Backstepping based Saturated Sliding Mode Joints ...
Transcript of Backstepping based Saturated Sliding Mode Joints ...
International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:20 No:06 97
202406-5959-IJMME-IJENS ยฉ December 2020 IJENS I J E N S
Backstepping based Saturated Sliding Mode Joints
Controller for Humanoid Walking Gait
Abstractโ Development of a joint controller for humanoid
robot is a challenging task since the dynamic system of the
humanoid robot has nonlinearity characteristic and is extremely
vulnerable to disturbance. This paper presents a robust back-
stepping sliding mode controller to track the pre-computed joint
trajectories in a commercial humanoid robot known as UXA-90.
The proposed controller utilizes the back-stepping technique to
decompose the dynamic equation of UXA-90 into reduced order
systems which is asymptotically stable, then the control law for
back-stepping SMC is chosen to guarantee that the systemโs
states are driven to the sliding manifold and stay there for all
future time. Furthermore, the control law utilizes saturation
function instead of sign function to reduce the chattering effect.
Finally, some simulation results are carried out through
MATLAB to evaluate the effectiveness of the proposed
controller in comparison to the previous controller.
Index Termโ Sliding mode controller, biped walking, back-
stepping, humanoid robot.
I. INTRODUCTION Human locomotion has been one of the most sophisticated
topics to scientists in different fields: biologists, physiologists, mathematicians and engineers [1, 2, 3]. Therefore, it is a critical task for humanoid robots to imitate human locomotion because unlike human, they are not equipped with visual system to recognize object, soft tissues in the leg to reduce impact force, the brain to maintain balance. Despite many efforts in performing function like biped walking, the problem has yet been solved satisfactorily.
Traditionally, there are two levels of controller when controlling a humanoid robot: the high-level walking pattern controller and the low-level joint controller [4]. The high-level controller ensures that the walking pattern is dynamically stable. Kajita et al. [5] proposed the preview control to modify COM trajectory while walking. Jun-Ho Oh et al. [6] utilized two high-level controllers: Landing Position Controller and Landing Timing Controller to adjust the walking trajectory. The low-level controller is then used to track the pre-computed joint trajectories, which are calculated by inverse kinematic. Various methods in designing the joint controller have been proposed in existing literatures. To track the pre-computed trajectories, Jun Ho Oh et al. [6] from Korea Advanced Institute of Science and Technology used PD controller to control position of each joint of KHR-3. CHARLI-2 from Virginia Tech utilized differential controller, based on the IMUโs feedback data to control the jointsโ positions [7]. Park and Kim successfully implemented the computed torque controller (CTC) to control a 7-link biped robot [8]. However, the dynamic of humanoid robot has many non-linear terms, uncertainty parameters, which degrade the performance of CTC and classic PID over time [9]. To avoid
this problem, it is essential to adopt an advance control strategy to control the joints such as robust control [10, 11]. The SMC is one of the suitable controllers for controlling humanoid robotโs dynamic system because it can guarantee robustness for system with strong uncertainty parameters, nonlinearity characteristics and even rejects disturbance [12, 13, 14]. Raibert et al. [15] has compared three controller schemes implemented on the 5-link planar biped robot: SMC, PID, CTC and the results showed that SMC has far smaller average tracking error and overshoot than the other two: PID and CTC. Moosavian et al. [16] modeled a 6 DoFs lower body and used a fuzzy regulated SMC on it to reduce the chattering effect, the results showed that the tracking error were considerably small even with major uncertainties. In-Gyu Park and Jin-Geol Kim used SMC to control each leg of a robot, the simulation results showed that the trajectories tracking was done very well [17].
In our previous study, the low-level joint controller for humanoid robot UXA-90 was proposed using sliding mode controller (SMC) to calculate the torque required for each joint. Although the SMC outperform the traditional controllers in tracking the pre-computed trajectories of the 12 joints, the control input of SMC is significantly suffered by the chattering phenomenon. Additionally, the pre-computed joint trajectories introduced peak and valley values at certain points, where the SMC was not able to drive the tracking error to zero. This problem can be realized by increasing the controller gain, however, it causes the chattering phenomenon worse. To overcome the aforementioned problems, an improved SMC should be implemented. The dynamic system of the humanoid robot is very complex. Therefore, the back-stepping technique provides an effective way to decompose the complex system into reduced order subsystems, then the SMC can guarantee their robustness despite the uncertainty parameters and external disturbances. As mentioned above, the SMC has proven its robustness against model uncertainty parameters, non-linear terms. However, the main drawback of the SMC is the use of the sign function, leading to the undesirable chattering phenomenon. This phenomenon reduces the performance of the system, damaging the mechanical parts in the long-time operation.
In this paper, a back-stepping SMC is proposed to control the joints in humanoid robot UXA-90. First, the dynamic system of the lower body is derived using Euler-Lagrange method. Secondly, the back-stepping method is employed to decompose the aforementioned dynamic system into subsystems, then the control law based on SMC is proposed to drive the subsystems to the desired sliding manifold. To reduce the chattering effect, the saturation function is adopted in the control law instead of the sign function. Finally, the
Xuan Tien Nguyen2,3, Tri Duc Tran1 , Huy Hung Nguyen1,3,* , Nhut Phuong Tong1 ,Thanh Phuong Nguyen2 and
Tan Tien Nguyen1,* 1National Key Laboratory of Digital Control and System Engineering (DCSELAB), HCMUT, VNU-HCM
2 Hutech Institute of Engineering, Hochiminh City University of Technology (HUTECH) 3Faculty of Electronics and Telecommunication, Saigon University, Vietnam
*Corresponding author: [email protected], [email protected]
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simulation is carried out through MATLAB to evaluate its effectiveness.
II. DYNAMIC MODEL
Table I
UXA-90 jointsโ constraints
๐ Constraint ๐ Constraint
๐ฝ๐ โ20ยฐ~20ยฐ ๐ฝ๐ โ40ยฐ~50ยฐ
๐ฝ๐ โ20ยฐ~50ยฐ ๐ฝ๐ โ120ยฐ~30ยฐ
๐ฝ๐ โ150ยฐ~0ยฐ ๐ฝ๐ โ40ยฐ~0ยฐ
๐ฝ๐ โ30ยฐ~120ยฐ ๐ฝ๐๐ 0ยฐ~150ยฐ
๐ฝ๐ 0ยฐ~40ยฐ ๐ฝ๐๐ โ50ยฐ~20ยฐ
๐ฝ๐ โ40ยฐ~50ยฐ ๐ฝ๐๐ โ20ยฐ~20ยฐ
UXA-90 has 6 DOFs leg, a total of 12 DOFs for its lower body. Each joint is actuated by DC servo motor from Robobuilder. Normally, the humanoid robot has three phases when running: single support phase, biped-in-the-air phase, double support phase (impact phase). Since UXA-90 currently has only a straight walking function, it has a total of two phases: single support phase, double support phase. To imitate a humanโs joints precisely, twelve joints in the lower body of UXA-90 are constrained according to the motion range in humanโs body [18] as shown in Table I.
By assigning coordinate frame as shown in Fig. 1, a 12 DOFs dynamic model can be derived by using Euler-Lagrange method. The Denavit-Hatenberg parameters and transformation matrixes are taken from [19].
0z1
0y1z2
1y
2z
2y
0x
1x
2x
3x
3z
3y
4z
4x
4y5y
5z
5x
3
456z
6x
6y7z7x
7y8x
8z
8y
9z
9y
9x
10
10z
10y
10x11
11z12
11y11x
8
7
6
9
12z12x
12y
fx fy
fz
, rollx , pitchy
, yawz
Fig. 1. Coordinates system of lower body
A. Swing phase model
This phase occurs when only one foot is in contact with
the ground as shown in Fig. 2, so the whole lower body can
be considered as a 12 DOFs robot manipulator.
By applying Euler-Lagrange method, the dynamic
model for a manipulator is given by [20]:
๐ซ(๐ฝ)๏ฟฝฬ๏ฟฝ + ๐ช(๐ฝ, ๏ฟฝฬ๏ฟฝ)๏ฟฝฬ๏ฟฝ + ๐(๐ฝ) = ๐ where
๐ฝ = [๐1, ๐2, โฆ , ๐12] โ โ12ร1 is the vector of joints.
๐ซ(๐ฝ) โ โ12ร12 is the inertia matrix.
๐ช(๐ฝ, ๏ฟฝฬ๏ฟฝ ) โ โ12ร12 is the Coriolis and centrifugal force matrix.
๐(๐ฝ) โ โ12ร1 is the vector of gravitational forces.
๐ = [๐1, ๐2, โฆ , ๐12] โ โ12ร1 is the vector of joint torques.
2m
3m
4m
5m
fm
hm
O x
z
Swing leg
Stance leg
Fig. 2. Swing phase
B. Impact phase model
This phase is activated when the swing leg comes to
contact with the ground as shown in Fig. 3, the duration of
this process is assumed to be infinitesimal time interval.
During this process, external forces acting on the robot appear
due to the contact made by the swing leg and the ground. The dynamic equation in this case is:
๐ซ(๐ฝ)๏ฟฝฬ๏ฟฝ + ๐ช(๐ฝ, ๏ฟฝฬ๏ฟฝ)๏ฟฝฬ๏ฟฝ + ๐(๐ฝ)
= ๐ + ๐น๐ญ๐๐๐
where
๐ฝ = [๐1, ๐2, โฆ , ๐12] โ โ12ร1 is the vector of joints.
๐ซ(๐ฝ) โ โ12ร12 is the inertia matrix.
๐ช(๐ฝ, ๏ฟฝฬ๏ฟฝ ) โ โ12ร12 is the Coriolis and centrifugal force matrix.
๐(๐ฝ) โ โ12ร1 is the vector of gravitational forces.
๐ = [๐1, ๐2, โฆ , ๐12] โ โ12ร1 is the vector of joint torques.
๐น๐ญ๐๐๐ โ โ12ร1 is the impulsive external forces vector acting on the robot.
extF
2m
3m 4m
5m
fm
O x
z
Fig. 3. Impact phase
By taking the integration of Eq. (2) over [๐ก, ๐ก + โ๐ก] ,
gives:
limโ๐กโ0
โซ ๐ซ(๐ฝ)
๐ก+โ๐ก
๐ก
๏ฟฝฬ๏ฟฝ๐๐ก + limโ๐กโ0
โซ [๐ช(๐ฝ, ๏ฟฝฬ๏ฟฝ)๏ฟฝฬ๏ฟฝ + ๐(๐ฝ) โ ๐]๐๐ก
๐ก+โ๐ก
๐ก
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The integration of ๐(๐ฝ) and ๐ over [๐ก, ๐ก + โ๐ก] are zero,
we have:
limโ๐กโ0
๐ซ(๐ฝ)(๏ฟฝฬ๏ฟฝ๐+โ๐ โ ๏ฟฝฬ๏ฟฝ๐) + limโ๐กโ0
๐ช(๐ฝ, ๏ฟฝฬ๏ฟฝ)(๐ฝ๐+โ๐ โ ๐ฝ๐)
= lim
โ๐กโ0โซ ๐น๐ญ๐๐๐
๐ก+โ๐ก
๐ก
๐๐ก
Since the contact duration is infinitesimal, the joint
positions remain unchanged, therefore:
limโ๐กโ0
๐ช(๐ฝ, ๏ฟฝฬ๏ฟฝ)(๐ฝ๐+โ๐ โ ๐ฝ๐) = 0
Then the final result when integrating Eq. (2) is:
๐ซ(๐ฝ)โ๏ฟฝฬ๏ฟฝ = ๐ญ๐๐๐ The differential of the joint angles โ๏ฟฝฬ๏ฟฝ can be calculated
using the following equation [9]:
โ๏ฟฝฬ๏ฟฝ = ๐ซโ๐๐ฑ๐ป(๐ฑ๐ซโ๐๐ฑ๐ป)โ๐โ๏ฟฝฬ๏ฟฝ where
โ๏ฟฝฬ๏ฟฝ is the velocity of the foot just before the
contact with the ground.
๐ฑ is the Jacobian matrix.
III. SLIDING MODE CONTROLLER DESIGN
To realize the biped walking process, pre-computed walking trajectories with constraints for the robot should be given first. Then by using inverse kinematic, a set of joint trajectories is deduced. Finally, a suitable controller is used to make the joints follow the pre-computed trajectories. The basic work flow to realize biped walking is shown in Fig. 4.
Inverse
Kinematic
Constraints
SMC SystemForward
Kinematic
referror u
+
-
realCOM
refCOM
Fig. 4. Simulation workflow
To implement the sliding mode controller, the dynamic model (2) should be expressed in the compact form:
๏ฟฝฬ๏ฟฝ = ๐(๐, ๐ก) + ๐(๐, ๐ก)๐ + ๐ (๐ก) where
๐ is the system state.
๐(๐, ๐ก), ๐(๐, ๐ก) are known non-linear functions.
๐ โ โ12ร1 is control inputs vector.
๐ (๐ก) โ โ12ร1 is external disturbances vector which is assumed to be bounded.
โ๐ (๐ก)โโ โค ๐ท๐
where ๐ท๐ is positive scalar constant
By defining ๐ฟ๐(๐ฝ) = [๐1, โฆ , ๐12]๐ , ๐ฟ๐(๏ฟฝฬ๏ฟฝ) =
[๏ฟฝฬ๏ฟฝ1, โฆ , ๏ฟฝฬ๏ฟฝ12]๐
, ๐ = [๐1, โฆ , ๐12]๐.
Eq. (2) can be rewritten by:
{
๏ฟฝฬ๏ฟฝ๐ = ๐ฟ๐
๏ฟฝฬ๏ฟฝ๐ = ๐ซโ๐(โ๐ช๐ฟ๐ โ ๐ฎ + ๐ + ๐น๐ญ๐๐๐)
By defining ๐(๐ฟ) = โ๐ซโ๐๐ช๐ฟ๐ โ ๐ซโ๐๐ฎ, ๐(๐ฟ) = ๐ซโ๐ ,
๐ (๐ก) = ๐ซโ๐๐น๐ญ๐๐๐, Eq. (10) can be summarized as follows:
{
๏ฟฝฬ๏ฟฝ๐ = ๐ฟ๐
๏ฟฝฬ๏ฟฝ๐ = ๐(๐ฟ) + ๐(๐ฟ)๐ + ๐ (๐ก)
Defines a joint trajectories vector as ๐ฝ๐ =[๐๐1, โฆ , ๐๐12]๐ , the difference between the current joint angles and the desired angle is defined as the error:
๐๐ = ๐ฟ๐ โ ๐ฝ๐
Step 1: The first time derivative of ๐๐ can be obtained as follow:
๏ฟฝฬ๏ฟฝ๐ = ๏ฟฝฬ๏ฟฝ๐ โ ๏ฟฝฬ๏ฟฝ๐
The candidate Lyapunov function (CLF) for considering the system of Eq. (13) is chosen as:
๐1 =1
2๐๐
๐ป๐๐
Considering Eq. (14) and taking the first time derivative
of ๐1, it yields :
๏ฟฝฬ๏ฟฝ1 = ๐๐๐ป๏ฟฝฬ๏ฟฝ๐ = ๐๐
๐ป(๐ฟ๐ โ ๏ฟฝฬ๏ฟฝ๐ )
Defines a vector ๐บ = [๐ 1, โฆ , ๐ 12]๐ as the sliding surfaces which is given in the following form
๐บ = ๐ฟ๐ + ๐๐๐๐ โ ๏ฟฝฬ๏ฟฝ๐
where ๐๐ โ โ12ร12 is a constant symmetric
positive-definite matrix
Substituting Eq. (16) into Eq. (15), it can be
expressed as:
๏ฟฝฬ๏ฟฝ1 = ๐๐๐ป๐บ โ ๐๐
๐ป๐๐๐๐
It can be observed that ๐๐ and ๐1 are bounded
when the following condition holds true : ๐บ โ ๐.
Step 2 : The augmented CLF for considering the
dynamic of sliding surface ๐บ is chosen as:
๐2 = ๐1 +1
2๐บ๐ป๐บ
Considering Eq. (18) and taking the first derivative of Eq. (16) gives:
๏ฟฝฬ๏ฟฝ = ๏ฟฝฬ๏ฟฝ๐ + ๐๐๏ฟฝฬ๏ฟฝ๐ โ ๏ฟฝฬ๏ฟฝ๐
Substituting the 2nd system of Eq. (11) into Eq. (19), it yields:
๏ฟฝฬ๏ฟฝ = ๐(๐ฟ) + ๐(๐ฟ)๐ + ๐ (๐ก) + ๐๐๏ฟฝฬ๏ฟฝ๐ โ ๏ฟฝฬ๏ฟฝ๐
Taking the first derivative of Eq. (18), it yields:
๏ฟฝฬ๏ฟฝ2 = ๏ฟฝฬ๏ฟฝ1 + ๐บ๐ป๏ฟฝฬ๏ฟฝ
= ๐๐๐ป๐บ โ ๐๐
๐ป๐๐๐๐ + ๐บ๐ป[๐(๐ฟ) + ๐(๐ฟ)๐
+ ๐ (๐ก) + ๐๐๏ฟฝฬ๏ฟฝ๐ โ ๏ฟฝฬ๏ฟฝ๐ ]
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In order to restrain the chattering phenomenon, the saturation function ๐ ๐๐ก is adopted instead of the ๐ ๐๐ . Therefore, the control input designed for system of Eq. (11) is as followed:
๐ = ๐(๐ฟ)โ๐[โ๐(๐ฟ) โ ๐๐๐บ โ ๐๐ โ ๐๐๏ฟฝฬ๏ฟฝ๐ + ๏ฟฝฬ๏ฟฝ๐
โ ๐ผ๐ ๐๐ก(๐บ)]
where ๐๐ โ โ12ร12 is a constant symmetric positive-definite matrix.
๐ผ โ โ12ร12 is a constant diagonal matrix.
Each element in the main diagonal of ๐ผ should satisfy the following:
๐๐ > ๐ท๐
where ๐ = 1 รท 12.
๐ ๐๐ก(โ) is the saturation function defined as:
๐ ๐๐ก(โ) = {
1, ๐๐ โ > โ
๐ ร โ , ๐๐ | โ | โค โ, ๐ = 1/โโ1, ๐๐ โ < โโ
where 0 < โ< 1 is the boundary layer constant.
Substituting Eq. (22) into Eq. (21), gives:
๏ฟฝฬ๏ฟฝ2 = โ๐๐๐ป๐๐๐๐ โ ๐บ๐ป๐๐๐บ + ๐บ๐ป๐ (๐ก)
โ ๐บ๐ป๐ผ๐ ๐๐ก(๐บ)
Since ๐๐, ๐๐ are constant symmetric positive-definite matrix, let us define:
๐ถ1 โ โ๐๐โโ โฅ 0, ๐ถ2 โ โ๐๐โโ โฅ 0, ๐ท๐ โ ๐๐๐๐ฅ
Taking the infinity norm of ๏ฟฝฬ๏ฟฝ2, it yields:
๏ฟฝฬ๏ฟฝ2 = โ๐ถ1โ๐๐โโ2 โ ๐ถ2โ๐บโโ
2 + โ๐บโโ๐๐๐๐ฅ
โ โ๐บโโโ๐ผโโ๐ ๐๐ก(โ๐บโโ)
Recall Eq. (9), Eq. (23) and Eq. (24) we obtain:
๏ฟฝฬ๏ฟฝ2 โค 0
With Eq. (28), we can conclude that ๐1 โ 0 and ๐2 โ 0 as ๐ก โ โ.
IV. SIMULATION RESULTS
To evaluate the effectiveness of the proposed controller,
a simulation was carried out. The proposed joint controller is
commanded to track the desired trajectories of the hips, knees
and ankles. The sampling period is 0.01 second and the pre-
computed trajectories take 10 seconds to complete. The
physical parameters are given by Table II.
Table II
Parameters of UXA-90
Link Mass (kg) Length (m)
๐2 0.077 0.21
๐3 0.211 0.21
๐4 0.211 0.21
๐5 0.077 0.21
๐โ 0.5 0.09
๐๐ 0.469 0.065
First, the trajectory of the center of mass for UXA-90 on
the planar ๐๐ is obtained such that it satisfies the given
constraint: the ๐ง value of the center of mass should stay at
0.4๐(400๐๐) since the assumed distance of the center of
mass of UXA-90 to the ground is approximately 0.4๐. Then,
the joint trajectories (๐1, โฆ , ๐12) were calculated using
inverse kinematic. The ๐ซ(๐ฝ) , ๐ช(๐ฝ, ๏ฟฝฬ๏ฟฝ) and ๐(๐ฝ) are
calculated using the physical parameters obtained from
UXA-90. The inertia matrixes for the links of UXA-90 are
shown as:
๐ผ3 = [0.000267879958 โ0.000039044428 0
โ0.000039044428 0.000987981853 00 0 0.000852226607
]
๐ผ6 = [0.001058274664 0 0
0 0.000906206447 โ0.000045279150 โ0.00004527915 0.0001523499
]
๐ผ9 = [0.000267879958 โ0.000039044428 0
โ0.000039044428 0.000987981853 00 0 0.000852226607
]
๐ผ10 = [0.000013103094 0 0
0 0.00004938225 00 0 0.000045965477
]
๐ผ12 = [0.001058274664 0 0
0 0.000906206447 โ0.000045279150 โ0.00004527915 0.0001523499
]
The angle trajectories tracking are shown on the left, the
tracking errors are shown on the right in Fig. 5 below.
a. ๐1 trajectory and tracking error
b. ๐2 trajectory and tracking error
c. ๐3 trajectory and tracking error
d. ๐4 trajectory and tracking error
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e. ๐5 trajectory and tracking error
f. ๐6 trajectory and tracking error
g. ๐7 trajectory and tracking error
h. ๐8 trajectory and tracking error
i. ๐9 trajectory and tracking error
j. ๐10 trajectory and tracking error
k. ๐11 trajectory and tracking error
l. ๐12 trajectory and tracking error
Fig. 5. Joint angle trajectories and errors
The simulation of joint trajectories as shown in Fig. 5
showed that the back-stepping SMC tracked the joint
trajectories closely. The errors range from โ1.5ยฐ to 1.25ยฐ.
The COM trajectories were calculated: ๐ฅ trajectory is shown
in Fig. 6, ๐ฆ trajectory is shown in Fig. 7, ๐ง trajectory is shown
in Fig. 8.
Fig. 6. ๐ฅ trajectory of COM
Fig. 7. ๐ฆ trajectory of COM
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Fig. 8. ๐ง trajectory of COM
The COM trajectories showed that the real COM has
successfully tracked the pre-computed trajectories. The ๐ง
motion oscillate around 0.4๐ , the error ranges from
โ0.002๐ (โ2๐๐) to 0.0016๐ (1.6mm). As a result, this
satisfied the constraint proposed before: the ๐ง value of the
center of mass maintains at 0.4๐.
a. Comparison between SMC and back-stepping SMC for ๐1 and ๐2
b. Comparison between SMC and back-stepping SMC for ๐3 and ๐4
c. Comparison between SMC and back-stepping SMC for ๐5 and ๐6
d. Comparison between SMC and back-stepping SMC for ๐7 and ๐8
e. Comparison between SMC and back-stepping SMC for ๐9 and ๐10
f. Comparison between SMC and back-stepping SMC for ๐11 and ๐12
Fig. 9. Control input comparison between SMC and back-stepping SMC
As shown in Fig. 9, the control inputs for 12 joints of the
proposed controller expressed in red solid line do not feature
the chattering phenomenon in comparison to that of
traditional SMC employed the sign function. At the time of
2 seconds, 4 seconds, 6 seconds and 8 seconds, the control inputs have peaks and valleys due to the abrupt change of the
sign of acceleration in a very short time interval (0.01s).
Overall, the proposed controller has successfully reduced the
chattering effect, thus showing its advantage over the
traditional SMC.
V. CONCLUSION This paper proposed a novel back-stepping sliding mode
controller to control the dynamic equation of humanoid robot
UXA-90. The back-stepping method decompose the dynamic
system into reduced order subsystems which can be easily
controlled. The SMC guarantees the robustness while the use
of saturation function reduces the chattering effect. Also, the
COM tracked closely the desired trajectories, satisfying the
constraint: the ๐ง value of the center of mass should maintain
around 0.4๐ throughout the walking phase. The simulation results revealed that the back-stepping SMC outperforms the
traditional SMC controller when used to control the
humanoid robot UXA-90.
ACKNOWLEDGMENT This research is funded by Vietnam National University
HoChiMinh City (VNU-HCM) under grant number B2019-
20-09 and TX2020-20B-01.
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