# Galloping Trajectory Generation of a Legged Transport Robot Based on Energy Consumption Optimization

Authors: Yaguang Zhu; Tong Guo
Journal: Journal of Robotics
 (2016)

---

## Abstract

Legged walking robots have very strong operation ability in the complex surface and they are very suitable for transportation of tools, materials, and equipment in unstructured environment. Aiming at the problems of energy consumption of legged transport robot during the fast moving, a method of galloping trajectory planning based on energy consumption optimization is proposed. By establishing transition angle polynomials of flight phase, lift-off phase, and stance phase and constraint condition between each state phase, the locomotion equations of the ellipse trajectory are derived. The transition angle of each state phase is introduced into the system energy consumption equations, and the energy optimization index based on transition angles is established. Inverse kinematics solution and trajectory planning in one gait cycle are applied to genetic algorithm process to solve the nonlinear programming problem. The results show that the optimized distribution of transition angles of state phases is more reasonable, and joint torques and system energy consumption are reduced effectively. Thus, the method mentioned above has a great significance to realize fast operation outdoors of transport robot.


---

## Body

## 1. Introductions

With the development of technology, upgrading the traditional construction machinery products by using the computer control technology and robot technology and so forth has become an inevitable trend. By increasing the high-tech content, the purposes of improving operational efficiency and work quality and reducing fuel consumption and construction cost can be achieved. Due to the fact that legged robot has much difference with wheeled and tracked robot in structure, which only needs some continuous and discrete foot tip during the motion, it has stronger operation ability in complex environments such as striding over obstacles and passing the rugged, unstructured, or muddy surface easily. That is very suitable for transportation of tools, materials, and equipment in unstructured environment. Because legged robot has the characteristics of multi-DOF, construction complexity, uneasy control, and easy overturn, most of legged robot ensures stability during the motion by keeping contact with surface with one or more legs [1]. However, high-speed operation will be limited by this type of exercise, which caused the fact that primary control strategies currently can only achieve fast gait such as work or jog. The speed can be improved effectively if the foot tip of robot can be made off the ground for a period of time in the process of moving. That will enable the biped robot and multilegged robot to achieve the running in the real meaning [2].In recent years, many researchers have engaged in research on multilegged robot running [3–5]. Kim and Park [6] proposed a leg trajectory that is generated by elliptic motion based on lip joint and keen joint. Meanwhile, the gallop was achieved by analyzing the movement rules of the robot during the contact with surface. More than that, all kinds of galloping styles can also be generated by controlling parameters such as the center of long and short axis of elliptical trajectory and rotation velocity. Step control and force redistribution strategy were proposed by Palmer III and Orin [7], which was used for controlling the locomotion of the robot. That allowed the quadruped to reach a speed of 4.75 m/s and turn at a rate of 20 degrees/s. Focchi et al. [8] were successful in trotting gait with elliptical trajectory generated by Central Pattern Generation (CPG) and controlled by reflection trajectory, which guarantees the stability combined with body posture controller. Marhefka et al. used fuzzy control techniques to control both bounding and galloping of multirobot [9], and they mapped inverse kinematics to a fuzzy system using intuitive heuristics. According to the question that the time of touch-down phase and swing phase are constant within the existing strategy of galloping trajectory, Capi et al. [10] reestablished the polynomial of trajectory and motion state of the robot, which was optimized by using genetic algorithm, and thus the more reasonable galloping trajectory was obtained. Roh and Park [11] proposed a simple and efficient galloping algorithm with elliptic trajectory. This method can effectively keep body height and body balance in the process of gallop based on angular momentum and fuzzy logic control, in which the desired reaction forces are obtained by angular momentum analysis and the fuzzy logic control used for keeping galloping stability.Generally, during the running and galloping, flight phase when the robot is completely off the ground and stance phase when the robot is in contact with the ground are repeated in sequence. During the fight phase, the robot is completely governed by the gravity. This indicated that the motion of the robot in flight is decided by the robot motion during the previous stance phase [12]. Of course, another issue is the energy consumption during the moving process. The battery life is an important indicator when transport robots work outdoors independently. However, energy consumption will increase with the speed increased. So, if we make energy consumption an optimization index, it can be effectively reduced by dividing duration of each state and range of motion reasonably. To sum up, aiming at the problems about energy consumption in trajectory generation of multilegged robot, this paper proposes a galloping trajectory strategy based on energy consumption optimization. By deriving the function of elliptic trajectory, we introduce transition angles of state phase into system energy consumption equation as variables. And this issue of nonlinear planning can be solved by using genetic algorithm. This method enables making energy consumption optimization as objective to divide each state phase reasonably. The advantages of this method are that it can reduce energy consumption effectively and increase cruising ability of robot.


## 2. Modeling and Galloping Trajectory Generation

### 2.1. Transport Robot Model with Mammalian Configuration

Due to the fact that transport robot needs capacity of loading, we adopt a kind of legged structure with mammalian configuration. On the other hand, the advantages of all the legs with the same design in terms of design cost, replacement, modularity, and so on are also considered [13]. Compared with the structure of imitated insects, the workspace and flexibility are reduced, but the advantages of low energy and high load are more suitable for transportation in the construction. The mechanical model of hexapod transport robot is shown in Figure 1. Each of the legs consists of three parts which are connected by a hip joint and a knee joint, and the leg mechanism is attached to the body via a root joint. Each of the legs has three joints, and the length of each link has been optimized [14], which can be regarded as a manipulator with three rotating joints attached to a stationary base (the robotic body). Hence, the establishment of kinematic model and the derivation of the kinematic equation can follow the traditional method of robot technology, which can be obtained by defining the reference coordinate system according to the Denavit–Hartenberg model. The body of the robot is made of aluminum-alloy material, which is used to load controller, power supply module, and load. Material of sponge is adhered to the foot tip, which can effectively prevent skid and reduce the impact force from ground.Figure 1
Mechanical model of hexapod transport robot.


### 2.2. Galloping Trajectory Generation

A cycle of animal movements is divided into two states: [15] swing phase and touch-down phase. But the position and speed during the gallop are the most important factors in effecting robot motion. Here, the swing phase is further divided to two phases: flight phase and lift-off phase. In this paper, we adopt elliptic trajectory to generate galloping trajectory [16]. The entire trajectory is made of three transition angles ψ
f
l
i
g
h
t, ψ
t
o
u
c
h
d
o
w
n, and ψ
l
i
f
t
o
f
f, which represent flight phase, touch-down phase, and and lift-off phase, respectively. ψ
1, ψ
2, and ψ
3 are initial angles of each state, which are shown in Figure 2. Those transition angles are obtained by Hermite interpolation algorithm and interconnected by continuous conditions. Thus, the gallop is achieved.Figure 2
State phases and transition angles of galloping.We assume the whole body of the robot is a mass. According to free fall conditions,(1)
y
=
1
2
g
t
2
,
t
free  fall
=
2
y
g
,in which t
f
r
e
e
f
a
l
l is the time of free fall and y is falling height of body. The condition of ψ
f
l
i
g
h
t section is from contact off to contact on, and the COG velocity is z
˙
C
O
G
<
0. The ellipse angle is determined by a cubic polynomial, and the 4 unknown parameters are determined by the following 4 constraints:(2)
ψ
i
n
i
t
1
=
ψ
1
,
ψ
f
i
n
a
l
1
=
ψ
2
,
ψ
˙
i
n
i
t
1
=
0
,
ψ
˙
f
i
n
a
l
1
=
v
b
o
d
y
b
.
t
f
i
n
a
l
1
=
t
f
r
e
e
f
a
l
lin which ψ
i
n
i
t
i and ψ
f
i
n
a
l
i are origin angle and end angle belonging to i state, ψ
˙
i
n
i
t
i and ψ
˙
f
i
n
a
l
i are origin velocity and end velocity belonging to i state, and v
b
o
d
y is the motion velocity of body. ψ and ψ
˙ of flight state are all determined by cubic polynomial, which is shown in (3). According to the 4 constraints, (4) can be obtained, and then to solve it, (3)
ψ
1
t
=
a
3
t
3
+
a
2
t
2
+
a
1
t
+
a
0
,
ψ
˙
1
t
=
3
a
3
t
2
+
2
a
2
t
+
a
1
,
(4)
t
i
n
i
t
i
a
l
3
t
i
n
i
t
i
a
l
2
t
i
n
i
t
i
a
l
1
1
3
t
i
n
i
t
i
a
l
2
2
t
i
n
i
t
i
a
l
1
0
t
f
i
n
a
l
3
t
f
i
n
a
l
2
t
f
i
n
a
l
1
1
3
t
f
i
n
a
l
2
2
t
f
i
n
a
l
1
0
a
3
a
2
a
1
a
0
=
ψ
i
t
i
n
i
t
i
a
l
ψ
˙
i
t
i
n
i
t
i
a
l
ψ
i
t
f
i
n
a
l
ψ
˙
i
t
f
i
n
a
l
.However, in theψ
t
o
u
c
h
d
o
w
n section, the feet tip of robot is keeping contact with ground all the time, and the COG velocity is z
˙
C
O
G
<
0
→
z
˙
C
O
G
>
0. The ellipse angle is determined by a first-order polynomial, and the 2 unknown parameters are determined by the equation of ψ and ψ
˙ of flight section when the foot first contacts ground: (5)
ψ
2
t
=
ψ
˙
f
i
n
a
l
1
·
t
2
+
ψ
f
i
n
a
l
1
ψ
˙
2
t
=
ψ
˙
f
i
n
a
l
1
t
2
=
t
f
i
n
a
l
.Inψ
l
i
f
t
o
f
f section, the condition of the foot tip of robot is from contact on to contact off, and the COG velocity is z
˙
C
O
G
>
0. The elliptic angle of this section is similar to the section of flight, which is determined by a cubic polynomial, and the 4 unknown parameters are determined by the following 4 constraints:(6)
ψ
i
n
i
t
3
=
ψ
f
i
n
a
l
2
,
ψ
f
i
n
a
l
3
=
ψ
i
n
i
t
i
a
l
1
,
ψ
˙
i
n
i
t
3
=
ψ
˙
f
i
n
a
l
2
,
ψ
˙
f
i
n
a
l
3
=
v
b
o
d
y
b
t
f
i
n
a
l
3
=
t
f
r
e
e
f
a
l
l
.Onψ
f
l
i
g
h
t section, each ψ and ψ
˙ are similar to ψ
l
i
f
t
o
f
f, which are determined by a cubic polynomial equation (7), and it can be solved by following 4 constraints (6):(7)
ψ
3
t
=
a
3
t
3
+
a
2
t
2
+
a
1
t
+
a
0
,
ψ
˙
3
t
=
3
a
3
t
2
+
2
a
2
t
+
a
1
.To sum up, each polynomial based on analysis above can be solved by corresponding constraints one by one, and the whole curve of the ellipse angle is determined, which is shown in Figure3. According to the change of ellipse angle and following ellipse equation, the ideal location of foot tip of the robot leg can be obtained: (8)
X
i
t
=
λ
1
·
cos
⁡
ψ
i
+
x
c
e
n
t
e
r
,
Z
i
t
=
λ
2
·
sin
⁡
ψ
i
+
z
c
e
n
t
e
rin which λ
1 and λ
2 are coefficients of long and short axis of ellipse, X and Z are front direction and vertical direction, and x
c
e
n
t
e
r and z
c
e
n
t
e
r are abscissa and ordinate of elliptical center.Figure 3
Ellipse trajectory of galloping.


## 2.1. Transport Robot Model with Mammalian Configuration

Due to the fact that transport robot needs capacity of loading, we adopt a kind of legged structure with mammalian configuration. On the other hand, the advantages of all the legs with the same design in terms of design cost, replacement, modularity, and so on are also considered [13]. Compared with the structure of imitated insects, the workspace and flexibility are reduced, but the advantages of low energy and high load are more suitable for transportation in the construction. The mechanical model of hexapod transport robot is shown in Figure 1. Each of the legs consists of three parts which are connected by a hip joint and a knee joint, and the leg mechanism is attached to the body via a root joint. Each of the legs has three joints, and the length of each link has been optimized [14], which can be regarded as a manipulator with three rotating joints attached to a stationary base (the robotic body). Hence, the establishment of kinematic model and the derivation of the kinematic equation can follow the traditional method of robot technology, which can be obtained by defining the reference coordinate system according to the Denavit–Hartenberg model. The body of the robot is made of aluminum-alloy material, which is used to load controller, power supply module, and load. Material of sponge is adhered to the foot tip, which can effectively prevent skid and reduce the impact force from ground.Figure 1
Mechanical model of hexapod transport robot.


## 2.2. Galloping Trajectory Generation

A cycle of animal movements is divided into two states: [15] swing phase and touch-down phase. But the position and speed during the gallop are the most important factors in effecting robot motion. Here, the swing phase is further divided to two phases: flight phase and lift-off phase. In this paper, we adopt elliptic trajectory to generate galloping trajectory [16]. The entire trajectory is made of three transition angles ψ
f
l
i
g
h
t, ψ
t
o
u
c
h
d
o
w
n, and ψ
l
i
f
t
o
f
f, which represent flight phase, touch-down phase, and and lift-off phase, respectively. ψ
1, ψ
2, and ψ
3 are initial angles of each state, which are shown in Figure 2. Those transition angles are obtained by Hermite interpolation algorithm and interconnected by continuous conditions. Thus, the gallop is achieved.Figure 2
State phases and transition angles of galloping.We assume the whole body of the robot is a mass. According to free fall conditions,(1)
y
=
1
2
g
t
2
,
t
free  fall
=
2
y
g
,in which t
f
r
e
e
f
a
l
l is the time of free fall and y is falling height of body. The condition of ψ
f
l
i
g
h
t section is from contact off to contact on, and the COG velocity is z
˙
C
O
G
<
0. The ellipse angle is determined by a cubic polynomial, and the 4 unknown parameters are determined by the following 4 constraints:(2)
ψ
i
n
i
t
1
=
ψ
1
,
ψ
f
i
n
a
l
1
=
ψ
2
,
ψ
˙
i
n
i
t
1
=
0
,
ψ
˙
f
i
n
a
l
1
=
v
b
o
d
y
b
.
t
f
i
n
a
l
1
=
t
f
r
e
e
f
a
l
lin which ψ
i
n
i
t
i and ψ
f
i
n
a
l
i are origin angle and end angle belonging to i state, ψ
˙
i
n
i
t
i and ψ
˙
f
i
n
a
l
i are origin velocity and end velocity belonging to i state, and v
b
o
d
y is the motion velocity of body. ψ and ψ
˙ of flight state are all determined by cubic polynomial, which is shown in (3). According to the 4 constraints, (4) can be obtained, and then to solve it, (3)
ψ
1
t
=
a
3
t
3
+
a
2
t
2
+
a
1
t
+
a
0
,
ψ
˙
1
t
=
3
a
3
t
2
+
2
a
2
t
+
a
1
,
(4)
t
i
n
i
t
i
a
l
3
t
i
n
i
t
i
a
l
2
t
i
n
i
t
i
a
l
1
1
3
t
i
n
i
t
i
a
l
2
2
t
i
n
i
t
i
a
l
1
0
t
f
i
n
a
l
3
t
f
i
n
a
l
2
t
f
i
n
a
l
1
1
3
t
f
i
n
a
l
2
2
t
f
i
n
a
l
1
0
a
3
a
2
a
1
a
0
=
ψ
i
t
i
n
i
t
i
a
l
ψ
˙
i
t
i
n
i
t
i
a
l
ψ
i
t
f
i
n
a
l
ψ
˙
i
t
f
i
n
a
l
.However, in theψ
t
o
u
c
h
d
o
w
n section, the feet tip of robot is keeping contact with ground all the time, and the COG velocity is z
˙
C
O
G
<
0
→
z
˙
C
O
G
>
0. The ellipse angle is determined by a first-order polynomial, and the 2 unknown parameters are determined by the equation of ψ and ψ
˙ of flight section when the foot first contacts ground: (5)
ψ
2
t
=
ψ
˙
f
i
n
a
l
1
·
t
2
+
ψ
f
i
n
a
l
1
ψ
˙
2
t
=
ψ
˙
f
i
n
a
l
1
t
2
=
t
f
i
n
a
l
.Inψ
l
i
f
t
o
f
f section, the condition of the foot tip of robot is from contact on to contact off, and the COG velocity is z
˙
C
O
G
>
0. The elliptic angle of this section is similar to the section of flight, which is determined by a cubic polynomial, and the 4 unknown parameters are determined by the following 4 constraints:(6)
ψ
i
n
i
t
3
=
ψ
f
i
n
a
l
2
,
ψ
f
i
n
a
l
3
=
ψ
i
n
i
t
i
a
l
1
,
ψ
˙
i
n
i
t
3
=
ψ
˙
f
i
n
a
l
2
,
ψ
˙
f
i
n
a
l
3
=
v
b
o
d
y
b
t
f
i
n
a
l
3
=
t
f
r
e
e
f
a
l
l
.Onψ
f
l
i
g
h
t section, each ψ and ψ
˙ are similar to ψ
l
i
f
t
o
f
f, which are determined by a cubic polynomial equation (7), and it can be solved by following 4 constraints (6):(7)
ψ
3
t
=
a
3
t
3
+
a
2
t
2
+
a
1
t
+
a
0
,
ψ
˙
3
t
=
3
a
3
t
2
+
2
a
2
t
+
a
1
.To sum up, each polynomial based on analysis above can be solved by corresponding constraints one by one, and the whole curve of the ellipse angle is determined, which is shown in Figure3. According to the change of ellipse angle and following ellipse equation, the ideal location of foot tip of the robot leg can be obtained: (8)
X
i
t
=
λ
1
·
cos
⁡
ψ
i
+
x
c
e
n
t
e
r
,
Z
i
t
=
λ
2
·
sin
⁡
ψ
i
+
z
c
e
n
t
e
rin which λ
1 and λ
2 are coefficients of long and short axis of ellipse, X and Z are front direction and vertical direction, and x
c
e
n
t
e
r and z
c
e
n
t
e
r are abscissa and ordinate of elliptical center.Figure 3
Ellipse trajectory of galloping.


## 3. Energy Consumption Model during Gallop

It can be known from above that as long asψ
1, ψ
2, and ψ
3 are determined, the whole galloping trajectory is determined too. Taking the condition of transport robots working outdoors independently into consideration, energy consumption is a very important factor. So, in this paper, energy consumption is made as the optimization index, and the degrees of each state are determined by minimum system. Firstly, we establish the model of energy consumption during the gallop.In this paper, the rotation of each joint of the robot leg is realized by brushless DC motor and reducer with corresponding motor controller. The related functions can be expressed as below [13].Voltage balance function of DC motor is(9)
U
a
t
=
R
a
i
a
t
+
L
a
d
i
a
t
d
t
+
U
g
t
,
U
g
t
=
k
E
θ
˙
m
t
.According to static characteristics of DC motor, we have(10)
τ
M
t
=
k
M
i
a
t
.For output torque of motor and joint driving torque, we have(11)
τ
M
t
=
τ
t
η
N
G
.According to speed of motor and joint speed, we obtain(12)
θ
˙
m
t
=
N
G
θ
˙
tin which k
E is the EMF constant, k
M is the torque constant, R
a and L
a are the linear resistance and inductance, U
g and i
a are the armature voltage and current, N
G is the gear reduction ratio, θ
˙
m
(
t
) and θ
˙
(
t
) are the speed of motor and joint, η is the motor and gear efficiency, τ
M
(
t
) is the electromagnetic torque, and τ
(
t
) is the joint torque.The instantaneous power of the motor is(13)
P
a
t
=
U
a
t
i
a
t
.Substituting (9) into (13), we obtain(14)
P
a
t
=
R
a
i
a
2
t
+
L
a
i
a
t
d
i
a
t
d
t
+
k
E
i
a
t
θ
˙
m
t
.Substituting (10) and (11) into (14), we obtain(15)
P
a
t
=
R
a
k
M
2
η
2
N
G
2
τ
2
t
+
θ
˙
t
τ
t
ηin which inductance L
a is the energy storage element and if the energy is invariant from the beginning to the end of a period, the energy loss of the inductance can be neglected. Then, the instantaneous power of the motor becomes [16] (16)
P
a
t
=
R
a
k
M
2
η
2
N
G
2
τ
2
t
+
Λ
θ
˙
t
τ
t
η
.Taking all joints in the leg into consideration, the total energy of a single leg in a walking gait cycle becomes [16](17)
E
p
=
∫
0
T
∑
j
=
1
3
Λ
θ
˙
j
t
τ
j
t
η
+
R
a
k
M
2
η
2
N
G
2
τ
j
2
t
d
tin which Λ
x is defined as(18)
Λ
x
=
x
i
f
x
>
0
0
i
f
x
≤
0
.Here,T is cycle time, the duration of each leg’s one step cycle in a periodic gait. R
a, k
M, η, and N
G are constant. The value of θ
˙
j
(
t
) depends on the trajectory planning. τ
j
(
t
) is derived by Lagrange’s dynamical equations.Since the transition angles of each state phaseψ
1, ψ
2, ψ
3 are the parameters to be optimized and they determine the final leg trajectory and ideal location of foot tip P
F
B, according to the inverse kinematics function, we have(19)
q
=
θ
1
P
ψ
-
B
,
t
θ
2
P
ψ
-
B
,
t
θ
3
P
ψ
-
B
,
t
T
∈
R
3
×
1in which(20)
P
B
=
P
x
B
P
y
B
P
z
B
T
,
ψ
-
=
ψ
1
ψ
2
ψ
3
.Usually,P
F
B is the position vector of foot tip in the leg reference frame and is given by the gait and trajectory planning. Joint angle, joint angular speed, and angular acceleration can be solved by inverse kinematics function, expressed as(21)
q
ψ
-
,
t
=
θ
1
ψ
-
,
t
θ
2
ψ
-
,
t
θ
3
ψ
-
,
t
,
q
˙
ψ
-
,
t
=
θ
˙
1
ψ
-
,
t
θ
˙
2
ψ
-
,
t
θ
˙
3
ψ
-
,
t
,
q
¨
ψ
-
,
t
=
θ
¨
1
ψ
-
,
t
θ
¨
2
ψ
-
,
t
θ
¨
3
ψ
-
,
t
.Joint angle, joint angular speed, and angular acceleration can be calculated through the initial and final state of the locomotion trajectory, but it is still a function with reference toψ
- and t.For the joint torque,(22)
τ
^
p
ψ
-
,
t
=
D
^
·
q
¨
ψ
-
,
t
+
H
^
·
q
˙
ψ
-
,
t
+
C
^
,
(23)
D
^
=
0
0
0
0
1
4
m
2
L
2
2
+
I
2
+
m
3
L
3
2
1
2
m
3
L
2
L
3
C
3
/
2
0
1
2
m
3
L
2
L
3
C
3
/
2
1
4
m
3
L
3
2
+
I
3
,
(24)
H
^
=
0
-
1
4
m
3
L
2
L
3
θ
˙
3
2
S
3
/
2
-
1
2
m
3
L
2
L
3
θ
˙
2
θ
˙
3
S
3
/
2
,
(25)
C
^
=
-
1
2
m
2
g
L
2
C
2
S
1
-
m
3
g
L
2
C
2
S
1
-
1
2
m
3
g
L
3
C
23
S
1
-
1
2
m
2
g
L
2
C
1
S
2
-
m
3
g
L
2
C
1
S
2
-
1
2
m
3
g
L
3
C
1
S
23
-
1
2
m
3
g
L
3
C
1
S
23
.D
^ is a symmetric matrix associated with acceleration, which involves the inertia tensor, H
^ is the Coriolis and centrifugal matrix, and C
^ is the gravity matrix. According to the derivation above, they are all the function with ψ
- and t. Hence, the power equation (16) and total energy consumption equation (17) of the three-jointed leg system can be formulated as(26)
P
˘
ψ
-
,
t
=
∑
j
=
1
3
P
j
ψ
-
,
t
=
τ
^
p
ψ
-
,
t
T
Q
^
p
τ
^
p
ψ
-
,
t
+
max
⁡
c
p
T
τ
^
p
ψ
-
,
t
,
0
,
(27)
E
˘
p
ψ
-
=
∫
0
T
P
ψ
-
,
t
d
t
=
∫
0
T
∑
j
=
1
3
P
j
ψ
-
,
t
d
t
=
∫
0
T
τ
^
p
ψ
-
,
t
T
Q
^
p
τ
^
p
ψ
-
,
t
+
max
⁡
c
^
p
T
τ
^
p
ψ
-
,
t
,
0
d
t
,where Q
^
p
=
2
R
/
k
M
2
η
2
N
G
2
E
3 and E
3 is a unit matrix of 3 by 3,(28)
c
^
p
=
q
˙
1
ψ
-
,
t
η
q
˙
2
ψ
-
,
t
η
q
˙
3
ψ
-
,
t
η
T
.It can be known from above that the whole galloping trajectory can be determined as long asψ
1, ψ
2, and ψ
3 are confirmed. According to the inverse kinematics function, the rotation trajectory of each joint can be determined, and then the joint torque and systematic energy consumption are obtained. The function with ψ
- and t can be obtained by substituting unknown transition angles of state phase into systematic energy consumption equation. ψ
1, ψ
2, and ψ
3 can be confirmed as long as the minimum energy consumption is obtained by optimization. Based on the equation above, taking the energy consumption as optimization index, the state phase transition angle that is meeting the requirement of energy consumption can be obtained by solving (29) which means that the galloping trajectory planning is completed:(29)
O
b
j
e
c
t
F
u
n
c
t
i
o
n
⟶
m
i
n
ψ
-
ϑ
=
∫
0
t
E
˘
p
ψ
-
(30)
S
u
b
j
e
c
t
t
o
C
ψ
-
≤
0
,where C
ψ
-
≤
0 is initial range of state phase transition angle.


## 4. Optimal Solutions with Genetic Algorithm

The genetic algorithm is used for solving this optimization problem that has linear or nonlinear constrained condition. It is based on natural selection, solved by biological evolution. It is widely used in various optimization problems [17]. This method is described in Figure 4.(1)
Firstly, we create a population with n size and then select the new state phase transition angle within the constraint and then put it into the galloping trajectory planner to calculate. In one gait period, the running speed and stride sizes are all determined by gait generator [16]. For each individual, the initial position and the end position are obtained by the trajectory planning. According to the initial and end condition, the initial and end joint angle, angular speed, and angular accelerations are obtained by inverse kinematics equation (21).
(2)
Joint torques can be obtained by joint angle, angular speed, angular accelerations, and (22), and total energy consumption in one gait period can be obtained by (27). According to constraint condition, punishment factors are given and fitness is valuated. The inferior individuals are removed and superior individual with a number of m are to be saved directly to the next generation.
(3)
If the stopping criterion is satisfied after evaluating the fitness, the optimization will stop. Otherwise, the new individual will be produced continually and create the next generation by selection, rebuilding, cross, and variation. During the optimization, cross probability isξ, variation probability is β, and penalty factor is ρ. And the variation of mean fitness is shown in Figure 5.Figure 4
Process of GA.Figure 5
Variation of mean fitness.In this paper, the parameters of the system are shown in Table1. The length of each joint is L
1, L
2, L
3, the long and short axis coefficient are λ
1, λ
2, and the boundary of the variations is(31)
0
π
2
3
π
2
≤
ψ
-
≤
π
2
π
2
π
.Table 1
Parameters of system.
Parameters
Value
Unit
k
E
13.4
mNm/A
R
a
9.17
Ω
L
a
0.219
mH
N
G
200
—
L
1
0.05
m
L
2
0.103
m
L
3
0.142
m
λ
1
0.08
—
λ
2
0.025
—
η
0.7
—
k
M
13.4
mV/rpm
I
2
0.375
10−3 Kg⋅m2
I
3
0.375
10−3 Kg⋅m2
n
20
—
m
4
—
ξ
0.6
—
β
0.2
—
ρ
2
—The change of the final fitness is smaller than 10−6 after 51 generations, and the final result is shown below:(32)
ψ
-
=
0.754
2.23
4.737
.


## 5. Simulations

In this section, the ellipse trajectory planning during the gallop is simulated by the software Matlab/Simulink. It is assume that the robot goes forward along thex-axis on the horizontal ground, the length of a single step is 0.15 m in a cycle time, the relative height between the root joint of leg and ground is 0.17 m, and the gait cycle is 2 s. Galloping trajectory is generated based on the parameters of D-H model established in Matlab/Simulink, which is shown in Table 1.According to the original and optimized state phase transition angle and galloping trajectory of the robot, the objective trajectory of foot tip based on root-joint coordinate is generated, which is shown in Figure6. The parameter ψ
~
=
0.628
2.512
5.338 is used before the optimization, which is obtained by deriving the ideal step-size. Compared with the optimized parameter ψ
-, both of the ellipse trajectories are similar basically and just differ in the division of each phase. After optimization, the angles of each state phase have a significant difference, flight phase increased, touch-down phase decreased, and the range of each state phase changed, which can be seen from the hip and knee joint movement angle (Figure 7). In the range of 0 to 0.65 seconds approximately, regardless of the optimization or not, the joint angles are all in the lift-off phase and unchanged basically. However, when near to 0.56 seconds, the optimized trajectory begins to transform with large degree and enters into the touch-down phase early. After that, the optimized trajectory becomes bigger than initial. It makes the distribution of work more reasonable during the motion.Figure 6
Ellipse trajectory of galloping.Figure 7
Variations of joint angles.
(a)
Hip joint
(b)
Knee jointThe torques curves of each joint are shown in Figure8. Joint torque almost does not work in 0–0.65 s, which is corresponding with the locomotion. The main reason it is that the left-off phase is free fall and do not need to overcome its own weight to do work. However, after 0.65 s, the optimized trajectory enters the flight phase firstly and do more work than before. But in the lift-off phase later, the work of optimized trajectory regardless of hip joint and knee joint is much smaller than nonoptimization.Figure 8
Variations of joint torques.
(a)
Hip joint
(b)
Knee jointThis is also reflected in the system energy consumption during the movement, which is shown in Figure9. Due to the existing of touch-down phase, the system energy consumption of optimized trajectory is larger than nonoptimization during 0.65 seconds to 1.5 seconds. But the optimized parameters are more reasonable obviously in the lift-off phase. After optimization, the system energy consumption is reduced by 23% in one gait period. It illustrates that the method of trajectory planning based on energy consumption optimization is reasonable and effective.Figure 9
System energy consumption of galloping.


## 6. Conclusions

In this paper, using the existed transportation robot model, the ideal foot tip function based on elliptic galloping trajectory is derived by establishing the transition angles of flight phase, touch-down phase, and lift-off phase and constraint condition of each phase. On the basic analysis of the robot leg dynamics and the joint drive motor model, the formula of joint torque is derived, system energy consumption model is established, and it introduces the galloping state phase transition angle of the robot into that model, and then the index of energy consumption optimization is proposed based on the transition angle. The genetic algorithm is used for optimization. We introduce the inverse kinematics solution and the galloping trajectory planning into the process of the genetic algorithm to solve. The result shows that the distribution of optimized state phase transition angles is more reasonable, and it can reduce joint torque and energy consumption effectively. Although the algorithm proposed in this paper is based on the motor model, it can also apply to other legged robots.



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*Source: 9645730-2016-07-05.xml*