squat are based on a kinesi- ologic analysis and the combination of a kinematic anal- ysis with EMG. These methods cannot be used for objec- tive conclusions on the momentum of the forces acting on particular joints during the respective stages of a deep squat. Therefore, the aim of this study is to create a model of an anthropomorphic mechanism that would enable such a biomechanical analysis of a deep squat and related forms of movements. The study focuses on the diagnosis of movement in the sagittal plane. Segment structure and parametrization of the model The model is created to diagnose the lateral movement projection of a person with the permanent support of both feet. It is based on a 3-segment 3D stick anthropomorphic mechanism when the feet, shins and shanks of both limbs are aligned, see Fig. 1. To derive motion equations, the weight of shank ms, thigh mt and half the weight of the upper body (head - h, upper limbs - a, trunk - r) is re- spected as a unit mb due to the symmetry. The weights and respective moments of inertia (Is, It, Ib) of particular segments may be estimated based on the method of Zatsiorsky et al. [5] from the body height (v [cm]) and total weight (m [kg]) of a person according to the equa- tions (1). [ ] { } ( ) [ ] [ ] ( ) { } 0 1 2 2 2 0 1 2 0 1 2 2 2 0 1 2 1 3 1 3 , ; , . ; , ; , ; , ; , . ; , , , b b m E E m E v kg I m k F F m F v kg m s t m E E m E v kg m m kg I m l F F m F v kg m ha r α α α α α α α α α α β β β β β β β β β β β β β β α β − − = + +   = + + + ∈   = + + =   = + + + ∈   ∑ ∑ ∑ ∑ (1)