2.1.1. The shortcoming of traditional potential field model

There are some problems when this method is applied to robot path planning:


#### 2.1.2. Improved measures

Aimed at the defects of traditional artificial potential field model, improved measures are proposed as below:


## 2.1.3. Improved potential model

According to above improved measures, the improved artificial potential field model is proposed. The attractive force model of the target to a full range of vehicle's body is:

$$\mathcal{U}\_{\rm att}(\mathbf{X}) = \mathbf{0}.5k\rho^2 \left(\mathbf{X}, \mathbf{X}\_{\rm g}\right) \tag{1}$$

where r X; Xg � � is the distance between the current location of the central point of mobile vehicle's body and target point; k is a proportional gain coefficient; X is the position ½ � x; y <sup>T</sup> of robot's central point in movement space; and Xg is the target point position xg; yg h i<sup>T</sup> .

Repulsive potential model of the i-th static obstacles on the full range of movement of the body is:

<sup>U</sup> <sup>¼</sup> Uattð Þþ <sup>X</sup> <sup>X</sup><sup>n</sup>

i¼1

<sup>U</sup> <sup>¼</sup> Uattð Þþ <sup>X</sup> <sup>X</sup><sup>n</sup>

2.2. Solving target point based on genetic trust region

can greatly improve the convergence speed of algorithm [9–11].

0

minU zð Þ¼ Uattð Þþ <sup>z</sup> <sup>X</sup><sup>n</sup>

¼ x þ R cos θ, z<sup>2</sup>

The points of the shaded part can show as x

0

field method.

assuming Uaddð Þz z<sup>1</sup>

i¼1

Urepsð Þþ Xi

Based on above model, every sampling period all regard the minimum point of potential field sum as a sub-goal point, and multiple sub-target is consisted of global optimization path. The sum of potential field is shown in (2). In order to avoid the case of target vibrating in the vicinity of local minimum point, vector synthesis method is used to judge whether the robot is in local minimum point, if in the local minimum point, potential field sum contains added potential, as shown in (3). Assuming the maximum speed of robot is vmax, the sampling period is t0, so when robot in the reachable range of every sampling period, it takes current location as a center, and vmaxt<sup>0</sup> is a radius of the circle. The speed of robot movement should not too big or too small in order to ensure the stability of robot movement and performing efficiency, so a sub-goal point can be chosen from an annular region, R∈ð Þ 2Vmaxt0=3; Vmaxt<sup>0</sup> , θ∈ ð Þ 0; 2π .

As Figure 1 shows, the shaded part of the annular region is the optional region of the subtarget point. Therefore, solving the sub-target issue is the key to improve artificial potential

In order to improve the efficiency of path planning, below two situations will be considered:

(1) When the robots move in the free space far away from obstacle ( ≤ r0), that solving the minimum point of the sum of annular region potential field intensity shown in Figure 1. The trust region method in optimal search algorithm has good reliability and fast convergence, which offers a new idea to solve the sub-target. However, its iteration speed sometimes is affected by the radial trust. When the iteration speed is affected by the radius of trust region, using genetic algorithms to solve a point, which is better than one in the current iteration point, then, based on the point the trust region is restarted until the optimized point is found, which

0

minimum point, (4) and added potential Uaddð Þz are regarded as the target function.

and (3) are the function about variables R and <sup>θ</sup>, <sup>R</sup>∈ð Þ <sup>2</sup>Vmaxt0=3; <sup>V</sup>maxt<sup>0</sup> , <sup>θ</sup><sup>∈</sup> <sup>0</sup>; <sup>2</sup><sup>∗</sup> ð Þ <sup>3</sup>:<sup>14</sup> :

algorithm to solve the target function of sub-goal point as (4) shows, in other words, that solving a class of linear constrained optimized problem, when that the robot is located in local

i¼1

Using quadratic approximation and constructing constrained trust region sub-problem:

¼ x þ R cos θ, y<sup>0</sup>

Urepsð Þþ zi

Xn r¼1

Urepsð Þþ Xi

Xm r¼1

Xm r¼1

Urepmð Þ Xr : (2)

Motion Planning for Mobile Robots http://dx.doi.org/10.5772/intechopen.76895 149

¼ y þ R sin θ. Therefore, Eqs. (2)

Urepmð Þ zr (4)

¼ y þ R sin θ, z ¼ ð Þ z1; z<sup>2</sup> . Using genetic trust region

Urepmð Þþ Xr Uaddð Þ X : (3)

$$U\_{r\text{opt}}(X\_{\prime}) = \begin{cases} 0.5\eta(\frac{1}{\rho(X,X\_{\prime})} - \frac{1}{\rho\_{0}})^{2} \left\| X - X\_{\varkappa} \right\|\_{2}, \\\\ 0, \\ \qquad \qquad \qquad \qquad \qquad \text{if } \rho(X,X\mathbb{i}) \text{ s.p0} \end{cases}$$

where i∈ 1; 2 000 ; <sup>n</sup> � �, n is the summation of static obstacles; <sup>r</sup>ð Þ <sup>X</sup>; Xi is the shortest distance of between current location of the center of mobile vehicle's body and the i-th obstacle; r<sup>0</sup> the effective effect distance of obstacle; and η is proportional position gain coefficient.

Dynamic obstacles are in motion, and it cannot reflect environment information completely if only taking location into consideration. Bring the relative speed of dynamic obstacles and robots into potential field function, Repulsive potential model of the i-th static obstacles on the full range of movement of the body is got:

$$\begin{aligned} \mathbf{U}\_{\
u \rho \nu}(X\_i) = \begin{cases} 0.5 \eta \rho \widetilde{(X\_i X\_i)} - \frac{1}{\rho\_0} \mathfrak{l}^2 \begin{bmatrix} \|X - X\_{\pi} \|\_{2} + \\ \zeta \|\nu - \nu\_r \sin(\varphi - \mathbf{O}) \| \end{bmatrix}, \\ \mathbf{0}, \end{cases} \end{aligned}$$

$$\begin{aligned} \text{iff } \rho(\mathbf{X}, \mathbf{X} \mathbf{i}) \succeq \rho \mathbf{o} \end{aligned}$$

where r∈ ð Þ 1; 2;…; m , m is the summation of mobile obstacles; ζ is proportional coefficient; V is the current speed of the mobile body, v∈ð Þ 2vmax=3; vmax ; vmaxt<sup>0</sup> is the current speed of the rth dynamic obstacles; ϕ is the current movement direction of the mobile body; and φ is the current movement direction of the r-th dynamic obstacle.

When the robot is in local minimum point, the "added potential" is brought to figure the problem of local minimum out, the added potential model is:

$$\mathcal{U}\_{add}(\mathbf{X}) = \begin{cases} s\rho^2(\mathbf{X}, \mathbf{X}\_{\S}) \, \rho(\mathbf{X}, \mathbf{X}\_{\S}) > \rho\_a; \\ 0 \, \rho\left(\mathbf{X}, \mathbf{X}\_{\S}\right) \le \rho\_a \end{cases}$$

where r<sup>a</sup> judgment distance of whether the mobile body reaches to a target point; s is a proportional coefficient.

Therefore, the whole potential field intensity of a range of mobile body is shown in (2). When the robot is in local minimum point, taking (2) plus the added potential as the total potential field value (as shown in (2)).

$$\mathcal{U} = \mathcal{U}\_{\mathcal{U}t}(\mathbf{X}) + \sum\_{i=1}^{n} \mathcal{U}\_{\text{reps}}(\mathbf{X}\_{i}) + \sum\_{r=1}^{m} \mathcal{U}\_{\text{repm}}(\mathbf{X}\_{r}).\tag{2}$$

$$\mathcal{U} = \mathcal{U}\_{\text{aff}}(\mathbf{X}) + \sum\_{i=1}^{n} \mathcal{U}\_{\text{reps}}(\mathbf{X}\_i) + \sum\_{r=1}^{m} \mathcal{U}\_{\text{repm}}(\mathbf{X}\_r) + \mathcal{U}\_{\text{add}}(\mathbf{X}).\tag{3}$$

Based on above model, every sampling period all regard the minimum point of potential field sum as a sub-goal point, and multiple sub-target is consisted of global optimization path. The sum of potential field is shown in (2). In order to avoid the case of target vibrating in the vicinity of local minimum point, vector synthesis method is used to judge whether the robot is in local minimum point, if in the local minimum point, potential field sum contains added potential, as shown in (3). Assuming the maximum speed of robot is vmax, the sampling period is t0, so when robot in the reachable range of every sampling period, it takes current location as a center, and vmaxt<sup>0</sup> is a radius of the circle. The speed of robot movement should not too big or too small in order to ensure the stability of robot movement and performing efficiency, so a sub-goal point can be chosen from an annular region, R∈ð Þ 2Vmaxt0=3; Vmaxt<sup>0</sup> , θ∈ ð Þ 0; 2π .

As Figure 1 shows, the shaded part of the annular region is the optional region of the subtarget point. Therefore, solving the sub-target issue is the key to improve artificial potential field method.

#### 2.2. Solving target point based on genetic trust region

Repulsive potential model of the i-th static obstacles on the full range of movement of the

; <sup>n</sup> � �, n is the summation of static obstacles; <sup>r</sup>ð Þ <sup>X</sup>; Xi is the shortest distance of

between current location of the center of mobile vehicle's body and the i-th obstacle; r<sup>0</sup> the

Dynamic obstacles are in motion, and it cannot reflect environment information completely if only taking location into consideration. Bring the relative speed of dynamic obstacles and robots into potential field function, Repulsive potential model of the i-th static obstacles on

where r∈ ð Þ 1; 2;…; m , m is the summation of mobile obstacles; ζ is proportional coefficient; V is the current speed of the mobile body, v∈ð Þ 2vmax=3; vmax ; vmaxt<sup>0</sup> is the current speed of the rth dynamic obstacles; ϕ is the current movement direction of the mobile body; and φ is the

When the robot is in local minimum point, the "added potential" is brought to figure the

� �, r X;X<sup>g</sup>

� � <sup>&</sup>gt; <sup>r</sup>a;

sr<sup>2</sup> X; Xg

8 < :

0, r X;X<sup>g</sup> � � <sup>≤</sup> <sup>r</sup><sup>a</sup>

where r<sup>a</sup> judgment distance of whether the mobile body reaches to a target point; s is a

Therefore, the whole potential field intensity of a range of mobile body is shown in (2). When the robot is in local minimum point, taking (2) plus the added potential as the total potential

effective effect distance of obstacle; and η is proportional position gain coefficient.

body is:

where i∈ 1; 2

000

148 Advanced Path Planning for Mobile Entities

proportional coefficient.

field value (as shown in (2)).

the full range of movement of the body is got:

current movement direction of the r-th dynamic obstacle.

problem of local minimum out, the added potential model is:

Uaddð Þ¼ X

In order to improve the efficiency of path planning, below two situations will be considered:

(1) When the robots move in the free space far away from obstacle ( ≤ r0), that solving the minimum point of the sum of annular region potential field intensity shown in Figure 1. The trust region method in optimal search algorithm has good reliability and fast convergence, which offers a new idea to solve the sub-target. However, its iteration speed sometimes is affected by the radial trust. When the iteration speed is affected by the radius of trust region, using genetic algorithms to solve a point, which is better than one in the current iteration point, then, based on the point the trust region is restarted until the optimized point is found, which can greatly improve the convergence speed of algorithm [9–11].

The points of the shaded part can show as x 0 ¼ x þ R cos θ, y<sup>0</sup> ¼ y þ R sin θ. Therefore, Eqs. (2) and (3) are the function about variables R and <sup>θ</sup>, <sup>R</sup>∈ð Þ <sup>2</sup>Vmaxt0=3; <sup>V</sup>maxt<sup>0</sup> , <sup>θ</sup><sup>∈</sup> <sup>0</sup>; <sup>2</sup><sup>∗</sup> ð Þ <sup>3</sup>:<sup>14</sup> : assuming Uaddð Þz z<sup>1</sup> 0 ¼ x þ R cos θ, z<sup>2</sup> 0 ¼ y þ R sin θ, z ¼ ð Þ z1; z<sup>2</sup> . Using genetic trust region algorithm to solve the target function of sub-goal point as (4) shows, in other words, that solving a class of linear constrained optimized problem, when that the robot is located in local minimum point, (4) and added potential Uaddð Þz are regarded as the target function.

$$\min \mathcal{U}(\mathbf{z}) = \mathcal{U}\_{\text{d}\mathcal{U}}(\mathbf{z}) + \sum\_{i=1}^{n} \mathcal{U}\_{\text{reps}}(\mathbf{z}\_{i}) + \sum\_{r=1}^{n} \mathcal{U}\_{\text{repm}}(\mathbf{z}\_{r}) \tag{4}$$

Using quadratic approximation and constructing constrained trust region sub-problem:

Figure 1. The optional range of sub-target point.

$$\begin{aligned} \min q\_k(d) &= g\_k^T d + 0.5d^T G\_k d\\ \text{s.t.} & \|d\|\_2 \le \Delta\_k, z\_k + d\_k \in \Omega \end{aligned} \tag{5}$$

Step 5: Choosing Δ<sup>k</sup>þ1, let it meet:

U ze kþ1

to step 8.

to step 8.

where

when solving ze

Δ<sup>k</sup>þ<sup>1</sup>

Step 7: If k k Ukþ<sup>1</sup> � Uk <sup>&</sup>gt; M Uk k <sup>k</sup>þ<sup>1</sup> � Uk , so order zkþ<sup>1</sup> <sup>≔</sup> ze

algorithm 1, using a genetic algorithm to figure minU z<sup>e</sup>

<sup>c</sup> <sup>¼</sup> ze

crossover; and mutation operator for the basic bit mutation.

ð Þ <sup>k</sup>þ<sup>1</sup> <sup>1</sup>, z<sup>e</sup>

The procedure of algorithm 2:

z1b, z2b: described as:

If

ze

<sup>k</sup><sup>1</sup> � a, f <sup>¼</sup> <sup>z</sup><sup>e</sup>

8 >><

>>:

Step 8: Using BFGS formula to modify Bk and gets Bkþ1, k ≔ k þ 1, back to step 2.

<sup>k</sup>þ<sup>1</sup> <sup>∈</sup> <sup>Ω</sup>, c <sup>&</sup>lt; <sup>z</sup><sup>2</sup>

ð Þ <sup>k</sup>þ<sup>1</sup> <sup>2</sup>, c, f , l, n are all known quantity.

obtained, need to be considered in accordance with the actual situation.

Δk; β3Δ<sup>k</sup> �� �, rk <sup>&</sup>gt; <sup>η</sup>2;

� �, rk <sup>∈</sup> <sup>η</sup>1; <sup>η</sup><sup>2</sup>

� �;

<sup>k</sup>þ<sup>1</sup>Gkþ<sup>1</sup> <sup>≔</sup> <sup>Δ</sup>0, back to step 2; or back

Motion Planning for Mobile Robots http://dx.doi.org/10.5772/intechopen.76895 151

β2Δk; Δ<sup>k</sup>

β1Δk; β2Δ<sup>k</sup> � �, rk <sup>&</sup>lt; <sup>η</sup>1:

Step 6: If Δ<sup>k</sup>þ<sup>1</sup> < ε2, Uk k <sup>k</sup>þ<sup>1</sup> � Uk < ε3, and using genetic algorithm to quickly solve min

ε2, ε3, M values can be adjusted to control the number of times the call number of sub-issues of a genetic algorithm to achieve the optimization algorithm speed, regulating ε<sup>1</sup> also can regulate the speed of convergence of the algorithm, but there are trade-offs to optimize the value

When the convergence speed is influenced by the radius of the trust region in step 6 of

<sup>k</sup><sup>1</sup> <sup>þ</sup> a, l <sup>¼</sup> ze

Fitness function: establishing the mapping relationship between the objective function and the function of moderate: G zð Þ¼ <sup>0</sup>:618U zð Þ; adopting the binary code to encode; replication strategy to preserve the best individual mixed roulette selection; crossover operator is single point

Step 1: Parameter initialization: population size N, crossover probability Pc, mutation probability Pm, current algebraic Tc ¼ 0, maximum algebraic Tmax, and 0 < k < 0:618. The coding initial solution produces a unit corresponding to the initial solution. And it randomly produces two l-bit binary string working as one unit, and it does not stop until it generates N units.

Step 2: Solving adaption value of every individual in the current group, to get the optimization

Tc ≤ Tmax

<sup>k</sup>þ<sup>1</sup>, l <sup>&</sup>lt; ze

kþ1

<sup>k</sup>þ<sup>1</sup> <sup>&</sup>lt; <sup>n</sup>:

<sup>k</sup><sup>2</sup> � b, n <sup>¼</sup> ze

� � out must meet:

<sup>k</sup><sup>2</sup> þ b,

� � so that an iteration point which is better than the current point, go to step 7, Or back

where gk <sup>¼</sup> <sup>∇</sup>U zð Þ<sup>k</sup> ; <sup>Δ</sup><sup>k</sup> is the radius of trust region; Gk <sup>¼</sup> <sup>∇</sup><sup>2</sup>U zð Þ<sup>k</sup> : solving Gk is very complicated, and using BFGS formula of quasi-Newton to structure Hessian matrix Bk, which is approximate to Gk. dk is the decline tentative step. Ω is the range of R and θ.

The algorithm contains, and remark shows: z ∈R<sup>2</sup> , zk ¼ ð Þ zk1; zk<sup>2</sup> , zkþ<sup>1</sup> ¼ zð Þ <sup>k</sup>þ<sup>1</sup> <sup>1</sup>; zð Þ <sup>k</sup>þ<sup>1</sup> <sup>2</sup> , ze <sup>k</sup>þ<sup>1</sup> ¼ ze ð Þ <sup>k</sup>þ<sup>1</sup> <sup>1</sup>; ze ð Þ kþ1 2 , yk <sup>¼</sup> gkþ<sup>1</sup> � gk, := value assignment, BBFGS <sup>k</sup>þ<sup>1</sup> <sup>¼</sup> Bk <sup>þ</sup> ykyT <sup>k</sup> =yT <sup>k</sup> dk � BkdkB<sup>T</sup> <sup>k</sup> <sup>=</sup>dT <sup>k</sup> Bkdk, the amount of actual decline in the ratio of the amount of pre-estimated decline is:

$$r\_k = \Delta \mathcal{U}\_k / \Delta q\_k = \frac{\mathcal{U}(z\_k) - \mathcal{U}(z\_k + d\_k)}{q\_k(0) - q\_k(d\_k)}.$$

The first step in the algorithm.

Step 1: Initialization, given: z0, Δ<sup>0</sup> > 0, ε<sup>1</sup> > 0, ε<sup>2</sup> > 0, ε<sup>3</sup> > 0, a > 0, b > 0M > 1.

$$0 < \eta\_1 < \eta\_2 < 1,\\
B\_0 =: I\_{2\times 2}. \;0 < \beta\_1 < \beta\_2 < 1 \le \beta\_3, \; k := 0.$$

Step 2: Calculating gk, if gk <sup>2</sup> < ε1, stop calculating, and output the result.

Step 3: Solving sub-problem in Eq. (5), and get the decline tentative step d1.

Step 4: Calculating rk, if rk > η<sup>1</sup> meanwhile zk þ dk ∈ Ω, so, zkþ<sup>1</sup> ≔ zk þ dk rectifies Bkþ1, or zkþ<sup>1</sup> ≔ zk, Bkþ<sup>1</sup> ≔ Bk.

Step 5: Choosing Δ<sup>k</sup>þ1, let it meet:

$$\Delta\_{k+1} \begin{cases} \{ [\Delta\_k, \beta\_3 \Delta\_k] \; ; \; r\_k > \eta\_2 \text{'} \\ \{ \beta\_2 \Delta\_k, \Delta\_k \} \; ; r\_k \in [\eta\_1, \eta\_2] \text{'} \\ \{ \beta\_1 \Delta\_k, \beta\_2 \Delta\_k \} \; ; r\_k < \eta\_1 \text{'} \end{cases}$$

Step 6: If Δ<sup>k</sup>þ<sup>1</sup> < ε2, Uk k <sup>k</sup>þ<sup>1</sup> � Uk < ε3, and using genetic algorithm to quickly solve min U ze kþ1 � � so that an iteration point which is better than the current point, go to step 7, Or back to step 8.

Step 7: If k k Ukþ<sup>1</sup> � Uk <sup>&</sup>gt; M Uk k <sup>k</sup>þ<sup>1</sup> � Uk , so order zkþ<sup>1</sup> <sup>≔</sup> ze <sup>k</sup>þ<sup>1</sup>Gkþ<sup>1</sup> <sup>≔</sup> <sup>Δ</sup>0, back to step 2; or back to step 8.

Step 8: Using BFGS formula to modify Bk and gets Bkþ1, k ≔ k þ 1, back to step 2.

ε2, ε3, M values can be adjusted to control the number of times the call number of sub-issues of a genetic algorithm to achieve the optimization algorithm speed, regulating ε<sup>1</sup> also can regulate the speed of convergence of the algorithm, but there are trade-offs to optimize the value obtained, need to be considered in accordance with the actual situation.

When the convergence speed is influenced by the radius of the trust region in step 6 of algorithm 1, using a genetic algorithm to figure minU z<sup>e</sup> kþ1 � � out must meet:

$$z\_{k+1}^{\varepsilon} \in \Omega, \ c < z\_{k+1}^2, l < z\_{k+1}^{\varepsilon} < n.$$

where

minqkð Þ¼ <sup>d</sup> <sup>g</sup><sup>T</sup>

approximate to Gk. dk is the decline tentative step. Ω is the range of R and θ.

the amount of actual decline in the ratio of the amount of pre-estimated decline is:

Step 1: Initialization, given: z0, Δ<sup>0</sup> > 0, ε<sup>1</sup> > 0, ε<sup>2</sup> > 0, ε<sup>3</sup> > 0, a > 0, b > 0M > 1.

Step 3: Solving sub-problem in Eq. (5), and get the decline tentative step d1.

The algorithm contains, and remark shows: z ∈R<sup>2</sup>

Figure 1. The optional range of sub-target point.

150 Advanced Path Planning for Mobile Entities

 

, yk <sup>¼</sup> gkþ<sup>1</sup> � gk, := value assignment, BBFGS

ze ð Þ <sup>k</sup>þ<sup>1</sup> <sup>1</sup>; ze

ð Þ kþ1 2

The first step in the algorithm.

Step 2: Calculating gk, if gk

zkþ<sup>1</sup> ≔ zk, Bkþ<sup>1</sup> ≔ Bk.

<sup>k</sup> <sup>d</sup> <sup>þ</sup> <sup>0</sup>:5dTGkd

qkð Þ� 0 qkð Þ dk

where gk <sup>¼</sup> <sup>∇</sup>U zð Þ<sup>k</sup> ; <sup>Δ</sup><sup>k</sup> is the radius of trust region; Gk <sup>¼</sup> <sup>∇</sup><sup>2</sup>U zð Þ<sup>k</sup> : solving Gk is very complicated, and using BFGS formula of quasi-Newton to structure Hessian matrix Bk, which is

rk <sup>¼</sup> <sup>Δ</sup>Uk=Δqk <sup>¼</sup> U zð Þ� <sup>k</sup> U zð Þ <sup>k</sup> <sup>þ</sup> dk

0 < η<sup>1</sup> < η<sup>2</sup> < 1, B<sup>0</sup> ≕ I2x2, 0 < β<sup>1</sup> < β<sup>2</sup> < 1 ≤ β3, k ≔ 0

Step 4: Calculating rk, if rk > η<sup>1</sup> meanwhile zk þ dk ∈ Ω, so, zkþ<sup>1</sup> ≔ zk þ dk rectifies Bkþ1, or

<sup>2</sup> < ε1, stop calculating, and output the result.

<sup>s</sup>:t:k k<sup>d</sup> <sup>2</sup> <sup>≤</sup>Δk, zk <sup>þ</sup> dk <sup>∈</sup> <sup>Ω</sup> (5)

, zk ¼ ð Þ zk1; zk<sup>2</sup> , zkþ<sup>1</sup> ¼ zð Þ <sup>k</sup>þ<sup>1</sup> <sup>1</sup>; zð Þ <sup>k</sup>þ<sup>1</sup> <sup>2</sup>

<sup>k</sup> =yT

<sup>k</sup>þ<sup>1</sup> <sup>¼</sup> Bk <sup>þ</sup> ykyT

:

, ze

<sup>k</sup> dk � BkdkB<sup>T</sup>

<sup>k</sup>þ<sup>1</sup> ¼

<sup>k</sup> <sup>=</sup>dT <sup>k</sup> Bkdk,

$$\mathcal{L} = \mathbf{z}\_{k1}^{\varepsilon} - a\_{\prime}\mathbf{f} = \mathbf{z}\_{k1}^{\varepsilon} + a\_{\prime}\mathbf{l} = \mathbf{z}\_{k2}^{\varepsilon} - b\_{\prime}\mathbf{n} = \mathbf{z}\_{k2}^{\varepsilon} + \mathbf{b}\_{\prime}$$

when solving ze ð Þ <sup>k</sup>þ<sup>1</sup> <sup>1</sup>, z<sup>e</sup> ð Þ <sup>k</sup>þ<sup>1</sup> <sup>2</sup>, c, f , l, n are all known quantity.

Fitness function: establishing the mapping relationship between the objective function and the function of moderate: G zð Þ¼ <sup>0</sup>:618U zð Þ; adopting the binary code to encode; replication strategy to preserve the best individual mixed roulette selection; crossover operator is single point crossover; and mutation operator for the basic bit mutation.

The procedure of algorithm 2:

Step 1: Parameter initialization: population size N, crossover probability Pc, mutation probability Pm, current algebraic Tc ¼ 0, maximum algebraic Tmax, and 0 < k < 0:618. The coding initial solution produces a unit corresponding to the initial solution. And it randomly produces two l-bit binary string working as one unit, and it does not stop until it generates N units.

Step 2: Solving adaption value of every individual in the current group, to get the optimization z1b, z2b: described as:

If

$$T\_c \le T\_{\text{max}}$$

If

$$G(z\_{\mathbb{H}}) - G(z\_k) > k \text{ and } z \in \Omega \text{, then } z\_{(k+1)1}^\varepsilon := z\_{1\mathbb{h}\mathcal{H}}$$

ze ð Þ <sup>k</sup>þ<sup>1</sup> <sup>2</sup> <sup>≔</sup> <sup>z</sup>2b, stopping calculating, and backing to the step1 of algorithm 1.

ELSE backing to step 3.

ELSE updating initial parameters, baking to step 1.

Step 3: Replacing the two worst individuals in the two optimizations of the current generation, then selecting N individuals by using roulette selection.

Step 4: Manipulating cross mutation to produces a new population. Backing to step 2.

#### 2.3. Simulation and experiment

We select the initial parameters of the algorithm:

△<sup>0</sup> <sup>¼</sup> <sup>0</sup>:05∥U zð Þ<sup>0</sup> <sup>∥</sup>, <sup>ε</sup><sup>1</sup> <sup>¼</sup> <sup>0</sup>:1, <sup>ε</sup><sup>2</sup> <sup>¼</sup> <sup>0</sup>:03, <sup>ε</sup><sup>3</sup> <sup>¼</sup> <sup>0</sup>:05, a <sup>¼</sup> <sup>b</sup> <sup>¼</sup> <sup>0</sup>:5, M <sup>¼</sup> <sup>1</sup>:5, <sup>η</sup><sup>1</sup> <sup>¼</sup> <sup>0</sup>:15, <sup>η</sup><sup>2</sup> <sup>¼</sup> <sup>0</sup>:3, B<sup>0</sup> <sup>¼</sup> <sup>I</sup>2x2, β<sup>1</sup> ¼ 0:35, β<sup>2</sup> ¼ 0:75, β<sup>3</sup> ¼ 1:25, :

N ¼ 20, Pc ¼ 0:99, Pm ¼ 0:05, Tmax ¼ 100, k ¼ 0:1, coding lengthl ¼ 32:vmax ¼ 0:3m=s, t<sup>0</sup> ¼ 3s, z<sup>0</sup> ¼ ð Þ 5vmaxt0=6:0 , k ¼ 1, η ¼ 2, ζ ¼ 0:1, n ¼ 4, m ¼ 1, :

The laser tracker can precisely measure the center point of the coordinate (x, y) when mobile robot is in motion. Then it can send the distance information to the robot controller through a local network and get the location of the robot in the space. The mobile robot is shaped like a cylinder, with 0.6 m in diameter, 12 ultrasonic distance detect sensors with 0.1–6.0 m detect distance, and 32 infrared distance sensors with 0.1–0.8 m detect distance. The angle between two adjacent sensors is 30�. The measurement time is 0.015 s. The initial position of the robot is (1.375, �2.328), and the target position is (1.248, 1.353), the velocity of moving obstacle is Vr = 0.08 m/s, and φ ¼ π=6. The velocity of the mobile robot is V = 0.05 m/s, the moving

Motion Planning for Mobile Robots http://dx.doi.org/10.5772/intechopen.76895 153

The initial parameters of the experiment are k = 1, n = 2, m = 1 η ¼ 2, ζ ¼ 0:1, r<sup>0</sup> ¼ 1m, r<sup>a</sup> ¼ 0:15m. The laser tracker recorded many points of the mobile robot, and the approximated trajectory can be obtained from these points as shown in Figures 4 and 5. And the relationship between the field strength and algorithm implementation time was listed in Table 2. From the experiment, we can see that the algorithm has some advantages compared with those tradi-

We presented an approach of mobile robot trajectory planning under the existence of moving obstacles by using improved artificial potential field method. To further verify the improved artificial potential field method for moving machines effectiveness of robots path planning, the use of the existing conditions, design path planning experiment = experimental scene shown in "laboratory equipment including full orientation of the mobile robot laser tracker digital

direction can be measured in real-time by the digital compass.

Table 1. Magnitude of field of the point at the trajectory.

tional methods. It is faster in implementation and computation.

2.4. Commits

s ¼ 0:3. The speed of mobile obstacles.

$$v\_r = 0.2(0.5) + r \text{ and } (m) \text{/s}\_{\prime\prime}$$

 $\varphi = \pi/2$ ,  $\rho\_0 = 2m$ ,  $\rho\_a = 0.15m$ 

The original point of robots is (0, 0), the target point is (20, 24), and the unit is m (meter).

The simulation results are shown in Figures 4 and 5. It can be seen from simulation, that to combine trust region algorithm and the improved artificial potential field method can optimize mobile robot path planning, and get a better solution to local minima and the unreachable target problem. Under the same conditions, a pure genetic algorithm for solving the sub-goal problem, the average convergence time comes to the second level, and the individual points solvers need more than 30 s, therefore, the proposed algorithm is much fast.

Robot trajectory simulation resulting potential field on a typical point 1 seen from Table 1, the optimal path in the intensity values obtained are shown in table potential field strength decreases rapidly, and the strong field in the times to reach the target point, the basic value to 30 s.

To verify the validity of the path planning algorithm, an experiment was made based on the current conditions as shown in Figures 2 and 3. The equipment used in the experiment include mobile robot, laser tracker sensor, distance detector sensor, digital compass, dynamics obstacles, and still obstacles.


Table 1. Magnitude of field of the point at the trajectory.

The laser tracker can precisely measure the center point of the coordinate (x, y) when mobile robot is in motion. Then it can send the distance information to the robot controller through a local network and get the location of the robot in the space. The mobile robot is shaped like a cylinder, with 0.6 m in diameter, 12 ultrasonic distance detect sensors with 0.1–6.0 m detect distance, and 32 infrared distance sensors with 0.1–0.8 m detect distance. The angle between two adjacent sensors is 30�. The measurement time is 0.015 s. The initial position of the robot is (1.375, �2.328), and the target position is (1.248, 1.353), the velocity of moving obstacle is Vr = 0.08 m/s, and φ ¼ π=6. The velocity of the mobile robot is V = 0.05 m/s, the moving direction can be measured in real-time by the digital compass.

The initial parameters of the experiment are k = 1, n = 2, m = 1 η ¼ 2, ζ ¼ 0:1, r<sup>0</sup> ¼ 1m, r<sup>a</sup> ¼ 0:15m. The laser tracker recorded many points of the mobile robot, and the approximated trajectory can be obtained from these points as shown in Figures 4 and 5. And the relationship between the field strength and algorithm implementation time was listed in Table 2. From the experiment, we can see that the algorithm has some advantages compared with those traditional methods. It is faster in implementation and computation.

#### 2.4. Commits

If

ze

ELSE backing to step 3.

152 Advanced Path Planning for Mobile Entities

2.3. Simulation and experiment

β<sup>1</sup> ¼ 0:35, β<sup>2</sup> ¼ 0:75, β<sup>3</sup> ¼ 1:25, :

vr ¼ 0.2(0.5) + r and (m)/s,

cles, and still obstacles.

s ¼ 0:3. The speed of mobile obstacles.

G zð Þ� <sup>b</sup> G zð Þ<sup>k</sup> <sup>&</sup>gt; <sup>k</sup> and <sup>z</sup> <sup>∈</sup> <sup>Ω</sup>, then ze

Step 3: Replacing the two worst individuals in the two optimizations of the current generation,

△<sup>0</sup> <sup>¼</sup> <sup>0</sup>:05∥U zð Þ<sup>0</sup> <sup>∥</sup>, <sup>ε</sup><sup>1</sup> <sup>¼</sup> <sup>0</sup>:1, <sup>ε</sup><sup>2</sup> <sup>¼</sup> <sup>0</sup>:03, <sup>ε</sup><sup>3</sup> <sup>¼</sup> <sup>0</sup>:05, a <sup>¼</sup> <sup>b</sup> <sup>¼</sup> <sup>0</sup>:5, M <sup>¼</sup> <sup>1</sup>:5, <sup>η</sup><sup>1</sup> <sup>¼</sup> <sup>0</sup>:15, <sup>η</sup><sup>2</sup> <sup>¼</sup> <sup>0</sup>:3, B<sup>0</sup> <sup>¼</sup> <sup>I</sup>2x2,

N ¼ 20, Pc ¼ 0:99, Pm ¼ 0:05, Tmax ¼ 100, k ¼ 0:1, coding lengthl ¼ 32:vmax ¼ 0:3m=s, t<sup>0</sup> ¼ 3s,

φ ¼ π=2, r<sup>0</sup> ¼ 2m, r<sup>a</sup> ¼ 0:15m

The simulation results are shown in Figures 4 and 5. It can be seen from simulation, that to combine trust region algorithm and the improved artificial potential field method can optimize mobile robot path planning, and get a better solution to local minima and the unreachable target problem. Under the same conditions, a pure genetic algorithm for solving the sub-goal problem, the average convergence time comes to the second level, and the individual points

Robot trajectory simulation resulting potential field on a typical point 1 seen from Table 1, the optimal path in the intensity values obtained are shown in table potential field strength decreases

To verify the validity of the path planning algorithm, an experiment was made based on the current conditions as shown in Figures 2 and 3. The equipment used in the experiment include mobile robot, laser tracker sensor, distance detector sensor, digital compass, dynamics obsta-

rapidly, and the strong field in the times to reach the target point, the basic value to 30 s.

The original point of robots is (0, 0), the target point is (20, 24), and the unit is m (meter).

solvers need more than 30 s, therefore, the proposed algorithm is much fast.

Step 4: Manipulating cross mutation to produces a new population. Backing to step 2.

ð Þ <sup>k</sup>þ<sup>1</sup> <sup>2</sup> <sup>≔</sup> <sup>z</sup>2b, stopping calculating, and backing to the step1 of algorithm 1.

ELSE updating initial parameters, baking to step 1.

We select the initial parameters of the algorithm:

z<sup>0</sup> ¼ ð Þ 5vmaxt0=6:0 , k ¼ 1, η ¼ 2, ζ ¼ 0:1, n ¼ 4, m ¼ 1, :

then selecting N individuals by using roulette selection.

ð Þ <sup>k</sup>þ<sup>1</sup> <sup>1</sup> <sup>≔</sup> <sup>z</sup>1b,

We presented an approach of mobile robot trajectory planning under the existence of moving obstacles by using improved artificial potential field method. To further verify the improved artificial potential field method for moving machines effectiveness of robots path planning, the use of the existing conditions, design path planning experiment = experimental scene shown in "laboratory equipment including full orientation of the mobile robot laser tracker digital

Figure 2. The simulation result of improved artificial potential field method.

Figure 4. Trajectory of the mobile robot.

Motion Planning for Mobile Robots http://dx.doi.org/10.5772/intechopen.76895 155

Figure 5. Trajectory planning and simulation (1–3).

Figure 3. Geometric representation for computing target potion.

distance sensor compass static obstacles and dynamic obstacles. Laser tracker can accurately measure mobile robot motion moving the coordinate value of the center point of the process (x, y), through a wireless local area network location information measured in real-time sent to the robot controller, robot positioning cylindrical omnidirectional mobile robot structure with a diameter 0.6 m, and surrounded by 12 uniform ultrasonic ranging.

Figure 4. Trajectory of the mobile robot.

Figure 5. Trajectory planning and simulation (1–3).

distance sensor compass static obstacles and dynamic obstacles. Laser tracker can accurately measure mobile robot motion moving the coordinate value of the center point of the process (x, y), through a wireless local area network location information measured in real-time sent to the robot controller, robot positioning cylindrical omnidirectional mobile robot structure with

a diameter 0.6 m, and surrounded by 12 uniform ultrasonic ranging.

Figure 3. Geometric representation for computing target potion.

Figure 2. The simulation result of improved artificial potential field method.

154 Advanced Path Planning for Mobile Entities


RRT algorithm on ROS platform for mobile robot path planning research and applied to the

The integrity constraints include a system of generalized coordinates derivative and integral constraints. The mobile robot is a typical nonholonomic constraint system. The mobile robot's

Mobile robot's state variables in configuration space <sup>x</sup>; <sup>y</sup>; <sup>θ</sup>; <sup>v</sup>; <sup>ϕ</sup> � �, let ð Þ <sup>x</sup>; <sup>y</sup> represent the center of mobile robot rear axle in the system coordinates, θ represent the angle between the forward direction of the mobile robot and the x-axis, φ represent the angle between the forward direction of the mobile robot and the front wheel direction, v as the speed of the mobile robot, due to the nonholonomic constraints, the wheel is point contact with the ground, and only

> x\_ ¼ v cos θ y\_ ¼ v sin θ <sup>θ</sup>\_ <sup>¼</sup> <sup>v</sup> tan <sup>ϕ</sup>=<sup>L</sup> v\_ ¼ u<sup>0</sup> <sup>ϕ</sup>\_ <sup>¼</sup> <sup>u</sup><sup>1</sup>

(4.1)

8 >>>>><

>>>>>:

v cos θ v sin θ v tan <sup>ϕ</sup>=<sup>L</sup> 0 0

1

1

1

CCCCCCA

u<sup>1</sup> (4.2)

Motion Planning for Mobile Robots http://dx.doi.org/10.5772/intechopen.76895 157

0

BBBBBB@

CCCCCCA u<sup>0</sup> þ

0

BBBBBB@

CCCCCCA þ

motion planning problem in the dense obstacles environment [27].

3.1. Nonholonomic constraints for mobile robot

simplified moving model is shown in the Figure 6.

dx=dt dy=dt <sup>d</sup>θ=dt dv=dt <sup>d</sup>ϕ=dt 1

0

BBBBBB@

CCCCCCA ¼

0

BBBBBB@

pure roll no relative sliding at contact.

And the constraint equations:

Figure 6. Geometrics of the mobile robot.

Table 2. Algorithm implementation time and field strength.
