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Abstract

This paper describes a new locomotion mode to use in a crawling robot, inspired of real inchworm. The crawling device is modelled as a mobile manipulator, and for each step of its motion, the associated dynamics relations are derived using Euler-Lagrange equations. Next, the Genetic Algorithm (GA) is utilized to optimize the trajectory of the free joints (active actuators) in order to minimize the consumed effort (e.g. integral of square of torques over the step time). In this way, the results show a reduction of 5 to 37 percent in torque consumption in comparison with the gradient based method. Finally, numerical simulation for each step motion is presented to validate the proposed algorithm.

Keywords

Crawling Locomotion Lagrangian Dynamics Optimal Trajectory Genetic Algorithm

1. Introduction

Nature, for engineers and roboticists, has been a rich source of metaphors providing uncountable examples of locomotion system. The crawling motions, inspired of the snakes, worms and other limbless animals, are good instants that develop remarkable functional locomotion systems (Ávila, 2006).

The main motive of developing the crawling devices, such as it is described in this work; significantly relate to enhance remote sensing capabilities emergency situations, such as rescue and inspection in areas where motion is constrained, i.e. in situations involving disaster such as earthquakes, fires, or even hostile environments. Hirose investigated the locomotion of limbless animals for the first time. He was inspired by their motions to design and control a mobile robot within articulated body (Hirose & Morishima, 1990). After him, many researchers started to investigate these types of motions (Saito & et al., 2002, Crespi & Ijspeert, 2006).

Merino et al. presented description of a crawling gait for a modular robot and discussed about pros and cons of crawling locomotion rather than quadruped system. In their described locomotion pattern, a kinematics chain of modules was considered in which motion course begins with an open loop sub-mechanism and progression is achieved by the propagation of an undulatory wave from the rear to the front of the robot. In fact, crawling motion was formed based on lifting in the vertical direction to advance segments of body forward, using friction (Merino & Tosunoglu, 2004).

In continuation, the dynamic model of it was developed based on Newton-Euler laws that were involved with joint forces and extra calculation (Spranklin, 2006). Moreover, in this same work a gradient based optimization method was used to obtain the optimal motion with minimum effort metric. However, these type optimization methods have two significant drawbacks as follows:

These methods can only guarantee local optimally and convergence.

They need gradient and Hessian information, which for problems involving even moderate systems, approximation of this information takes a complicated calculation and may lead to ill condition, instability, and poor convergence behaviour.

Instead, the Genetic Algorithm (GA) is an efficient heuristic method for optimization problems, and does not require the derivative information of objective function, and has robust characteristic for avoidance of being trapped in local optima.

Recently, GA has been successfully applied to various motion synthesis problems. For example, genetic algorithm is applied to solve the position and movement of an end-effecter on the tip of a two-joint robot arm (Yano & Tooda, 1999). In other work, GA determines the parameters, which are the interior points to be interpolated to formulate the polynomial representing the trajectory, to minimize the fitness of the desired objective function (Lianfang & Curtis, 2004) Also, optimal trajectory selection for a three DOF manipulator with minimum energy consumption using GA is done by (Ata, 2007). In this work the angle of last link made with the x-axis is considered as cost function and needs to be optimized.

This work includes three main segments; first description of crawling gait, second, development of its dynamic model and third finding the optimal motion with minimum effort metric utilizing the GA. The remainder of this paper is organized as follows: Section 2 describes the crawling gait and some assumptions to develop the dynamic model In section 3 the Lagrangian formulation of a planner manipulator with revolute joints is derived including a closed loop chain and reaction forces due to connection between the tip of last link and the ground By introducing B-spline curves in section 4 we discuss. about trajectory planning and optimization In section 5 we present a brief review of a simple GA and its advantages The best results of simulation which obtained by GA brought in section 6 and finally in section 7 the contribution of the paper is summarized.

2. Gait Description

In this section basic motion pattern is described and assumptions of model in order to develop dynamics formulation, for each step motion, are explained.

In figure 1, a sequence of the joint configurations is shown that it should result in a forward motion if the robot is driven through them (Ghanbari & et. al, 2008) The distance travelled by the robot per complete cycle will be:

$x = 2 l (1 - \cos α)$ (1)

Figure 1.

The basic motion pattern [11].

According to the shown figure, this basic motion pattern can be resolved in four sub-mechanisms M₁, M₂, M‘₂, and M’₁. The joint angles that should be reached at the end of each step are given in Table 1.

Table 1.

Joint angles at end of each step.

	θ^*₁	θ^*₂	θ^*₃	θ^*₄
start	0	0	0	0
M₁	0	α	−α	−α
M₂	α	−α	−α	α
M‘₂	−α	−α	α	0
M‘₁	0	0	0	0

In each sub-mechanism, we consider friction is provided sufficiently that at least one link is non-moving or ground link. These ground links are shown with dark colour in all figures.

Also, with due attention to figure 1, a symmetry condition in motion sequences is cleared. Thus, only both first sub-mechanisms are sufficient for modelling as two manipulators according to figures 2 and 3. Then for last both sub-mechanisms, we reverse coordinate system and use opposite trajectory directions. It should be noted that in these models the angles are measured in a local joint coordinate system, and after solving each sub-problem, those should be mapped to their global coordinate system.

Figure 2.

Manipulator model of M₁ [11].

Figure 3.

Manipulator model of M₂ [11].

Sub-mechanisms M₁ is the first part of gait sequence and it is considered as a 3-R planar manipulator with two DOF, because the tip of the third link remains in constant contact with the ground during the motion. In other words we have a redundantly actuated closed chained system. It is worth noting that, a system is redundantly actuated when the number of actuated joints is greater than the degree of freedom of the mechanism.

Considering identical link length, the geometric constrain of systems can be formulated as:

$\sum_{i = 1}^{n} \sin φ_{i} = 0$ (2)

where n is the number of the joints in each sub-mechanism and φ_i is the angle of i^th link respect to positive direction of horizontal axes. In other words,

$φ_{0} = 0, φ_{i} = φ_{i - 1} + θ_{i}$ (3)

3. Dynamic Analysis

In the following, we develop the dynamic model of a 3-DOF planner manipulator with revolute joints, including a closed loop chain and reaction forces, due to connection between the tip of last link and the ground. The manipulator is supposed to move in the vertical plane x-y as shown in figures 2 and 3.

Let P_i and v_i be position and velocity of origin of i^th link frame, (P₀ = v₀ = 0). Thus, considering position of centriod of each link is at its midpoint, velocity of centriod point of p^th link and its square will be (the length, mass, and moment of inertia about centeriod of i^th link are noted by l_i, m_i, and Ī_i, respectively):

${\bar{v}}_{p} = v_{p} + \frac{l_{p}}{2} {\dot{φ}}_{p} (- sin φ_{p} \overset{⌢}{i} + cos φ_{p} \hat{j})$ (4)

${\bar{v}}_{p}^{2} = \frac{l_{p}^{2}}{4} {\dot{φ}}_{p}^{2} + \sum_{i = 1}^{p} \sum_{j = 1}^{p - 1} l_{i} l_{j} {\dot{φ}}_{i} {\dot{φ}}_{j} C_{i j}$ (5)

where and henceforward, for abbreviation we use C_ij = cos(φ_i - φ_j) and S_ij = sin(φ_i - φ_j).

Also, height of centriod point of p^th link is equal to:

$\begin{array}{l} {\bar{h}}_{p} = P_{p + 1} \cdot \hat{j} - \frac{1}{2} l_{p} \sin φ_{p} \\ = \sum_{j = 1}^{p} (l_{j} \sin φ_{j}) - \frac{l_{p}}{2} \sin φ_{p} \end{array}$ (6)

The dynamic model is based on the Lagrangian formulation. Hence, calculating the total of kinematic energy, total of gravity potential energy, and generalized forces will be needed. First, we derive each of them.

Considering identical link and Ī = ml²/12, the total of kinetic energy is calculated as:

$\begin{array}{l} T = \frac{1}{2} \sum_{p = 1}^{3} (m_{p} {\bar{v}}_{p}^{2} + {\bar{I}}_{p} {\dot{φ}}_{p}^{2}) \\ = \frac{1}{24} m l^{2} (28 {\dot{φ}}_{1}^{2} + 16 {\dot{φ}}_{2}^{2} + 4 {\dot{φ}}_{3}^{2} \\ + 36 {\dot{φ}}_{1} {\dot{φ}}_{2} C_{12} + 12 {\dot{φ}}_{2} {\dot{φ}}_{3} C_{23} + 12 {\dot{φ}}_{3} {\dot{φ}}_{1} C_{31}) \end{array}$ (7)

The total of gravity potential energy will be:

$\begin{array}{l} V = \sum_{p = 1}^{3} m_{p} g {\bar{h}}_{p} \\ = \frac{1}{2} m g l (5 \sin φ_{1} + 3 \sin φ_{2} + \sin φ_{1}) \end{array}$ (8)

Non-conservative forces acing on the system are represented by generalized forces, Q_k, in Lagrangian formulation. Let τ be a 3-dimensional vector including the joint torques acting on the links. Also, F_R and F_f be the normal and frictional reaction forces act on the tip of the last link. Considering a Coulomb friction model, the virtual work done by all the non-conservative forces acting on the system will be:

$\begin{array}{l} δ W = τ \cdot δ Θ + (F_{f} \hat{i} + F_{R} \hat{j}) \cdot δ P_{p + 1} \\ = \sum_{p = 1}^{3} τ_{p} δ (φ_{p} - φ_{p - 1}) \\ + F_{R} (\bar{μ} \sum_{p = 1}^{3} l_{p} \sin φ_{p} δ φ_{p} + \sum_{p = 1}^{3} l_{p} \cos φ_{p} δ φ_{p}) \\ = \sum_{p = 1}^{3} (τ_{p} - τ_{p - 1} + l_{p} (\cos φ_{p} + \bar{μ} \sin φ_{p}) F_{R}) δ φ_{p} \\ \equiv \sum_{p = 1}^{3} (τ_{p} - τ_{p - 1} + λ_{p} F_{R}) δ φ_{p} \equiv \sum_{p = 1}^{3} Q_{p} δ φ_{p} \end{array}$ (9)

where, λ_p = l_p(cosφ_p + μsinφ_p), and μ̄ = μ sgn (v_tip η e_t). It should be noted that the φ_i, i = 1,‥n, are not an independent coordinate system and in fact F_R is the Lagrange's coefficient appeared due to the geometric constrain of system. Also, we know φ₀ = δφ₀ = 0 and τ₄ = 0.

Now, using the Euler-Lagrange equations, motion equations of the dynamic system can be written as:

$\frac{d}{d t} (\frac{\partial T}{\partial {\dot{φ}}_{k}}) - \frac{\partial T}{\partial φ_{k}} + \frac{\partial V}{\partial φ_{k}} = Q_{k}$ (10)

Substituting the Eqs. (7), (8), and (9) into Eq. (10) we will have:

$\begin{array}{l} \frac{m l^{2}}{6} (\begin{matrix} 14 & 9 C_{12} & 3 C_{13} \\ 6 C_{12} & 8 & 3 C_{23} \\ 3 C_{13} & 3 C_{23} & 2 \end{matrix}) (\begin{matrix} {\overset{..}{φ}}_{1} \\ {\overset{..}{φ}}_{2} \\ {\overset{..}{φ}}_{3} \end{matrix}) \\ + \frac{m l^{2}}{6} (\begin{matrix} 0 & 9 S_{12} & 3 S_{13} \\ - 9 S_{12} & 0 & 3 S_{23} \\ - 3 S_{13} & - 3 S_{23} & 0 \end{matrix}) (\begin{matrix} {\dot{φ}}_{1}^{2} \\ {\dot{φ}}_{2}^{2} \\ {\dot{φ}}_{3}^{2} \end{matrix}) \\ + \frac{m g l}{2} (\begin{matrix} 5 \cos φ_{1} \\ 3 \cos φ_{2} \\ \cos φ_{3} \end{matrix}) = (\begin{matrix} 1 & - 1 & 0 \\ 0 & 1 & - 1 \\ 0 & 0 & 1 \end{matrix}) (\begin{matrix} τ_{1} \\ \begin{array}{l} τ_{2} \\ τ_{3} \end{array} \end{matrix}) + (\begin{matrix} λ_{1} \\ λ_{2} \\ λ_{3} \end{matrix}) F_{R} \end{array}$ (11)

The motion equations of the sub-mechanism M₂ are also derived similarly, and results obtained as:

$\begin{array}{l} \frac{m l^{2}}{6} (\begin{matrix} 20 & 15 C_{12} & 9 C_{13} & 3 C_{14} \\ 15 C_{12} & 14 & 9 C_{23} & 3 C_{24} \\ 9 C_{13} & 9 C_{23} & 8 & 3 C_{34} \\ 3 C_{14} & 3 C_{24} & 3 C_{34} & 2 \end{matrix}) (\begin{array}{l} \begin{matrix} {\overset{..}{φ}}_{1} \\ {\overset{..}{φ}}_{2} \\ {\overset{..}{φ}}_{3} \end{matrix} \\ {\overset{..}{φ}}_{4} \end{array}) \\ + \frac{m l^{2}}{6} (\begin{matrix} 0 & 15 S_{12} & 9 S_{13} & 3 S_{14} \\ - 15 S_{12} & 0 & 9 S_{23} & 3 S_{24} \\ - 9 S_{13} & - 9 S_{23} & 0 & 3 S_{34} \\ - 3 S_{14} & - 3 S_{24} & - 3 S_{34} & 0 \end{matrix}) (\begin{array}{l} \begin{matrix} {\dot{φ}}_{1}^{2} \\ {\dot{φ}}_{2}^{2} \\ {\dot{φ}}_{3}^{2} \end{matrix} \\ {\dot{φ}}_{4}^{2} \end{array}) \\ + \frac{m g l}{2} (\begin{matrix} \begin{matrix} 7 \cos φ_{1} \\ 5 \cos φ_{2} \\ 3 \cos φ_{3} \end{matrix} \\ \cos φ_{4} \end{matrix}) = (\begin{matrix} 1 & - 1 & 0 & 0 \\ 0 & 1 & - 1 & 0 \\ 0 & 0 & 1 & - 1 \\ 0 & 0 & 0 & 1 \end{matrix}) (\begin{matrix} τ_{1} \\ \begin{array}{l} τ_{2} \\ τ_{3} \\ τ_{4} \end{array} \end{matrix}) + (\begin{matrix} λ_{1} \\ λ_{2} \\ λ_{3} \\ λ_{4} \end{matrix}) F_{R} \end{array}$ (12)

4. Trajectory Planning

4.1 Cost Function Definition

In previous section we derived the dynamic equations for our crawling gait model in a close form, which they can be summarized as follows:

$M (Θ) \overset{..}{Θ} + C (Θ) {\dot{Θ}}^{2} + G (Θ) = D τ + λ (Θ) F_{R}$ (13)

where, M_nxn is the mass matrix of manipulator, C_nxn is the matrix of centrifugal coefficients, G_nx1 is vector of gravity terms, and, D_nxn is the subtraction matrix with D_pq = δ_pq - δ_p(q-1), where δ is Kronecker Delta.

Now, in this section, we intend to use this model to generate optimal joint trajectories that minimize a certain metric as cost function as:

$J = ψ (Θ (t)) + \int_{0}^{t_{f}} f (Θ (t)) d t$ (14)

subject the boundary conditions:

$\begin{array}{l} Θ (t = 0) = \dot{Θ} (t = 0) = 0, \\ Θ (t = t_{f}) = \dot{Θ} (t = t_{f}) = 0 \end{array}$ (15)

where Ψ penalizes deviations from the constrains and t_f is final time that it can be assumed fixed. For our problem, the effort, f = 1/2_||τ||², captures desired to minimize the exerted joint torques. Also, the constrains of our model are satisfying the Eq. (2) as an equality constrain, and as an inequality constrain, each joint of the mechanism must not lie below the position y=0, because we has taken the assumption about ground contact.

It is noted that with this definition for cost function, the last joint of both sub-mechanisms can be allowed to be passive, and by this way, considering a set of trajectories for n-1 first joints in the mechanisms and using the associated constrain relationship, the left side of Eqs. (11) and (12) will be known. Hence, the unknown terms remain in the right side of Eqs. (11) and (12) will become the reaction force, R, and n-1 first joint torques, that they will also be obtained recursively.

4.2 B-spline Curves

In this stage, we use B-spline curves to define the trajectory of the actuators, that in our formulation it has been transferred into joint space. B-spline curves can be thought of as a method for defining a sequence of degree (k-1) Bézier curves that join automatically with C^k-2 - continuity, regardless of where the control points are placed. The B-spline curves are typically specified in terms of a set of control points, P = {p₀, p₁, …,p_p}, and a knot vector, T = {t₀,…,t_k-1 ↑ t_k,…,t_p ↑ t_p+1,…,t_p+k}. Notice that there are k-1 extra knots prepended and appended to the knot vector that control the end conditions of the B-spline curve; Choosing a k-fold knot at each end, then curve interpolates the end control point and is tangent to the control polygon at its endpoints. Hence, trajectory of each active joint can be parameterized by a set of control points, as:

$θ (P, t) = \sum_{i = 0}^{p} B_{i, k} (t) p_{i}$ (16)

where k is the order of B-spline (for cubic B-spline k = 4 or for quadruplet B-spline k = 5) and B_i,k(t) is the basic function given by the recurrence relations as:

$\begin{array}{l} B_{i, k} (t) = \frac{t - t_{i}}{t_{i + k - 1} - t_{k}} B_{i, k - 1} (t) \\ + \frac{t_{i + k} - t}{t_{i + k} - t_{i + 1}} B_{i + 1, k - 1} (t) \\ B_{i, k} (t) = i f (t_{i} < t < t_{i + 1}, 1; o t h e r w i s e, 0) \end{array}$ (17)

Since B_i,k(t) = 0 ∀t ∉ [t_i,t_i+k], therefore the control points of the B-spline have only a local effect on the curve geometry, so given any t in knot span [t_i, t_i+k], there are only k nonzero B_i,k(t) in this knot span (i.e. B_i,k-1,…, B_i-k,k-1, B_i-(k-1),k-1). In addition, the convex hull property of B-splines makes them useful for smoothing or approximating data. Also, the fact that ∑^p_i=0 B_i,k(t) = 1 gives the desirable property that limits on joint displacements can be imposed through limits on the spline parameters p_i. That is, if one constrain p_i ⩾ p_max, then it follows that (Park, 2005):

$θ (P, t) \leq p_{\max} \forall t \in [0, t_{f}]$ (18)

By parameterizing the trajectories in term of B-splines, the original optimal control reduces to a parametric optimization problem of the form:

$\begin{array}{l} m i n i m i z e J (P) = ψ (P) + \int_{0}^{t_{f}} f (P) d t \\ s u b j e c t e d t o p_{\min} \leq p_{i} \leq p_{\max} \end{array}$ (19)

Here, our problem has been converted to a parametric optimization problem and we require an efficient heuristic optimization method that be used to minimize the cost function.

5. Genetic Algorithm

Genetic Algorithm (GA) is a sort of population-oriented search techniques which uses probabilistic transition rules to evolve multiple potential solutions. GA does not require the derivative information of objective function and constraints and has robust characteristic for the problems with multiple local minima. These properties of GA can give advantages over the other conventional optimal trajectory planners of a robot manipulator (Lee, 1999). The genetic algorithm based on the principles of genetics and natural selection so that repeatedly changes a population of individual solutions to evolve a state that maximizes the “fitness” (i.e., minimizes the cost function) (Haupt, 2004).

A simple GA commonly uses binary coded strings for parameter representation, which is also called a “chromosome”, might look like the figure 4.

Figure 4.

Schema of a chromosome.

At each cycle, as shown in figure 5, the GA selects individuals at random from the current population to be parents and uses them to generate the children for the next generation. Over successive generations, the population evolves toward an optimal solution.

Figure 5.

Flowchart of a simple GA.

The GA uses three main types of rules at each step to create the next generation from the current population (Chettibi, 2006):

Selection rules select the individuals, called parents, which contribute to the population at the next generation.

Crossover rules combine two parents to form children for the next generation.

Mutation rules apply random changes to individual parents to form children.

It is mentioned that, similar to other heuristic optimization methods, GA for a constrained optimization problem is also fundamentally based upon a penalty function approach.

6. Simulation Study

Our simulation includes generating optimal motion of a redundantly actuated closed chain using GA, which follows the shown algorithm in Fig. 6. Note that an exactly actuated closed chain can be considered as a special case of this general case.

Figure 6.

Fitness Evaluation algorithm.

In this algorithm, the evaluation of fitness of individuals involves calculating of effort metric for a discrete sampling of a motion obtained by a set of certain joint trajectories. For this end we approximate the effort metric as:

$\begin{array}{l} \int_{0}^{t_{f}} f (τ (P)) d t = \frac{1}{2} \int_{0}^{t_{f}} {‖ τ ‖}^{2} d t \\ ≃ \frac{1}{2} {\sum_{i = 1}^{n} \sum_{j = 1}^{s m p} (\frac{τ_{i (j + 1)} - τ_{i j}}{2})}^{2} \frac{t_{f}}{s m p} \end{array}$ (20)

where smp is number of sampling. Each of the existing links in sub-mechanisms is treated as a thin rod with uniform mass distribution. The specification of links and dynamic parameters are used by simulating process is given in Table 2, as same as those are considered by (Spranklin, 2006).

Table 2.

Specification of dynamic parameters.

mass (m)	link length (l)	gait angle (α)	friction coef. (μ)	time step (tf)
15 gr	0.14 m	45 deg	0.4	1 sec

In both sub-mechanisms last joint is treated as passive joint and its displacement and time derivatives are calculated using the constraint relationship. The simulation is run for each sub-mechanism using the GA parameters given in Table 3. The quadruplet B-splines are considered to make joint angle profiles and five bits are allocated to encode each variable parameter (control point value). Also, the random pairing method is chosen as selection rule of GA. Finally, the obtained cost values are given in Table 3. The results achieved by GA for each sub-mechanism show reduction in torque consumption in compression with the results achieved by Spranklin, who have used gradient based method. The reason of this reduction is that GA is a global search method which is robust against trapping in local optimality point.

Table 3.

Genetic Algorithm parameters and results.

	sub-mech. M₁	sub-mech. M₂
Population	8	10
Generation	200	200
Selection rate	0.5	0.5
Mutation rate	0.07	0.06

Cost obtained	1.09e-3 N²m²/s	5.43e-3 N²m²/s
Cost reported by Spranklin	1.74e-3 N²m²/s	5.70e-3 N²m²/s
Cost decreasing	37 %	5 %

The best control points set of each sub-problem is obtained as below. It is worth noting that, the boundary values of joint angles are repeated to achieve the desired boundary conditions.

Sub-mechanism M₁:

Joint 1 {0,043.594,40.781,36.56245,45}

Joint 2 {0,0-16.875,-40.781,-16.875-45,-45}

Sub-mechanism M₂:

Joint 1 {0,036.563,5.625,21.09445,45}

Joint 2 {45,45-14.063,16.875,-5.625-45,-45}

Joint 3 {-45,-45-37.969,-1.406,-7.031-45,-45}

The displacement profile of each free joint in sub-mechanism M₁, according to obtained optimal trajectory, and its associated consumed torque is shown in figures 7 and 8 and figures 10 and 11, respectively. Snapshots of motion are also illustrated in figure 9. Similarly, results of the sub-mechanism M₂ is shown in figures 12 to 18, according to its optimal solution.

Figure 7.

Optimal joint displacements in M1, joint 1.

Figure 8.

Optimal joint displacements in M₁, joint 2.

Figure 9.

Snapshots of optimal motion in sub-mech. M₁.

Figure 10.

required torque according to optimal joint trajectories in sub-mechanism M₁, joint 1.

Figure 11.

required torque according to optimal joint trajectories in sub-mechanism M₁, joint 2.

Figure 12.

Optimal joint displacements in M₂, joint 1.

Figure 13.

Optimal joint displacements in M₂, joint 2.

Figure 14.

Optimal joint displacements in M₂, joint 3.

Figure 15.

Snapshots of optimal motion in sub-mech. M₂.

Figure 16.

required torque according to optimal joint trajectories in sub-mechanism M₂, joint 1.

Figure 17.

required torque according to optimal joint trajectories in sub-mechanism M₂, joint 2.

Figure 18.

required torque according to optimal joint trajectories in sub-mechanism M₂, joint 3.

7. Conclusion

In this paper a new crawling gait to develop in a modular robot was described. Then each of gait steps was modeled as a manipulator and derived the dynamic formulation of them by Lagrangian method in explicit form. Imposing the geometric constrains of system, a redundantly actuated closed chain mechanism was analyzed and expressed an algorithm to solve it. Next, the optimal joint trajectories for a motion with minimum effort metric were presented utilizing GA. B.spline curves are used to convert the original optimal control problem into a parametric optimization problem. Finally our models were simulated and achieved results were represented for optimal motion of each of them. Efficiency of GA to search a global solution was shown by comparing the results with obtained results in previous similar work.

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Optimal Trajectory Planning for Design of a Crawling Gait in a Robot Using Genetic Algorithm