**4.2 Comparative control results**

where *ηi*∣*i*¼2*::*<sup>5</sup> and*σi*∣*i*¼1*::*<sup>5</sup> are positive learning rates.

*Overview of the gain-learning backstepping controller for low-level subsystems.*

Proof of *Theorem 2* could be referred in *Appendix E*.

control error. The form of Eq. (E.4) reveals that the learning rates

theorem is given. **Theorem 2:**

**Figure 1.**

hardware.

**4.1 Setup**

**42**

tion gains are obtained.

*Collaborative and Humanoid Robots*

ing mechanism proposed.

presented in **Figure 2**.

mated by Euler backward methods. Two systematic parameters *a*1*<sup>j</sup>*, *a*2*<sup>j</sup>*

**4. Real-time experiments**

To investigate the control performance of the learning control system, a new

If applying the control gains updated using Eq. (13) to a closed-loop system satisfying *Lemma 2*, asymptotic convergences of the state control error and varia-

*Remark 4:* Overview of the proposed controller is sketched in **Figure 1**. As stated in Theorem 1, the stability of the closed-loop system is ensured in a robust control framework, and as proven in Theorem 2, the adaptation of the control structure is highlighted by all the control gains learning for minimizing the tracking

(*σ<sup>i</sup>*∣*i*¼1*::*<sup>5</sup> and *η<sup>i</sup>*∣*i*¼2*::*5) can be employed with predefined values for specific control

*Remark 5*: In real-time applications, the proposed algorithm will be deployed in a discrete-time environment, the control errors will converge to arbitrary vicinities around zero. The desired control range can be however minimized under the learn-

In this section, control performance of the intelligent controller is discussed based on verification results carried out in a real-time legged 2DOF robot. The experimental leg included one hip joint and one knee joint which were actuated by two BLDC motors. The mechanical design and a photograph of the actual leg are

Incremental encoders were used to measure the joint angles, while a force sensor

*<sup>j</sup>*¼*h*,*<sup>k</sup>* of the low-level systems could be

was placed in the shank of the robot to evaluate the ground contact force. The velocity signal was calculated from filtered backward differentiation of the position data. The robot was setup to freely move in both *x* and *y* directions. Total weight of the robot was about 15.74 kg. The proposed control algorithm was deployed in a NI Electrical Controller throughout LABVIEW software with a sampling time of 2 ms. The time derivative and integral terms in real-time implementation were approxi-

determined as *a*1*<sup>h</sup>* ¼ 2*:*5; *a*2*<sup>h</sup>* ¼ 12*:*25; *a*1*<sup>k</sup>* ¼ 0*:*5; *a*2*<sup>k</sup>* ¼ 15*:*

estimated offline or online using a model-based identification method derived in previous works [27, 31, 32]. Nominal values of the parameters were approximately

Both the hip and knee joints were controlled at the same time using the same control algorithm proposed. The controller was also compared with an adaptive robust extended-state-observer-based (ARCESO) controller, a robust integral-signerror (RISE) controller, and another case of itself with fixed gains (nominal gains) in Eq. (8), which is denoted as the robust backstepping (RB) controller.

The ARCESO controller was designed based on a previous work [30] wherein their control gains were chosen as

$$\begin{cases} k\_{1k} = 100; k\_{2k} = 100; \boldsymbol{\mu}\_{0k} = 60; \boldsymbol{\theta}\_{\text{min}} = [6, 1]^T; \boldsymbol{\theta}\_{\text{max}} = [25, 5]^T; \boldsymbol{\hat{\theta}}\_{k}(0) = [12.25, 2.5]^T; \\ k\_{1k} = 80; k\_{2k} = 75; \boldsymbol{\mu}\_{0k} = 40; \boldsymbol{\theta}\_{\text{min}} = [5, 0.2]^T; \boldsymbol{\theta}\_{\text{max}} = [30, 1.5]^T; \boldsymbol{\hat{\theta}}\_{k}(0) = [15, 0.5]^T; \end{cases}$$

The RISE controller was implemented based on a robust integral theory [13] to control the studied system Eq. (2) without considering the measurement noise ð Þ*υ* . Its control signal was:

$$\begin{cases} e\_1 = \mathbf{x}\_1 - \mathbf{x}\_{1d}; e\_2 = \dot{e}\_1 + k\_{\text{RISEI}} e\_1 \\ \quad \boldsymbol{u}\_{\text{RISE}} = -\mathbf{a}\_2^{-1} \left( -a\_1 \mathbf{x}\_2 - \ddot{\mathbf{x}}\_{1d} + k\_{\text{RISEI}} (\mathbf{x}\_2 - \dot{\mathbf{x}}\_{1d}) + k\_{\text{RISEI}} e\_2 + \int\_0^t (k\_{\text{RISEI}} e\_2 + k\_{\text{RISEI}} \text{sgn} \, (e\_2)) d\tau \right) \end{cases} \tag{14}$$

The RISE control gains were set to be:

$$\begin{cases} k\_{\text{RISE}h} = \text{53}; k\_{\text{RISE}2h} = \text{85.4}; k\_{\text{RISE}3h} = \text{20}; k\_{\text{RISE}4h} = \text{250}; \\\\ k\_{\text{RISE}1k} = \text{45}; k\_{\text{RISE}2k} = \text{65}; k\_{\text{RISE}3k} = \text{17}; k\_{\text{RISE}4k} = \text{235}; \end{cases}$$

The nominal gains of the proposed controllers were chosen to be:

$$\begin{cases} \overline{k}\_{1h} = 1.5; \overline{k}\_{2h} = 72.5; \overline{k}\_{3h} = 2000; \overline{k}\_{4h} = 50; \overline{k}\_{5h} = 200; \\\\ \overline{k}\_{1k} = 5; \overline{k}\_{2k} = 72.5; \overline{k}\_{3k} = 2000; \overline{k}\_{4k} = 50; \overline{k}\_{5k} = 200; \end{cases}$$

The excitation signals ð Þ *ε* and *φ* of the learning laws Eq. (13) were directly synthesized from the control error (*e*1) and its high-order time derivatives based on Eq. (D.1):

**Figure 2.** *Design and setup of the experimental testing system.*

$$\begin{cases} \varepsilon = \dot{e}\_1 + k\_1 e\_1 \\ \rho = (\ddot{e}\_1 + k\_2 \dot{e}\_1) + \left(k\_3 + (k\_2 - k\_1)k\_1 - \text{sat}\left(\ddot{k}\_1\right)\right) e\_1 \end{cases} \tag{15}$$

From the nominal control gains selected, the feasible ranges of the variation gains were then chosen to gratify the constraint Eq. (9):

*k ^* <sup>1</sup>*<sup>h</sup>* min ¼ �1*:*4; *k ^* <sup>2</sup>*<sup>h</sup>* min ¼ �66*:*5; *k ^* <sup>3</sup>*<sup>h</sup>* min ¼ �1000; *k ^* <sup>4</sup>*<sup>h</sup>* min ¼ �49; *k ^* <sup>1</sup>*<sup>h</sup>* max ¼ 3*:*5; *k ^* <sup>2</sup>*<sup>h</sup>* max ¼ 500; *k ^* <sup>3</sup>*<sup>h</sup>* max ¼ 4000; *k ^* <sup>4</sup>*<sup>h</sup>* max ¼ 1500; *k ^* <sup>5</sup>*<sup>h</sup>* min ¼ �199; *k ^* <sup>5</sup>*<sup>h</sup>* max ¼ 1500; 8 < : *k ^* <sup>1</sup>*<sup>k</sup>* min ¼ �4*:*9; *k ^* <sup>2</sup>*<sup>k</sup>* min ¼ �66*:*5; *k ^* <sup>3</sup>*<sup>k</sup>* min ¼ �1000; *k ^* <sup>4</sup>*<sup>k</sup>* min ¼ �49; *k ^* <sup>1</sup>*<sup>k</sup>* max ¼ 5; *k ^* <sup>2</sup>*<sup>k</sup>* max ¼ 500; *k ^* <sup>3</sup>*<sup>k</sup>* max ¼ 4000; *k ^* <sup>4</sup>*<sup>k</sup>* max ¼ 1500; *k ^* <sup>5</sup>*<sup>k</sup>* max ¼ 1500; *k ^* <sup>5</sup>*<sup>k</sup>* min ¼ �199; 8 < : 8 >>>>>>>>>< >>>>>>>>>:

The learning rates (*σ<sup>i</sup>*∣*i*¼1*::*<sup>5</sup> and *η<sup>i</sup>*∣*i*¼2*::*5) were then set to comply with the condition Eq. (12) and to ensure the variation gains freely varying inside their predetermined ranges. For simplicity, the relaxation rates (*η<sup>i</sup>*∣*i*¼2*::*5) could be chosen to be *1* or *2*. Finally, the rates tuned were as

$$\begin{cases} \sigma\_{1h} = 1000; \sigma\_{2h} = 10^{-2}; \eta\_{2h} = 1; \sigma\_{3h} = 2 \times 10^{-4}; \eta\_{2h} = 1; \sigma\_{4h} = 0.05; \\\eta\_{4h} = 2; \sigma\_{5h} = 0.03; \eta\_{5h} = 1; \\\sigma\_{1k} = 10000; \sigma\_{2k} = 10^{-1}; \eta\_{2k} = 1; \sigma\_{3k} = 2 \times 10^{-3}; \eta\_{2k} = 1; \sigma\_{4k} = 0.05; \\\eta\_{4k} = 2; \sigma\_{5k} = 0.03; \eta\_{5k} = 1; \end{cases}$$

#### *4.2.1 Simple verification*

In this validation series, the proposed controller was only applied for positiontracking control of the hip joint. A sinusoidal signal of *x*1*dh* ¼ 14 sin 4ð Þ *πt* ð Þ deg was chosen as the desired trajectory of the test. The leg was put to move freely in the air to eliminate the external disturbance. **Figure 3(a)** presents the experimental data obtained by the comparative controllers. The ARCESO controller produced a very small control error of �0.14 deg. (�1.0%) in the high-speed tracking control thanks to the use of an effective adaptive-disturbance learning mechanism. The ARCESO control performance was still however limited with fast-variation disturbances [30]. By adopting the integral-robust control signal Eq. (14) to compensate for the lumped disturbance (*d*) in the low-level system Eq. (2), the RISE controller also exhibited a high control accuracy (control error: [�0.16; 0.14] deg. (�1.14%)). In fact, in real-time applications, improper control gains selected or large measurement noise (*υ*) could degrade the RISE control performance. As operating under the highly robust design Eq. (8) against all the disturbances, the RB technique provided better control precision (control error: �0.138 deg. (� 0.98%)). Theoretically, the control performance could be further increased if the best control gains were found, but it may be a time-consuming work. As a solution, the gain-tuning process could be supported by the learning mechanism Eqs. (11) and (13) proposed. Indeed, the control quality was intuitively enhanced by applying GARB control method, which yielded the smallest control error of �0.085 deg. (�0.6%).

*4.2.2 Complex verification*

*learning of the GARB controller.*

*Self-Learning Low-Level Controllers*

*DOI: http://dx.doi.org/10.5772/intechopen.96732*

**Figure 3.**

**Table 1.**

**Figure 4.**

**45**

To deeper challenge to the special properties of the proposed controller, the robot was controlled to perform a squatting exercise in three different working cases: in the air, on the ground, and with ground contact. The frequency and amplitude of the squatting motion were selected to be 2 Hz and 80 mm, respectively. These tests are normal working cases of the leg in real-time missions. The desired trajectories ð Þ *x*1*dh* and *x*1*dk* of the two robot joints (hip and knee) are plotted

*Experimental results of the single-joint test. (a). Comparative control errors of the testing controllers. (b). Gain*

**Control error ARCESO RISE RB GARB** MA 0.140 0.160 0.138 0.085 RMS 0.080 0.074 0.072 0.030

in **Figure 4**. The trajectories were derived from desired foot motion of

*Performance comparison of the controllers for the single-joint validation.*

*Desired profiles of the robot joints in the multiple-joint tests.*

The gain-learning behaviors are illustrated in **Figure 3(b)**. As seen in the figure, the variation gains were automatically changed in various ways under the adaptation laws to minimize the control error. The maximum-absolute (MA) and rootmean-squares (RMS) values of the control errors from after system was stable (from 2 s to 5 s) are summarized in **Table 1**. Herein, the proposed controller shows outperformance as comparing to the previous methods.

*Self-Learning Low-Level Controllers DOI: http://dx.doi.org/10.5772/intechopen.96732*

#### **Figure 3.**

*ε* ¼ *e*\_<sup>1</sup> þ *k*1*e*<sup>1</sup>

gains were then chosen to gratify the constraint Eq. (9):

*^*

*^*

*^*

<sup>2</sup>*<sup>h</sup>* min ¼ �66*:*5; *k*

<sup>2</sup>*<sup>k</sup>* min ¼ �66*:*5; *k*

*^*

<sup>2</sup>*<sup>h</sup>* max ¼ 500; *k*

<sup>2</sup>*<sup>k</sup>* max ¼ 500; *k*

to be *1* or *2*. Finally, the rates tuned were as

*<sup>σ</sup>*1*<sup>h</sup>* <sup>¼</sup> 1000; *<sup>σ</sup>*2*<sup>h</sup>* <sup>¼</sup> <sup>10</sup>�<sup>2</sup>

*η*4*<sup>h</sup>* ¼ 2; *σ*5*<sup>h</sup>* ¼ 0*:*03; *η*5*<sup>h</sup>* ¼ 1; *<sup>σ</sup>*1*<sup>k</sup>* <sup>¼</sup> 1000; *<sup>σ</sup>*2*<sup>k</sup>* <sup>¼</sup> <sup>10</sup>�<sup>1</sup>

*η*4*<sup>k</sup>* ¼ 2; *σ*5*<sup>k</sup>* ¼ 0*:*03; *η*5*<sup>k</sup>* ¼ 1;

yielded the smallest control error of �0.085 deg. (�0.6%).

outperformance as comparing to the previous methods.

(

*Collaborative and Humanoid Robots*

*^*

*^*

*^*

*^*

*k ^*

8 < :

8 >>>>>>>>><

>>>>>>>>>:

**44**

*k ^*

*k ^*

8 < :

*k ^*

<sup>1</sup>*<sup>h</sup>* min ¼ �1*:*4; *k*

<sup>1</sup>*<sup>h</sup>* max ¼ 3*:*5; *k*

<sup>1</sup>*<sup>k</sup>* min ¼ �4*:*9; *k*

<sup>1</sup>*<sup>k</sup>* max ¼ 5; *k*

8 >>><

>>>:

*4.2.1 Simple verification*

*φ* ¼ €*e*<sup>1</sup> þ *k*2*e*\_ ð Þþ <sup>1</sup> *k*<sup>3</sup> þ ð Þ *k*<sup>2</sup> � *k*<sup>1</sup> *k*<sup>1</sup> � sat *k*

From the nominal control gains selected, the feasible ranges of the variation

<sup>3</sup>*<sup>h</sup>* min ¼ �1000; *k*

<sup>3</sup>*<sup>k</sup>* min ¼ �1000; *k*

*^*

The learning rates (*σ<sup>i</sup>*∣*i*¼1*::*<sup>5</sup> and *η<sup>i</sup>*∣*i*¼2*::*5) were then set to comply with the condi-

predetermined ranges. For simplicity, the relaxation rates (*η<sup>i</sup>*∣*i*¼2*::*5) could be chosen

; *<sup>η</sup>*2*<sup>k</sup>* <sup>¼</sup> 1; *<sup>σ</sup>*3*<sup>k</sup>* <sup>¼</sup> <sup>2</sup> � <sup>10</sup>�<sup>3</sup>

In this validation series, the proposed controller was only applied for positiontracking control of the hip joint. A sinusoidal signal of *x*1*dh* ¼ 14 sin 4ð Þ *πt* ð Þ deg was chosen as the desired trajectory of the test. The leg was put to move freely in the air to eliminate the external disturbance. **Figure 3(a)** presents the experimental data obtained by the comparative controllers. The ARCESO controller produced a very small control error of �0.14 deg. (�1.0%) in the high-speed tracking control thanks to the use of an effective adaptive-disturbance learning mechanism. The ARCESO control performance was still however limited with fast-variation disturbances [30]. By adopting the integral-robust control signal Eq. (14) to compensate for the lumped disturbance (*d*) in the low-level system Eq. (2), the RISE controller also exhibited a high control accuracy (control error: [�0.16; 0.14] deg. (�1.14%)). In fact, in real-time applications, improper control gains selected or large measurement noise (*υ*) could degrade the RISE control performance. As operating under the highly robust design Eq. (8) against all the disturbances, the RB technique provided better control precision (control error: �0.138 deg. (� 0.98%)). Theoretically, the control performance could be further increased if the best control gains were found, but it may be a time-consuming work. As a solution, the gain-tuning process could be supported by the learning mechanism Eqs. (11) and (13) proposed. Indeed, the control quality was intuitively enhanced by applying GARB control method, which

The gain-learning behaviors are illustrated in **Figure 3(b)**. As seen in the figure, the variation gains were automatically changed in various ways under the adaptation laws to minimize the control error. The maximum-absolute (MA) and rootmean-squares (RMS) values of the control errors from after system was stable (from 2 s to 5 s) are summarized in **Table 1**. Herein, the proposed controller shows

*^*

<sup>3</sup>*<sup>h</sup>* max ¼ 4000; *k*

<sup>3</sup>*<sup>k</sup>* max ¼ 4000; *k*

tion Eq. (12) and to ensure the variation gains freely varying inside their

*^* 1

*e*1

<sup>5</sup>*<sup>h</sup>* min ¼ �199; *k*

<sup>5</sup>*<sup>k</sup>* max ¼ 1500; *k*

; *η*2*<sup>k</sup>* ¼ 1; *σ*4*<sup>k</sup>* ¼ 0*:*05;

*^*

*^*

(15)

<sup>5</sup>*<sup>h</sup>* max ¼ 1500;

<sup>5</sup>*<sup>k</sup>* min ¼ �199;

� � � �

*^*

*^*

<sup>4</sup>*<sup>k</sup>* max ¼ 1500; *k*

<sup>4</sup>*<sup>h</sup>* min ¼ �49;

<sup>4</sup>*<sup>k</sup>* min ¼ �49;

; *<sup>η</sup>*2*<sup>h</sup>* <sup>¼</sup> 1; *<sup>σ</sup>*3*<sup>h</sup>* <sup>¼</sup> <sup>2</sup> � <sup>10</sup>�4; *<sup>η</sup>*2*<sup>h</sup>* <sup>¼</sup> 1; *<sup>σ</sup>*4*<sup>h</sup>* <sup>¼</sup> <sup>0</sup>*:*05;

*^*

*^*

<sup>4</sup>*<sup>h</sup>* max ¼ 1500; *k*

*Experimental results of the single-joint test. (a). Comparative control errors of the testing controllers. (b). Gain learning of the GARB controller.*


#### **Table 1.**

*Performance comparison of the controllers for the single-joint validation.*

#### *4.2.2 Complex verification*

To deeper challenge to the special properties of the proposed controller, the robot was controlled to perform a squatting exercise in three different working cases: in the air, on the ground, and with ground contact. The frequency and amplitude of the squatting motion were selected to be 2 Hz and 80 mm, respectively. These tests are normal working cases of the leg in real-time missions. The desired trajectories ð Þ *x*1*dh* and *x*1*dk* of the two robot joints (hip and knee) are plotted in **Figure 4**. The trajectories were derived from desired foot motion of

**Figure 4.** *Desired profiles of the robot joints in the multiple-joint tests.*

*Py* <sup>¼</sup> <sup>460</sup> <sup>þ</sup> 40 sin 4ð Þ *<sup>π</sup><sup>t</sup>* ; *Px* <sup>¼</sup> <sup>0</sup> mm using simple inverse-kinematics computation as noted in *Appendix F*.

#### *4.2.2.1 Verification with minor external disturbances*

Although the robot worked in the air, the disturbances affecting the control joints were large due to high-speed control and interaction forces between the joints during the system movement. The dynamical and statical control results obtained by the validated controllers are respectively shown in **Figure 5** and **Table 2**. In spite of operating with faster motions (192.2 (deg/s) and 324.8 (deg/s) for the hip and knee joints) and in harder internal disturbance conditions, the ARCESO controllers

maintained high control outcome thanks to the strong adaptation ability: �0*:*8degð Þ � 4*:*8% and �1*:*5 degð Þ � 4*:*1% for the hip and knee joints.

*Performance comparison of the validated controllers in the small disturbance tests.*

create a better control performance as comparing to the others

control errors).

**Table 2.**

*Self-Learning Low-Level Controllers*

*DOI: http://dx.doi.org/10.5772/intechopen.96732*

forces as well. The RB and RISE controllers stabilized the control errors inside acceptable ranges: the errors for hip joint and knee joint are respectively ½ � �0*:*5 ! 1 degð Þ � 6% and ½ � �2 ! 1 degð Þ � 5*:*5% with the RB, while those are ½ � �0*:*8 ! 1*:*1 degð Þ � 6*:*6% and ½ � �2*:*3 ! 1*:*2 degð Þ � 6*:*3% with the RISE. In the new working conditions, excellent control errors were also resulted in by the GARB controller based on a new set of the control gains found. **Figure 5(d)** depicts the variation gains that were incorporated with the proposed robust design Eq. (11) to

(½ � �0*:*2 ! 0*:*35 degð Þ � 2*:*1% and ½ � �0*:*8 ! 0*:*5 degð Þ � 2*:*2% for the hip and knee

advantages of the proposed controller as comparing to other controllers.

acceptable control accuracy: ½ � �0*:*35 ! 0*:*68 degð Þ � 4*:*08% and

½ � �1*:*5 ! 1*:*1 degð Þ � 4*:*1% for the hip and knee joints.

confirmed via this investigation.

**47**

*4.2.2.2 Verification with large external disturbances*

Comparison of the control power required to conduct the high-speed control motions is shown in **Figure 5(b)**. Although the control efforts of the controllers were almost same for this mission. Only minor disparate nonlinearities in the control signals would lead to the divergence on control performances. The figure also reveals that the GARB controllers generated applicable control inputs even though the learning gains were moderated in a risk of the high-order measurement noise. The benefit comes from the low-pass-filter-like nature of the gain-learning algorithm proposed. External force affecting the leg measured in the shank using the GARB controllers is presented in **Figure 5(c)**. The coordinate of the measured force is sketched in **Figure 2**. This experiment shows the higher control accuracies and demonstrates the

In this experiment, the robust adaptive ability of the proposed controller was harshly investigated under conditions of heavy external load. The robot was put on the ground and supported by sliders in both the *x* and *y* directions. To avoid damage for the robot, only the proposed controller was used in the verification. The control results obtained are plotted in **Figure 6**. In this test, the external forces reacting from environment were significantly increased from 10 N to 390 N. The data presented in **Figure 6(a)** however implies that the controller still provided

As demonstrated in **Figure 6(b)**, in this case the system used larger energy than in the second one to execute the fast-tracking control under critical conditions. As presented in **Figure 6(d)**, the control gains were also automatically changed to higher values to deal with large disturbances for a smallest possible control error. Hence, the strong robustness and fast adaptability of the proposed method can be

As seen in **Figure 5(a)**, the robust backstepping designs coped with the reaction

**Control error ARCESO RISE RB GARB**

RMS 0.5 0.78 0.75 0.17

RMS 0.74 1.17 1.15 0.37

HIP MA 0.8 1.1 1 0.35

KNEE MA 1.5 2.3 2 0.8

#### **Figure 5.**

*Experimental results of the testing controllers for the multiple-joint test in case of small external disturbance. (a). Control errors of the comparative controllers. (b). Control inputs generated by the comparative controllers. (c) Forces measured at the shank with respect to the GARB controllers. (d) Gain learning of the GARB controllers.*



#### **Table 2.**

*Py* <sup>¼</sup> <sup>460</sup> <sup>þ</sup> 40 sin 4ð Þ *<sup>π</sup><sup>t</sup>* ; *Px* <sup>¼</sup> <sup>0</sup> mm using simple inverse-kinematics computation

Although the robot worked in the air, the disturbances affecting the control joints were large due to high-speed control and interaction forces between the joints during the system movement. The dynamical and statical control results obtained by the validated controllers are respectively shown in **Figure 5** and **Table 2**. In spite of operating with faster motions (192.2 (deg/s) and 324.8 (deg/s) for the hip and knee joints) and in harder internal disturbance conditions, the ARCESO controllers

*Experimental results of the testing controllers for the multiple-joint test in case of small external disturbance. (a). Control errors of the comparative controllers. (b). Control inputs generated by the comparative controllers. (c) Forces measured at the shank with respect to the GARB controllers. (d) Gain learning of the GARB*

as noted in *Appendix F*.

*Collaborative and Humanoid Robots*

**Figure 5.**

*controllers.*

**46**

*4.2.2.1 Verification with minor external disturbances*

*Performance comparison of the validated controllers in the small disturbance tests.*

maintained high control outcome thanks to the strong adaptation ability: �0*:*8degð Þ � 4*:*8% and �1*:*5 degð Þ � 4*:*1% for the hip and knee joints.

As seen in **Figure 5(a)**, the robust backstepping designs coped with the reaction forces as well. The RB and RISE controllers stabilized the control errors inside acceptable ranges: the errors for hip joint and knee joint are respectively ½ � �0*:*5 ! 1 degð Þ � 6% and ½ � �2 ! 1 degð Þ � 5*:*5% with the RB, while those are ½ � �0*:*8 ! 1*:*1 degð Þ � 6*:*6% and ½ � �2*:*3 ! 1*:*2 degð Þ � 6*:*3% with the RISE. In the new working conditions, excellent control errors were also resulted in by the GARB controller based on a new set of the control gains found. **Figure 5(d)** depicts the variation gains that were incorporated with the proposed robust design Eq. (11) to create a better control performance as comparing to the others (½ � �0*:*2 ! 0*:*35 degð Þ � 2*:*1% and ½ � �0*:*8 ! 0*:*5 degð Þ � 2*:*2% for the hip and knee control errors).

Comparison of the control power required to conduct the high-speed control motions is shown in **Figure 5(b)**. Although the control efforts of the controllers were almost same for this mission. Only minor disparate nonlinearities in the control signals would lead to the divergence on control performances. The figure also reveals that the GARB controllers generated applicable control inputs even though the learning gains were moderated in a risk of the high-order measurement noise. The benefit comes from the low-pass-filter-like nature of the gain-learning algorithm proposed. External force affecting the leg measured in the shank using the GARB controllers is presented in **Figure 5(c)**. The coordinate of the measured force is sketched in **Figure 2**. This experiment shows the higher control accuracies and demonstrates the advantages of the proposed controller as comparing to other controllers.

### *4.2.2.2 Verification with large external disturbances*

In this experiment, the robust adaptive ability of the proposed controller was harshly investigated under conditions of heavy external load. The robot was put on the ground and supported by sliders in both the *x* and *y* directions. To avoid damage for the robot, only the proposed controller was used in the verification. The control results obtained are plotted in **Figure 6**. In this test, the external forces reacting from environment were significantly increased from 10 N to 390 N. The data presented in **Figure 6(a)** however implies that the controller still provided acceptable control accuracy: ½ � �0*:*35 ! 0*:*68 degð Þ � 4*:*08% and ½ � �1*:*5 ! 1*:*1 degð Þ � 4*:*1% for the hip and knee joints.

As demonstrated in **Figure 6(b)**, in this case the system used larger energy than in the second one to execute the fast-tracking control under critical conditions. As presented in **Figure 6(d)**, the control gains were also automatically changed to higher values to deal with large disturbances for a smallest possible control error. Hence, the strong robustness and fast adaptability of the proposed method can be confirmed via this investigation.

#### **Figure 6.**

*Experimental results of the GARB controllers for the multiple-joint test in case of large external disturbance. (a) Control errors of the GARB controllers. (b) Control inputs generated by the GARB controllers. (c) Measurement of ground reaction forces. (d) Gain learning of the proposed mechanisms.*

parameters were varied to properly adapt to change of the new working conditions. **Figure 7(d)** shows that the new ranges of the control gains were found by the proposed algorithm, and **Figure 7(b)** presents the required energy for the new test.

*Experimental results of the GARB controllers for the multiple-joint test in case of fast-variation external disturbance. (a) Control errors of the GARB controllers. (b) Control inputs generated by the GARB controllers.*

*(c) Measurement of ground reaction forces. (d) Gain learning of the proposed mechanisms.*

The RMS values of the control errors, control signals (*u*), and the groundreaction forces for the hip and knee joints of the complex validation process are noted in **Table 3**. The data imply that the GARB controller was able to result in good control performances with the preset learning rates in the high-speed task under different working conditions. The learning mechanism and robust control technique generated proper power for each test case to effectively realize the control objective. Some snapshots of the robot movement in the last experiment are shown

*4.2.3 Additional Statical note*

*Self-Learning Low-Level Controllers*

*DOI: http://dx.doi.org/10.5772/intechopen.96732*

in **Figure 8**.

**49**

**Figure 7.**

#### *4.2.2.3 Verification with fast-variation external disturbances*

In this case study, transient behaviors of the designed controller were carefully validated by using fast-variation external disturbances. The robot was still controlled to conduct the same squatting work. Harder testing conditions were constituted by two consecutive distinguished phases of one working cycle: a groundcontact phase and ground-release phase. **Figure 7(c)** shows the ground-reaction forces measured during the test. The nature of the external disturbance in this case was different from those in the previous cases. Fast variation of the reaction forces may make the system instable. The control system designed had however showed the concrete robustness and impressive adaptation in real-time control again.

As presented in **Figure 7(a)**, the closed-loop system provided good performance: ½ � �0*:*22 ! 0*:*76 degð Þ � 4*:*56% for the hip joint and ½ � �0*:*8 ! 0*:*7 deg ð Þ � 2*:*2% for the knee joint. The control energy and control *Self-Learning Low-Level Controllers DOI: http://dx.doi.org/10.5772/intechopen.96732*

#### **Figure 7.**

*4.2.2.3 Verification with fast-variation external disturbances*

**Figure 6.**

*Collaborative and Humanoid Robots*

**48**

performance: ½ � �0*:*22 ! 0*:*76 degð Þ � 4*:*56% for the hip joint and

½ � �0*:*8 ! 0*:*7 deg ð Þ � 2*:*2% for the knee joint. The control energy and control

In this case study, transient behaviors of the designed controller were carefully validated by using fast-variation external disturbances. The robot was still controlled to conduct the same squatting work. Harder testing conditions were constituted by two consecutive distinguished phases of one working cycle: a groundcontact phase and ground-release phase. **Figure 7(c)** shows the ground-reaction forces measured during the test. The nature of the external disturbance in this case was different from those in the previous cases. Fast variation of the reaction forces may make the system instable. The control system designed had however showed the concrete robustness and impressive adaptation in real-time control again. As presented in **Figure 7(a)**, the closed-loop system provided good

*Experimental results of the GARB controllers for the multiple-joint test in case of large external disturbance. (a) Control errors of the GARB controllers. (b) Control inputs generated by the GARB controllers. (c) Measurement of ground reaction forces. (d) Gain learning of the proposed mechanisms.*

*Experimental results of the GARB controllers for the multiple-joint test in case of fast-variation external disturbance. (a) Control errors of the GARB controllers. (b) Control inputs generated by the GARB controllers. (c) Measurement of ground reaction forces. (d) Gain learning of the proposed mechanisms.*

parameters were varied to properly adapt to change of the new working conditions. **Figure 7(d)** shows that the new ranges of the control gains were found by the proposed algorithm, and **Figure 7(b)** presents the required energy for the new test.

#### *4.2.3 Additional Statical note*

The RMS values of the control errors, control signals (*u*), and the groundreaction forces for the hip and knee joints of the complex validation process are noted in **Table 3**. The data imply that the GARB controller was able to result in good control performances with the preset learning rates in the high-speed task under different working conditions. The learning mechanism and robust control technique generated proper power for each test case to effectively realize the control objective. Some snapshots of the robot movement in the last experiment are shown in **Figure 8**.


conventional form of the SMC scheme in which *e2* is the sliding surface. Hence, users have various options in adoption of the designed controller, which could be

Note also that the input gain constant (*a2*) could be selected with an arbitrarily positive constant while the nominal dynamical constant (*a1*) could be zero or any bounded value. Their deviations could be counted into the lumped disturbance (*d*) or extended disturbance (*h*). One possible way to determine such the terms is use of the model-based identification method presented in previous works [27, 30, 31]. As comparing to other intelligent gain-learning algorithms such as neural network or fuzzy logic engines, the computational burden and fast response are noteworthy advantages [9–11, 37, 38]. However, in some cases, one does not need to use the nominal dynamics or (a1 = 0), and at that time, overall design of the proposed

The experimental results have confirmed the outperformance of the gain-learning controllers over other robust adaptive nonlinear controllers, such as ARCESO and RISE [13, 30], thanks to a high-degree-of-learning mechanism. Furthermore, the designed controller has been improved from the former controller [27] to increase the real-time applicability by removing third-order time-derivation terms in the control signal. From the above analyses, the flexibility of the designed controller in terms of working efficiency and user implementation are intuitively observed. Its feasibility

This chapter presents a gain-adaptive robust position-tracking controller for low-level subsystems of large robotic systems. The mathematical model of the system dynamics was reviewed to provide necessary information for the controller design. To realize the tracking control objective, a robust control signal based on the backstepping scheme was adopted. In fact, this design is a nonlinear extension of ordinary PID controller or conventional sliding mode controller. New adaptation laws were developed to automatically tune the control gains for different working conditions. The learning mechanism was activated by various forms of the control

Stability of the overall system was concretely maintained by proper Lyapunovbased constraints. Extended real-time experiments were conducted to verify the performance of the proposed controller. The results achieved confirmed the advantages on the robustness, adaptation, high accuracy, and fast response of the proposed controller. Depending on the usage purpose of user, the controller could be simplified to become a

*h* ¼ �*x*€1*<sup>d</sup>* þ *υ*\_ þ *k*2*υ* þ *d* (A.1)

(A.2)

ðð Þ *k*4*ε* þ *k*<sup>5</sup> sgn ð Þ *e*<sup>1</sup> *dς*Þ þ *h*

gain-learning PID controller or an adaptive robust sliding mode controller.

Also synthesize a new state variable and lumped term as follows:

*<sup>φ</sup>* ¼ � <sup>Ð</sup>*<sup>t</sup>* 0

*ε* ¼ *υ* þ *e*<sup>2</sup>

in movable robots have been also confirmed by intensive experiments.

easily switched to basic control options [6, 8, 28, 35].

*Self-Learning Low-Level Controllers*

*DOI: http://dx.doi.org/10.5772/intechopen.96732*

control method becomes a model-free controller.

error and deactivated by the relaxation functions.

Let define the following new disturbance:

8 < :

**Appendix A. Proof of Lemma 1**

**6. Summary**

**51**

**Table 3.**

*Performance comparison of the garb controllers in the multiple-joint tests.*

**Figure 8.** *Snapshots of the leg motion in the large external disturbance test.*
