Backstepping based Saturated Sliding Mode Joints

International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:20 No:06 97

202406-5959-IJMME-IJENS © December 2020 IJENS I J E N S

Backstepping based Saturated Sliding Mode Joints

Controller for Humanoid Walking Gait

Abstract— 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 system’s

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 Term— 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 IMU’s feedback data to control the joints’ 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 robot’s 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]



simulation is carried out through MATLAB to evaluate its effectiveness.

II. DYNAMIC MODEL

Table I

UXA-90 joints’ constraints

𝛉 Constraint 𝛉 Constraint

𝜽𝟏 −20°~20° 𝜽𝟕 −40°~50°

𝜽𝟐 −20°~50° 𝜽𝟖 −120°~30°

𝜽𝟑 −150°~0° 𝜽𝟗 −40°~0°

𝜽𝟒 −30°~120° 𝜽𝟏𝟎 0°~150°

𝜽𝟓 0°~40° 𝜽𝟏𝟏 −50°~20°

𝜽𝟔 −40°~50° 𝜽𝟏𝟐 −20°~20°

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 human’s joints precisely, twelve joints in the lower body of UXA-90 are constrained according to the motion range in human’s 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]:

𝑫(𝜽)�̈� + 𝑪(𝜽, �̇�)�̇� + 𝒈(𝜽) = 𝝉 where

𝜽 = [𝜃1, 𝜃2, … , 𝜃12] ∈ ℛ12×1 is the vector of joints.

𝑫(𝜽) ∈ ℛ12×12 is the inertia matrix.

𝑪(𝜽, �̇� ) ∈ ℛ12×12 is the Coriolis and centrifugal force matrix.

𝒈(𝜽) ∈ ℛ12×1 is the vector of gravitational forces.

𝝉 = [𝜏1, 𝜏2, … , 𝜏12] ∈ ℛ12×1 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:

𝑫(𝜽)�̈� + 𝑪(𝜽, �̇�)�̇� + 𝒈(𝜽)

= 𝝉 + 𝜹𝑭𝒆𝒙𝒕

where

𝜽 = [𝜃1, 𝜃2, … , 𝜃12] ∈ ℛ12×1 is the vector of joints.

𝑫(𝜽) ∈ ℛ12×12 is the inertia matrix.

𝑪(𝜽, �̇� ) ∈ ℛ12×12 is the Coriolis and centrifugal force matrix.

𝒈(𝜽) ∈ ℛ12×1 is the vector of gravitational forces.

𝝉 = [𝜏1, 𝜏2, … , 𝜏12] ∈ ℛ12×1 is the vector of joint torques.

𝜹𝑭𝒆𝒙𝒕 ∈ ℛ12×1 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 [𝑡, 𝑡 + ∆𝑡] ,

gives:

lim∆𝑡→0

∫ 𝑫(𝜽)

𝑡+∆𝑡

𝑡

�̈�𝑑𝑡 + lim∆𝑡→0

∫ [𝑪(𝜽, �̇�)�̇� + 𝒈(𝜽) − 𝝉]𝑑𝑡

𝑡+∆𝑡

𝑡



The integration of 𝒈(𝜽) and 𝝉 over [𝑡, 𝑡 + ∆𝑡] are zero,

we have:

lim∆𝑡→0

𝑫(𝜽)(�̇�𝒕+∆𝒕 − �̇�𝒕) + lim∆𝑡→0

𝑪(𝜽, �̇�)(𝜽𝒕+∆𝒕 − 𝜽𝒕)

= lim

∆𝑡→0∫ 𝜹𝑭𝒆𝒙𝒕

𝑡+∆𝑡

𝑡

𝑑𝑡

Since the contact duration is infinitesimal, the joint

positions remain unchanged, therefore:

lim∆𝑡→0

𝑪(𝜽, �̇�)(𝜽𝒕+∆𝒕 − 𝜽𝒕) = 0

Then the final result when integrating Eq. (2) is:

𝑫(𝜽)∆�̇� = 𝑭𝒆𝒙𝒕 The differential of the joint angles ∆�̇� can be calculated

using the following equation [9]:

∆�̇� = 𝑫−𝟏𝑱𝑻(𝑱𝑫−𝟏𝑱𝑻)−𝟏∆�̇� where

∆�̇� is the velocity of the foot just before the

contact with the ground.

𝑱 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:

�̇� = 𝒇(𝒙, 𝑡) + 𝒈(𝒙, 𝑡)𝒖 + 𝒅(𝑡) where

𝒙 is the system state.

𝒇(𝒙, 𝑡), 𝒈(𝒙, 𝑡) are known non-linear functions.

𝒖 ∈ ℛ12×1 is control inputs vector.

𝒅(𝑡) ∈ ℛ12×1 is external disturbances vector which is assumed to be bounded.

‖𝒅(𝑡)‖∞ ≤ 𝐷𝑐

where 𝐷𝑐 is positive scalar constant

By defining 𝑿𝟏(𝜽) = [𝜃1, … , 𝜃12]𝑇 , 𝑿𝟐(�̇�) =

[�̇�1, … , �̇�12]𝑇

, 𝒖 = [𝜏1, … , 𝜏12]𝑇.

Eq. (2) can be rewritten by:

{

�̇�𝟏 = 𝑿𝟐

�̇�𝟐 = 𝑫−𝟏(−𝑪𝑿𝟐 − 𝑮 + 𝒖 + 𝜹𝑭𝒆𝒙𝒕)

By defining 𝒇(𝑿) = −𝑫−𝟏𝑪𝑿𝟐 − 𝑫−𝟏𝑮, 𝒈(𝑿) = 𝑫−𝟏 ,

𝒅(𝑡) = 𝑫−𝟏𝜹𝑭𝒆𝒙𝒕, Eq. (10) can be summarized as follows:

{

�̇�𝟏 = 𝑿𝟐

�̇�𝟐 = 𝒇(𝑿) + 𝒈(𝑿)𝒖 + 𝒅(𝑡)

Defines a joint trajectories vector as 𝜽𝒅 =[𝜃𝑑1, … , 𝜃𝑑12]𝑇 , the difference between the current joint angles and the desired angle is defined as the error:

𝒆𝟏 = 𝑿𝟏 − 𝜽𝒅

Step 1: The first time derivative of 𝒆𝟏 can be obtained as follow:

�̇�𝟏 = �̇�𝟏 − �̇�𝒅

The candidate Lyapunov function (CLF) for considering the system of Eq. (13) is chosen as:

𝑉1 =1

2𝒆𝟏

𝑻𝒆𝟏

Considering Eq. (14) and taking the first time derivative

of 𝑉1, it yields :

�̇�1 = 𝒆𝟏𝑻�̇�𝟏 = 𝒆𝟏

𝑻(𝑿𝟐 − �̇�𝒅)

Defines a vector 𝑺 = [𝑠1, … , 𝑠12]𝑇 as the sliding surfaces which is given in the following form

𝑺 = 𝑿𝟐 + 𝒄𝟏𝒆𝟏 − �̇�𝒅

where 𝒄𝟏 ∈ ℛ12×12 is a constant symmetric

positive-definite matrix

Substituting Eq. (16) into Eq. (15), it can be

expressed as:

�̇�1 = 𝒆𝟏𝑻𝑺 − 𝒆𝟏

𝑻𝒄𝟏𝒆𝟏

It can be observed that 𝒆𝟏 and 𝑉1 are bounded

when the following condition holds true : 𝑺 → 𝟎.

Step 2 : The augmented CLF for considering the

dynamic of sliding surface 𝑺 is chosen as:

𝑉2 = 𝑉1 +1

2𝑺𝑻𝑺

Considering Eq. (18) and taking the first derivative of Eq. (16) gives:

�̇� = �̇�𝟐 + 𝒄𝟏�̇�𝟏 − �̈�𝒅

Substituting the 2nd system of Eq. (11) into Eq. (19), it yields:

�̇� = 𝒇(𝑿) + 𝒈(𝑿)𝒖 + 𝒅(𝑡) + 𝒄𝟏�̇�𝟏 − �̈�𝒅

Taking the first derivative of Eq. (18), it yields:

�̇�2 = �̇�1 + 𝑺𝑻�̇�

= 𝒆𝟏𝑻𝑺 − 𝒆𝟏

𝑻𝒄𝟏𝒆𝟏 + 𝑺𝑻[𝒇(𝑿) + 𝒈(𝑿)𝒖

+ 𝒅(𝑡) + 𝒄𝟏�̇�𝟏 − �̈�𝒅]



In order to restrain the chattering phenomenon, the saturation function 𝑠𝑎𝑡 is adopted instead of the 𝑠𝑔𝑛 . Therefore, the control input designed for system of Eq. (11) is as followed:

𝒖 = 𝒈(𝑿)−𝟏[−𝒇(𝑿) − 𝒄𝟐𝑺 − 𝒆𝟏 − 𝒄𝟏�̇�𝟏 + �̈�𝒅

− 𝜼𝑠𝑎𝑡(𝑺)]

where 𝒄𝟐 ∈ ℛ12×12 is a constant symmetric positive-definite matrix.

𝜼 ∈ ℛ12×12 is a constant diagonal matrix.

Each element in the main diagonal of 𝜼 should satisfy the following:

𝜂𝑖 > 𝐷𝑐

where 𝑖 = 1 ÷ 12.

𝑠𝑎𝑡(∙) is the saturation function defined as:

𝑠𝑎𝑡(∙) = {

1, 𝑖𝑓 ∙ > ∆

𝑘 × ∙ , 𝑖𝑓 | ∙ | ≤ ∆, 𝑘 = 1/∆−1, 𝑖𝑓 ∙ < −∆

where 0 < ∆< 1 is the boundary layer constant.

Substituting Eq. (22) into Eq. (21), gives:

�̇�2 = −𝒆𝟏𝑻𝒄𝟏𝒆𝟏 − 𝑺𝑻𝒄𝟐𝑺 + 𝑺𝑻𝒅(𝑡)

− 𝑺𝑻𝜼𝑠𝑎𝑡(𝑺)

Since 𝒄𝟏, 𝒄𝟐 are constant symmetric positive-definite matrix, let us define:

𝐶1 ≜ ‖𝒄𝟏‖∞ ≥ 0, 𝐶2 ≜ ‖𝒄𝟐‖∞ ≥ 0, 𝐷𝑐 ≜ 𝑑𝑚𝑎𝑥

Taking the infinity norm of �̇�2, it yields:

�̇�2 = −𝐶1‖𝒆𝟏‖∞2 − 𝐶2‖𝑺‖∞

2 + ‖𝑺‖∞𝑑𝑚𝑎𝑥

− ‖𝑺‖∞‖𝜼‖∞𝑠𝑎𝑡(‖𝑺‖∞)

Recall Eq. (9), Eq. (23) and Eq. (24) we obtain:

�̇�2 ≤ 0

With Eq. (28), we can conclude that 𝑒1 → 0 and 𝑒2 → 0 as 𝑡 → ∞.

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)

𝑙2 0.077 0.21

𝑙3 0.211 0.21

𝑙4 0.211 0.21

𝑙5 0.077 0.21

𝑙ℎ 0.5 0.09

𝑙𝑓 0.469 0.065

First, the trajectory of the center of mass for UXA-90 on

the planar 𝑋𝑌 is obtained such that it satisfies the given

constraint: the 𝑧 value of the center of mass should stay at

0.4𝑚(400𝑚𝑚) since the assumed distance of the center of

mass of UXA-90 to the ground is approximately 0.4𝑚. Then,

the joint trajectories (𝜃1, … , 𝜃12) were calculated using

inverse kinematic. The 𝑫(𝜽) , 𝑪(𝜽, �̇�) and 𝒈(𝜽) are

calculated using the physical parameters obtained from

UXA-90. The inertia matrixes for the links of UXA-90 are

shown as:

𝐼3 = [0.000267879958 −0.000039044428 0

−0.000039044428 0.000987981853 00 0 0.000852226607

]

𝐼6 = [0.001058274664 0 0

0 0.000906206447 −0.000045279150 −0.00004527915 0.0001523499

]

𝐼9 = [0.000267879958 −0.000039044428 0

−0.000039044428 0.000987981853 00 0 0.000852226607

]

𝐼10 = [0.000013103094 0 0

0 0.00004938225 00 0 0.000045965477

]

𝐼12 = [0.001058274664 0 0

0 0.000906206447 −0.000045279150 −0.00004527915 0.0001523499

]

The angle trajectories tracking are shown on the left, the

tracking errors are shown on the right in Fig. 5 below.

a. 𝜃1 trajectory and tracking error

b. 𝜃2 trajectory and tracking error

c. 𝜃3 trajectory and tracking error

d. 𝜃4 trajectory and tracking error



e. 𝜃5 trajectory and tracking error

f. 𝜃6 trajectory and tracking error

g. 𝜃7 trajectory and tracking error

h. 𝜃8 trajectory and tracking error

i. 𝜃9 trajectory and tracking error

j. 𝜃10 trajectory and tracking error

k. 𝜃11 trajectory and tracking error

l. 𝜃12 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 −1.5° to 1.25°.

The COM trajectories were calculated: 𝑥 trajectory is shown

in Fig. 6, 𝑦 trajectory is shown in Fig. 7, 𝑧 trajectory is shown

in Fig. 8.

Fig. 6. 𝑥 trajectory of COM

Fig. 7. 𝑦 trajectory of COM



Fig. 8. 𝑧 trajectory of COM

The COM trajectories showed that the real COM has

successfully tracked the pre-computed trajectories. The 𝑧

motion oscillate around 0.4𝑚 , the error ranges from

−0.002𝑚 (−2𝑚𝑚) to 0.0016𝑚 (1.6mm). As a result, this

satisfied the constraint proposed before: the 𝑧 value of the

center of mass maintains at 0.4𝑚.

a. Comparison between SMC and back-stepping SMC for 𝜏1 and 𝜏2

b. Comparison between SMC and back-stepping SMC for 𝜏3 and 𝜏4

c. Comparison between SMC and back-stepping SMC for 𝜏5 and 𝜏6

d. Comparison between SMC and back-stepping SMC for 𝜏7 and 𝜏8

e. Comparison between SMC and back-stepping SMC for 𝜏9 and 𝜏10

f. Comparison between SMC and back-stepping SMC for 𝜏11 and 𝜏12

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 𝑧 value of the center of mass should maintain

around 0.4𝑚 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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Documents

Backstepping based Saturated Sliding Mode Joints