Applied Bionics and Biomechanics 10 (2013) 113–124DOI 10.3233/ABB-130075IOS Press
113
What strategy central nervous system uses toperform a movement balanced?Biomechatronical simulation of human lifting
Ali Leylavi Shoushtari∗Department of Computer Engineering, Shoushtar Branch, Islamic Azad University, Shoushtar, Iran
Abstract. How does the central nervous system control the body posture during various tasks despite a redundancy? It’s awell-known question existed in such fields of study as biomechanics and bioengineering. Some techniques based on muscle andtorques synergies are presented to study the function which Central Nervous System uses to addresses the kinetic redundancyin musculoskeletal system. The human body with its whole numerous joints considered as a hyper redundant structure whichcaused to be seemed that it is impossible for CNS to control and signal such system. To solve the kinematic redundancy inprevious studies it is hypothesize that CNS functions as an optimizer, such of that are the task-based algorithms which search tofind optimal solution for each specific task. In this research a new objective function based on ankle torques during movementis implemented to guarantee the stability of motion. A 2D 5DOF biomechatronical model of human body is subjected to liftingtask simulation. The simulation process implements inverse dynamics as major constraint to consider the dynamics of motionfor predicted postures. In the previous optimization-based techniques which are used to simulate the human movements, themotion stability was guaranteed by a nonlinear inequality constraint which restricts the total moment arm of the links to anupper and lower boundary. In this method, there is no need to use this constraint. The results show that the simulated posturesare normal and the predicted motion is performed completely balanced.
Multibody dynamic model of human body is usedin an extensive area of research contains robotics,biomechatronics and biomedical engineering since itcan provide an approach to find some variables thatare not possible to be measured like: torques andinternal forces of joints, stress exerted to joint’s softtissues. These mechanical parameters are so importantto understand joint disease initiation and progression,like osteoarthritis [4, 17]. In addition to pathological
∗Corresponding author: Ali Leylavi Shoushtari, Department ofComputer Engineering, Shoushtar Branch, Islamic Azad University,Shoushtar, Iran. E-mail: st a [email protected].
aspects, the graphical simulation purposes are one ofthe major applications of human body modeling.
In order to know how the body postures vary duringdifferent movements, a dynamic model of the wholehuman body is applied in the movement simulationprocess. The motion simulation algorithms are usedto analyze human movements; these are commonlyimplemented for athletics in order to improve the per-formance of the motion and to prevent injuries dueto the incorrect movements [11]. Some abnormalitiesoccurred in parts of the musculoskeletal system whichare resulted in inaccurate function of the muscle acti-vation and control [16, 18, 29]. Therefore, it is veryimportant to know how the central nervous system
114 A. Leylavi Shoushtari / What strategy central nervous system uses to perform a movement balanced?
(CNS) controls the variation of body posture duringdifferent tasks.
There is a need to large number of degrees offreedom to model human body more exactly and accu-rately. From a designer point of view, the redundancyis a profit because it makes such systems maneuver-able, while from a analyzer view point it is a problem.A system which is designed with numerous degrees offreedom is able to do complicated movements skill-fully and this is the benefit of redundancy and theproblem is difficultness to analysis the motion of thissystem. On the other hand, the multiplicity of thejoint space variables (DOFs) causes the model maneu-verable and creates the redundancy problem too. Weface the redundancy problem when the number ofDOFs is more than needed to perform a task. Thisproblem is also existed in robotics which is classi-fied based on kinematics, dynamics and control types[8, 20–22]. The model of human body usually imple-mented in computational algorithm by considering alarge number of DOFs for motion simulation purposes.Optimization-based approaches are good solutions toovercome with the redundancy problem [23]. Someof these techniques are applied to robotic manipula-tor models with redundant DOFs [6, 9, 22, 26]. Thesemethods is used for predict the posture of humanoidrobots during lifting task based on the algorithmwhich CNS uses to controls the human postures [15].Optimization-based solutions are completely suitableto solve the problem with a large number of variables,because such methods use a small amount of data asthe inputs to result a large number of variables as theoutput set. The inputs contain two sets of constraintswhich are considered in motion simulation process: (1)the constraints which are calculated by using dynamicequations of motion, and (2) sets of algebraic equationswhich define the kinematics of the system.
CNS controls the configuration of body segments toperform movement with balance such as walking, run-ning, sitting and lifting. These daily living activitiesare good examples of the tasks which are performedcompletely balanced involuntarily. Actually CNS usesan unknown algorithm to control such tasks uncon-sciously. Optimization-based simulation methods havean analogous performance with CNS function causingbalanced movements [19]. These approaches use anobjective function description subjected to be mini-mized; it is hypothesis as the function of CNS. On theother hand, to simulate a motion as similar as whathuman does, it is assumed that optimization approach
chooses the set of answers which minimizes the objec-tive function which also is considered by CNS to beminimized too.
However there are different definitions of stabilityin medicine and engineering, but they have the samemeaning. In engineering, the stability has a mathe-matical definition commonly described by Lyapunovlaws. According to such definition, a stable systemhas a minus energy performance which is called “Lya-punov Candidate Function” but the criteria of stabilityin medicine defines as pain threshold. A stable sys-tem in engineering is the one that has the convergenceand bounded outputs. Based on this statement, theinstable system has divergence outputs. In medicine,crossing the pain threshold results undesirable motionsthat CNS can’t controls musculoskeletal system toperform the tasks appropriately. In this optimizationapproaches, the stability of the motion is guaranteed byusing constraints which restrict the total moment arms(TMA) of segments of human body between horizontalposition of heel and toe [10, 24]. In fact, this constraintprevents of figurate the postures which will causethe falling forward and backward. In this researchthe optimization-based algorithm named predictivedynamics [27] with the objective function consistedof ankle torque squares summation during lifting timeare used. In such approach, the inverse dynamics, lim-itation of joints torque and joints ranges of motion areused as constraints to construct the motion similar toreal lifting movement. By and large, in this algorithmtwo kinds of constraints are used: 1- The constraintswhich shape the simulated motion similar to liftingmovement consisted of two classes of constraint: a)Kinematical governing constraints such ones whichdetermine initial and final position of box, body col-lision avoidance and constraints which guarantee themoving up motion of the box. b) Dynamic governingconstrains which actually is a set of differential equa-tions of motion in inverse dynamic form implementedas equality constraint to govern the dynamics of themotion in the simulation process. 2- The second typenamed “bounder constraint” determine of the varia-tion of the designed variables. This classification isillustrated in Fig. 1.
Ankle torque amplitude is considered as the indexof boundary stability and the optimization algorithmtries to minimize the integral of ankle torque squaresduring the lifting time. A five DOF biomechatroni-cal model of the whole human body represented inpart 2 is designed in Denavit-Hartenberg framework.
A. Leylavi Shoushtari / What strategy central nervous system uses to perform a movement balanced? 115
Fig. 1. Constraints classification.
The dynamics of the motion formulated based on theLagrangian method which results the equation of themotion in inverse dynamics form. In part 3 the simu-lation process is described and parts 4 and 5 presentresults of predicted motion and conclusion remarksrespectively.
2. Modeling
A planar model with 5DOF in sagittal plane is uti-lized to represent kinematics and dynamic of humanbody. All the limbs as shank, thigh, spine, arm andforearm are subjected to the modeling ore consid-ered as rigid bars with mass points at the center ofthe mass of each link which are named l1, l2, l3, l4,l5 respectively. For human major joints: ankle, knee,hip, shoulder and elbow are considered joint angles tofigurate human body posture and represented by thenames as q1, q2, q3, q4, q5 respectively. The box isassumed to be jointed at the wrist with constant hor-izontal orientation. Forward kinematics of this openkinematic chain which represents Cartesian position ofthe end-effector (wrist) according to the joints anglesis calculated based on D-H method [12].
2.1. Forward kinematics
The kinematical governing constraints consist ofsets of algebraic equations are obtained from forwardkinematics of mentioned model. To calculate the for-ward kinematics of an open kinematic chain, firstallocate the coordinates should be allocated at each
Fig. 2. 5DOF model of human body with coordination systemsattached to each link. q1, q2, q3, q4, q5 shows the angular positionof ankle, knee, hip, shoulder, and elbow respectively.
link, then the relational rotation of each adjacent coor-dinate system is calculated, and finally the forwardkinematics will obtains by post multiply the trans-lational matrixes. Biomechatronical model of humanbody with coordination systems attached to limbs illus-trated by Fig. 2.
The (x0, y0) is the global coordination system allo-cated to the ankle joint. The D-H parameters relatedto the coordination systems of the presented model areshown in Table 1.
116 A. Leylavi Shoushtari / What strategy central nervous system uses to perform a movement balanced?
Table 1D-H parameters of the human body model which is presented in
Fig. 2
i - 1i T I �i−1 ai−1 di θi01T 1 0◦ 0 0 q112T 2 0◦ 11 0 q223T 3 180◦ 12 0 q334T 4 180◦ 13 0 q445T 5 0◦ 14 0 q556T 6 0◦ 15 0 0
Fig. 3. Numbering adjacent joints and bodies to obtain D-H param-eters [25].
Figure 3 shows how the D-H parameters are namedand chosen for the two adjacent links. These parame-ters are: αi−1, ai−1, di, θi which are used to calculate thetranslational matrix i−1
i T related to each two adjacentcoordination systems i-1 and i.
D-H parameters help to calculate the translationalmatrixes related to the adjacent coordinates calculatedas (1):
i−1i T =
⎡⎢⎢⎢⎢⎣
Cθi −Sθi 0 ai
SθiCαi CθiCαi −Sαi −Sαidi
SθiSαi CθiSαi Cαi Cαidi
0 0 0 1
⎤⎥⎥⎥⎥⎦ (1)
In (1) Cθi and Sθi represent cos(θi) and sin(θi), andCαi, Sαi are the symbol of cos(αi) and sin(αi). Finally,by using the transformation matrix i−1
i T the forwardkinematics are calculated as 0
6T matrix which inter-
prets the motion of the wrist in global coordinationsystem connected to the ground as (2). The detailsof the transformational matrixes are presented inappendix.
06T = 0
1T · 12T · 2
3T · 34T · 4
5T · 56T (2)
2.2. Inverse dynamics
The whole human body commonly models as openkinematics chain as like as robot manipulators as men-tioned before [13, 14], so the method used to model thedynamics of the motion of this kinematical chain is likeones used for the robotic manipulators. In this approachthe Lagrangian of system is calculates, then by exter-mizing the integral of the lagrangian of the system, theequations which govern the dynamics of motion willbe obtained. The kinetic energy of the model whichwas presented before [28] is defines as (3).
K = 1
2qT D (q) q (3)
D (q) =5∑
i=1
(miJVci
T JVci + Jωi
T R0i IiR
0i ) (4)
In Equation (3) q is 5 × 1 vector of the angular veloc-ities of the joints, and qT is the transpose matrix of q,D(q) is 5 × 5 matrix related to the mass and inertialproperties of the model. All of the mass and inertialproperties, length of and COM the links related tothe subjected model are presented in Table 3 in theappendix. JVci , Jωi are 3 × 5 Jacobin matrix whichtranslates the linear and angular velocities of COMof i’th link to the universal coordinate system respec-tively. Ro
i is the rotational matrix transformation whichinterprets the orientation of the i’th links from its coor-dinate to the ground coordination. mi is the mass ofi’th link. By considering g as gravitational accelera-tion vector, and rci as the height of the COM of i’thlink respect to the ankle position, potential energy ofthe system is defines as bellow:
V =5∑
i=1
(migT rci ) (5)
The functional of the system’s energy which is calledLagrangian, is calculated as (6).
L = K − V = 1
2qT D(q)q −
5∑i=1
(migT rci ) (6)
A. Leylavi Shoushtari / What strategy central nervous system uses to perform a movement balanced? 117
Table 2Parameters and their values of lifting task
Lifting parameters Values
Box depth 0.370 (m)Box height 0.365 (m)Box weight 9 (kg)Initial height 0.365 (m)Final height 1.37 (m)Initial horizontal position 0.490 (m)Final horizontal position 0.460 (m)Lifting time duration 1.2 (sec)
Body segments parameters of biomechatronicalmodel presented in Fig. 1 are as the following:
Table 3Parameters of body segments, l6 is the symbol of link which demon-
strate the head. The l1 to l5 are illustrated in Fig. 1
Segments Length (m) Mass (Kgr) COM (m) Inertia (N.m2)
A functional of system’s energy named E is definedas the integral of system Lagrangian during the liftingtime interval [0 T ], as follow:
E =T∫
t=0
L(q, q, t)dt (7)
d
dt
(∂L
∂q
)− ∂L
∂q= � (8)
Euler-Lagrange formulation represented in (8)is ananalytical method to extremize the integral of E toobtain equations of the motion. In (8) � is a gener-alized joints torque vector inserted to this formulationas an external generalized force vector. Finally the gen-eral form of the motion equations will be obtained as(9).
D(q)q + C(q, q)q + V (q) = � (9)
In (9) C(q, q) is a term related to the centrifugaland coriolis forces and V (q) is the gravitational forcesvector. These terms are calculated as follow:
C(q, q) = D(q) − 1
2qT
(∂D(q)
∂q
)(10)
V (q) =(
∂V
∂q
)T
(11)
The generalized joint torque represented in (9) isdivided into two parts: 1. The torques resulted in mus-cle forces and 2. The torques due to the load of thebox which are exerted on the wrist. These kinds areobtained as (12):
� = τmuscle − τbox; τbox = JT (mbox gT ) (12)
In (12) JT is the transpose of the Jacobian matrixwhich projects the box load to the joints.mbox is the boxmass and gT is the transpose of the gravity accelerationvector.
3. Optimization-based simulation
In this paper the lifting movement simulation is con-sidered as the optimization procedure which CNS doesas well. In this problem an objective function is sub-jected to be minimized with some constraints whichlimit the range of allowable motions to a feasible rangeto construct the motion realistically. In other words, it isassumed that CNS tries to minimize a particular func-tion value to perform each task, and musculoskeletalsystem imposes some constraints on the motion, too.This assumption is implemented in some biomechani-cal researches to simulate the CNS action to predict themuscle forces and limb motions [1–3, 19]. The predic-tive dynamics is a novel approach used for the motionsimulation [27, 28]; it implements inverse dynamicequations as a major constraint to model the motion’sdynamics in the simulation process. The joints torquesand angles are the designed variables, so by using thismethod the joint angles and torques could obtain as out-puts, according to the task parameters which are usedas inputs. Profiles of joint torques and angles are sub-jected as optimization variables. Simulation process isplanned as a nonlinear constrained optimization prob-lem. The optimization algorithm tries to minimize theobjective function and satisfy the constraints, too. Theequation of motion which is obtained from the previoussection is written in difference form by using the finitedifferent method. The profiles of joints angles are con-sidered as 5 splines with the 5 control points. The firstand 5th control points are maintained at the initial andfinal time respectively to predict the optimal initial andfinal postures. So, there are 25 variables related to thejoints profiles and 50 variables related to the torques
118 A. Leylavi Shoushtari / What strategy central nervous system uses to perform a movement balanced?
Fig. 4. a) 5DOF model of human body for lifting motion task, b) inverted pendulum equivalent model for the lifting task.
profiles of the joints for 10 time sequences. Simulationelements are described in the following sections.
Min F (τ, q, t)τ, q
such that
⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩
C(τ, q) ≤ 0
Ceq(τ, q) = 0
τlb ≤ τ ≤ τub
qlb ≤ q ≤ qub
(13)
The presented structure in (13) shows the form ofoptimization process. In (13) C(τ, q) is inequality con-straint, Ceq(τ, q) is equality constraints and τlb and τub
are lower and upper boundaries of limitation of jointstorque respectively. qlb and qub are lower and upperboundaries of range of motion of joints. For optimiza-tion process a computer programming code is designedby using M-file environment of MATLAB software.
3.1. Objective function
Considering the lifting task as a simple inverted pen-dulum motion, can represent good insight to analysisof motion stability. On the other hand, if the liftingmotion be modeled as an inverted pendulum (Fig. 4) [5]It could be concluded that the magnitude of the pendu-lum joints torque has direct relation to the amount of thedeviation from the stability position (θ = 0◦). There-fore, a particular function which is constructed in termof the ankle torque could be used as the motion stabilityindex. This function is the integral of the ankle torquesquares in each time sequence (14). (14-A) shows acontinuous form of objective function and (14-B) illus-trates the digitalized form of this function. In (14-B)
the j is the number of time sequences which Fj iscalculated for.
F (q, τ, t) =T∫
t=0
τ2ankle dt (14–A)
Fj(q, τ)=j∑
i=1
(τankle(i))2 which j = 1, 2, 3 ∼ 10
(14–B)
3.2. Constraints
The constraints used in this research are: jointstorques and angles limitations, initial and final posi-tion of the box, elevating constraint, inverse dynamics,and body collision avoidance constraint. The verticalposition of the wrist (15) is a function of the joint anglesq(t) calculated by the forward kinematics (1):
Ywrist(t) = y(q(t)) (15)
In each time sequence Ywrist should be higher thanthe Ywrist for the previous sequence:
Ywrist(i) > Ywrist(i − 1) (16)
So the elevating constraint is defined by (17).
y (q(t − 1)) − y (q(t)) < 0 (17)
In fact the elevating constraint is considered to guar-anty the moving up motion of the box during the timesequences. The equality constraints which determine
A. Leylavi Shoushtari / What strategy central nervous system uses to perform a movement balanced? 119
the initial and final position of the box are definedby (18). ⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
x (q(0)) − xinitial = 0
y (q(0)) − yinitial = 0
x (q(T )) − xfinal = 0
y (q(T )) − yfinal = 0
(18)
Xwrist(t) andYwrist(t) are respectively horizontal andvertical positions of the wrist considered to be fixedin the box. According to the fourfold constraints set(18) should be placed at the initial position xinitial,yinitial and final position xfinal, yfinal of the box att = 0 and t = T respectively. These initial and finalpositions of the box actually are of task parameters.Inverse dynamics constraint is expressed as below:
τ − τinvd = 0; τinvd = f (q, t) (19)
In Equation (19) τ is the joints torque vector thatmust be predicted, and τinvd is the joints torque vec-tor obtained from the inverse dynamics. Body collisionavoidance which is implemented in this simulation is a
Fig. 5. Penetration of the box with the body, box line, body line andpenetration aria.
systematic method used to check the penetration valueof the box into the body in each iteration of the opti-mization process. This process is described as brief asbellow.
Fig. 6. The variation of human body postures during lifting task. The vertical axis shows the height in meter and the horizontal axis is timesequences.
120 A. Leylavi Shoushtari / What strategy central nervous system uses to perform a movement balanced?
The collision avoidance issue is considered in theoptimization process as a constraint to prevent thepenetration of the box with the body. It is inequalityconstraint that is defined as a term of the sufficient hor-izontal distance dx that the wrist should be displaced toprevent the collision box with the body. The distancethat the wrist should move to arrive at the horizon-tal boundary position Xbounndary is represented by dx,and the boundary position is a horizontal position ofthe place where the box edge touches the body.
Xbounndary − Xwristd < 0
Xbounndary = Xwristpr + dx (20)
Xwristd is the desired horizontal position of the wristwhich should be greater than Xbounndary. Xwristpr isthe horizontal position of the wrist obtained fromthe optimization algorithm in the current iteration.According to Fig. 5 the penetration value of the boxin the body d will calculate via the Equation (21).The values of Xbody and Xedge obtained from thebody line and the box line respectively. The maximumpenetration value determines the horizontal distancethat the wrist should move to arrive at the boundaryposition. “d” is the penetration index, so “dx” is defineas the maximum value of “d” (22).
d = Xbody − Xedge (21)
dx = MAX(d) (22)
4. Results
The optimization process is designed for 10 evenlydistributed time sequences. By considering the 5 jointsangles and torques, we have 75 variables subjected tobe optimized. Inertial properties of human segmentsare considered as the data used previously [7]. Theexperimental data which used in the [28], is imple-mented to validate the result of simulation process.The subjected population of experiment had an aver-age height 5′7′′ and the average weight of 143 lbs, themean age of the participants was 34 years. The liftingtask parameters are presented in Table 2.
The total moment arm (TMA) of the all links iscalculated as (23). It is calculated from the momentsrelated to the all links weight for each configurationof body segments in time t.mall is the total weight ofthe body and xi(t) is the horizontal position of center
Fig. 7. Joints torque profile.
A. Leylavi Shoushtari / What strategy central nervous system uses to perform a movement balanced? 121
of mass of i′th links at time t, and N is the number oflinks. Optimal profiles of joint angles show that howthe body posture varies during the lifting task, it isillustrated in Fig. 6.
TMA(t) = 1
mall
N∑i=1
mixi(t) (23)
The profiles of the joints torque profiles are shownin Fig. 7.
The predicted and experimental profile of humanjoints for lifting task is presented by Fig. 8. The solidlines are the predicted and the broken lines show theexperimental results.
TMA which is calculated based on (22) is plottedin Fig. 9 for all the 10 time sequences. The brokenlines show the upper and lower boundaries of the stableregion of the motion (base of support). In fact, theseare the horizontal position of the heel and toe in sagittalplane. In case of crossing from these boundaries, thehuman body will fall forward or backward.
5. Discussion
Simulation process has implemented a biomecha-tronical 5DOF model of the human body to simulate thelifting motion by using the unified optimization-basedapproach. The constraints applied to these processes,limit the range of the motions to a region that the actualhuman limbs could move through it. The major con-straint which is named inverse dynamics exerts thedynamics of the motion to the simulation process andfinally minimizing the objective function forms theshapes of the postures. The 10 time segments with 75variables are considered in optimization process. Fig-ure 6 shows that: (1) the posture variation is formedin a natural configuration with the extremely uprisingmotion of the box, (2) accurate initial and final positionof the box and (3) no collision of the box with the body.Figure 9 illustrates the TMA values during the time lift-ing and its boundaries. According to Fig. 9 the liftingmovement is performed completely balanced thanks toTMA has a value between upper and lower boundariesof stability. In other words, minimizing the summation
Fig. 8. The optimal profiles of the joints angles in comparison withexperimental results. The solid and broken lines are experimentaland predicted results respectively. The vertical axis are angles indegree and the horizontal axis are time sequences in 0.25 sec scale.
122 A. Leylavi Shoushtari / What strategy central nervous system uses to perform a movement balanced?
Fig. 9. TMA values during lifting time with its boundaries and relation with heel and toe. The solid lines are the calculated profile of TMA ofsegment of subjected model for simulated motion and broken lines are boundaries of base of support considered as margins of stability.
of squares of the ankle torque can guarantee the motionbalancing.
In Fig. 8 the simulation results are compared withthe experimental results due to the CNS. On the otherhand, Fig. 8 illustrate a comparison between optimiza-tion approach in contrast with CNS function to controlthe body posture. Since this model has less number ofDOFs than the actual human body, it is expected thatthe predicted motion profiles of the joints angles havethe amplitude greater than the amplitude of experimen-tal profiles. The predictive angles of the ankle, knee andhip joints have a good compatibility with the experi-mental results, and the results for the elbow are goodto some extent as well. But against these joints, theshoulder’s results haven’t a good correlation with theexperimental results. It is caused by the shoulder com-plex structure which needs to be modeled more exactlyand considered as suitable and sufficient constraints.The implemented model is consisting of 5 serial link-ages which the error of predicted angle in lower jointsbe added to the upper joints. Tanks to this reason thepredicted angles of shoulder and elbow have not goodcorrelation with the experimental results.
6. Conclusion
In sequences 6 to 10, the profile of the ankle torqueand TMA of the links have the same trends. It is con-cluded that in the standing posture, the ankle torquehas a direct relation with TMA of the links. So by min-imizing the ankle torque in standing postures the TMAalso will also be minimized. This strategy can be used
to guarantee the balancing of the motion in the standingpostures.
According to Fig. 6 it is concluded that: in thesequences 1 to 6, the box is lifted by the action ofthe joints of the lower limbs: ankle, knee and hip. Inthe rested sequences the lifting motion is continuedby the action of the shoulder and elbow joints. On theother hand, this simulation approach is able to simu-late the leg lift (squat) motion accurately. It can be usedto construct the skillful movement for athletics basedon the properties of their bodies which are presentedin Table 2 and the lifting task parameters presentedin Table 1. It can also be implemented to improve theperformance of the athletic movements and the injurypresentation using the algorithm to produce standardpattern for each given task.
Appendix
Transformer matrixes used to calculate forwardkinematics are as below:
01T =
⎡⎢⎢⎢⎢⎣
Cθ1 −Sθ1 0 0
Sθ1 Cθ1 0 0
0 0 1 0
0 0 0 1
⎤⎥⎥⎥⎥⎦
12T =
⎡⎢⎢⎢⎢⎣
Cθ2 −Sθ2 0 11
Sθ2 Cθ2 0 0
0 0 1 0
0 0 0 1
⎤⎥⎥⎥⎥⎦
A. Leylavi Shoushtari / What strategy central nervous system uses to perform a movement balanced? 123
23T =
⎡⎢⎢⎢⎢⎣
Cθ2 −Sθ2 0 12
−Sθ2 −Cθ2 0 0
0 0 −1 0
0 0 0 1
⎤⎥⎥⎥⎥⎦
34T =
⎡⎢⎢⎢⎢⎣
Cθ4 −Sθ4 0 13
−Sθ4 −Cθ4 0 0
0 0 −1 0
0 0 0 1
⎤⎥⎥⎥⎥⎦
45T =
⎡⎢⎢⎢⎢⎣
Cθ5 −Sθ5 0 14
Sθ5 Cθ5 0 0
0 0 1 0
0 0 0 1
⎤⎥⎥⎥⎥⎦
56T =
⎡⎢⎢⎢⎢⎣
1 0 0 15
0 1 0 0
0 0 1 0
0 0 0 1
⎤⎥⎥⎥⎥⎦
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