Research on Lower Limb Exoskeleton Trajectory Tracking Control Based on the Dung Beetle Optimizer and Feedforward Proportional–Integral–Derivative Controller

Abstract

The lower limb exoskeleton (LLE) plays an important role in production activities requiring assistance and load bearing. One of the challenges is to propose a control strategy that can meet the requirements of LLE trajectory tracking in different scenes. Therefore, this study proposes a control strategy (DBO–FPID) that combines the dung beetle optimizer (DBO) with feedforward proportional–integral–derivative controller (FPID) to improve the performance of LLE trajectory tracking in different scenes. The Lagrange method is used to establish the dynamic model of the LLE rod, and it is combined with the dynamic equations of the motor to obtain the LLE transfer function model. Based on the LLE model and target trajectory compensation, the feedforward controller is designed to achieve trajectory tracking in different scenes. To obtain the best performance of the controller, the DBO is utilized to perform offline parameter tuning of the feedforward controller and PID controller. The proposed control strategy is compared with the DBO tuning PID (DBO–PID), particle swarm optimizer (PSO) tuning FPID (PSO–FPID), and PSO tuning PID (PSO–PID) in simulation and joint module experiments. The results show that DBO–FPID has the best accuracy and robustness in trajectory tracking in different scenes, which has the smallest sum of absolute error (IAE), mean absolute error (MEAE), maximum absolute error (MAE), and root mean square error (RMSE). In addition, the MEAE of DBO–FPID is lower than 1.5 degrees in unloaded tests and lower than 3.6 degrees in the hip load tests, with only a few iterations, showing great practical potential.

1. Introduction

In production activities, human beings usually need to carry weights and work in different postures, which burdens the musculoskeletal system, leading to the occurrence of musculoskeletal diseases [1,2]. To reduce the labor intensity, the LLE was created. It can transfer weight to the ground, and it can also apply torque to the legs during the walking process to provide assistance. In recent years, the LLE has been applied more and more in scenes requiring load bearing and support [3,4,5], which effectively improves work efficiency and reduces the occurrence of injuries.

To meet the load and assistance requirements of the LLE in different scenes, a superior trajectory tracking performance is indispensable. The design of the controller is the key to LLE trajectory tracking, which needs to have high accuracy and strong robustness. PID controllers are widely used in the field of exoskeleton control because of their universality and reliability [6,7,8]. However, the PID controller has the disadvantages of being unable to deal with external interference and lacking parameter self–tuning capabilities [9]. To improve the capacity of the LLE to resist interference, the researchers tried to enhance conventional PID. The PID controller based on fuzzy logic is used in the control of the LLE and has achieved excellent control performance [10,11]. Liu et al. and Belkadi et al. proposed an adaptive PID control strategy based on improved particle swarm optimizer, which enhanced the anti–interference ability of the controller [12,13]. The above methods can tune PID parameters online to cope with external changes, but as they are model–free controllers, it is difficult to achieve high control accuracy. The feedforward controller is a model–based controller that can adjust the control output in advance according to the change of the target trajectory to adapt to different tracking scenes. Yuan et al. combined PID, integral sliding mode control, and feedforward control for the LLE [14]. Merola et al. combined PID and feedforward control in the control of pneumatic muscle–driven exoskeletons [15]. Their feedforward controller was obtained through dynamic modeling, achieving impressive performance. Similarly, a combination of PID and neural network feedforward controllers has also been used in the control of the LLE [16,17,18]. The above research establishes the feedforward controller through mathematical modeling or neural networks and confirms that the controller structure combining feedforward and feedback is applicable in the LLE. It is worth noting that the feedforward controller relies on the accuracy of the controlled object model. Therefore, establishing a more accurate feedforward controller is still an area that needs to be continuously explored and optimized.

The parameter tuning of the controller is crucial for its performance. Ziegler–Nichols tuning PID (ZN–PID) is a classical experimental tuning method, but ZN–PID cannot achieve satisfactory results in the control of exoskeletons [19]. In recent years, meta–heuristic algorithms have increasingly been applied to PID parameter tuning for the LLE, which provides a better way to obtain controller parameters. Sharma et al. adopted the dragonfly algorithm to optimize fuzzy PID [20]. Amiri et al. proposed a strategy of combining ZN–PID and adaptive particle swarm optimization to tune PID [21]. PSO is a popular meta–heuristic algorithm widely used in the field of exoskeleton controller parameter optimization. For example, Zhao et al. used PSO to optimize five parameters of fuzzy PID, and the simulation shows that the proposed control strategy has a fast response speed and strong anti–interference ability [22]. Amiri et al. proposed a lyapunov adaptive swarm–fuzzy logic control strategy, whose parameters were optimized by PSO, and whose effectiveness was verified by experiments [23]. Similarly, PSO has been used to optimize the neural network controller [24], active disturbance rejection control [25], adaptive fuzzy proportional derivative controller [26], etc. Although PSO is mature in the field of controller parameter optimization, it has the problems of slow convergence speed and becoming trapped in local optima, which prompts us to explore a new optimizer. A newly found meta–heuristic technology is the DBO, which realizes the global exploration and local exploitation of optimization problems through different populations and behavior patterns [27]. At present, the DBO has been successfully used for UAV three-dimensional path planning [28], the path planning of underwater joint robot [29], and other fields. Compared with conventional optimizers, the DBO has fewer parameters and there is no limitation on the setting of the initial parameters. Multiple behavior patterns and search areas enable the DBO to show stronger global optimization capability. Therefore, the DBO is suitable for complex and multi–parameter LLE controller parameter optimization problems.

In this paper, a DBO–FPID LLE trajectory tracking control strategy is proposed based on the excellent global optimization ability of the DBO and the model–based advantages of FPID. The purpose is to ensure that the LLE in different scenes can maintain excellent tracking effect. Firstly, the LLE transfer function model is established, which contains the dynamics equations of LLEs connecting rods and motors. The structure of the feedforward controller is designed based on the LLE transfer function model and target trajectory compensation, being combined with the PID feedback controller. In the parameter tuning process, the DBO is used to obtain FPID parameters instead of the popular PSO to obtain better controller performance. Specifically, the DBO is used to tune PID parameters. Then, the previously designed feedforward controller is added to the PID controller that has completed parameter tuning. The DBO performs parameter tuning on the feedforward controller to enable the precise compensation of the target trajectory. To ascertain the efficacy and practicability of the proposed DBO–FPID, simulation experiments and joint module experiments are designed. DBO–FPID, DBO–PID, PSO–FPID, and PSO–PID are utilized to track target trajectories in different scenes measured by the attitude sensors, and the tracking effect is evaluated comprehensively by IAE, MEAE, MAE, and RMSE.

2. LLE Mathematical Model

The thigh and calf components of the LLE can be equivalent to two straight and uniform rods. In Figure 1, l_i, p_i, m_i, and θ_i (i = 1, 2) represent the length, length of the center of mass, mass, and joint angle of the rod.

The dynamics equation of the LLE rod is derived from the Lagrange equation:

$$T_l=M\left(\theta \right)\ddot{\theta }+C\left(\theta ,\dot{\theta }\right)+G\left(\theta \right)$$

(1)

where T_l = (T_l1, T_l2)^T, which is a 2 × 1 joint torque vector; M is a 2 × 2 inertial matrix; C is a 2 × 1 matrix of Coriolis force and centrifugal force; and G is a 2 × 1 gravity vector. The LLE joint is driven by DC motors, and the differential equations of the motor armature current are as follows:

$$\left.\begin{array}{l}V=L{\displaystyle \frac{d{i}_a}{dt}}+R{i}_a+V_b\hfill \\ V_b=K_b{\displaystyle \frac{d{\theta }_m}{dt}}\hfill \end{array}\right\}$$

(2)

where L is the armature inductance; i_a is the armature current; R is the armature resistance; V is the armature voltage; V_b is the back electromotive force; K_b is the back electromotive force constant; and θ_m is the position of the rotor. The motion equation of the DC motor is as follows:

$$\left.\begin{array}{l}{J}_m{\displaystyle \frac{d^2{\theta }_m}{d{t}^2}}+B_m{\displaystyle \frac{d{\theta }_m}{dt}}=T_m-{\displaystyle \frac{T_l}{r}}\hfill \\ T_m=K_m{i}_a\hfill \end{array}\right\}$$

(3)

where J_m is the moment of inertia; B_m is the friction coefficient; r is the deceleration ratio; T_m is the motor torque; and K_m is the torque constant. The relationship between the position of the motor rotor θ_mi and the rotation angle of the LLE joint θ_i is as follows:

$${\theta }_{mi}=r_i{\theta }_i$$

(4)

By combining the Equations (2)–(4), the differential equation of the motor can be obtained as follows:

$$r_i{\displaystyle \frac{K_{mi}{V}_i}{R_i}}=r_i^2{J}_{mi}{\ddot{\theta }}_i+\left(B_{mi}+{\displaystyle \frac{K_{bi}{K}_{mi}}{R_i}}\right)r_i^2{\dot{\theta }}_i+T_{li}$$

(5)

By combining the Equations (1) and (5), the dynamical equations of the LLE can be obtained as follows:

$$V=M\left(\theta \right)\ddot{\theta }+C\left(\theta ,\dot{\theta }\right)+G\left(\theta \right)$$

(6)

where V = (V₁, V₂)^T, which is a 2 × 1 voltage vector. Although the dynamical model of the LLE is a nonlinear equation, within the range of motion, by rotating one joint while keeping the others frozen, the rotating one can be considered as a third-order linear system [30]. Therefore, each joint of the LLE can be approximated by a third-order transfer function as follows:

$${\displaystyle \frac{{\Theta }_i(s)}{V_i(s)}}={\displaystyle \frac{1}{a{s}^3+b{s}^2+cs+d}}$$

(7)

where a, b, c, and d are constants. To make the simulation experiments closer to the real situation, this study adopts the models obtained in the literature [30], which are identified by the MATLAB system identification toolbox. The transfer functions are as follows:

$$\left.\begin{array}{l}{G}_1\left(s\right)={\displaystyle \frac{{\Theta }_1(s)}{V_1(s)}}={\displaystyle \frac{52.1017}{0.001s^3+0.4396s^2+3.1604s+8.9409}}\hfill \\ G_2\left(s\right)={\displaystyle \frac{{\Theta }_2(s)}{V_2(s)}}={\displaystyle \frac{52.1017}{0.001s^3+0.1321s^2+1.1838s+3.4727}}\hfill \end{array}\right\}$$

(8)

where G₁(s) and G₂(s) are the transfer functions of the hip and knee joints.

3. Control Strategy

After completing the modeling of the LLE, this section will proceed with the design of control strategies. Section 3.1 will design the FPID controller; Section 3.2 will introduce the DBO optimizer; and Section 3.3 will introduce the DBO into the FPID controller for parameter tuning.

3.1. FPID Controller

The PID controller is represented as below:

$$u_p\left(k\right)=k_{p}e\left(k\right)+k_i{\displaystyle \sum _{j=1}^{k}e\left(j\right)}+k_d\Delta e\left(k\right)$$

(9)

where u_p(k) is the output of the PID for the kth time; k_p, k_i, and k_d are proportional, integral, and differential parameters; e(k) is the kth tracking error; and Δe(k) is the change in the kth tracking error, which are given by the following equation:

$$\left.\begin{array}{l}e\left(k\right)=r(k)-\theta \left(k\right)\hfill \\ \Delta e\left(k\right)=e\left(k\right)-e\left(k-1\right)\hfill \end{array}\right\}$$

(10)

where r(k) and θ(k) represent the kth target angle and measurement angle. Conventional PID is a feedback controller based on error correction, which only takes effect when deviations occur. For the LLE with complex working conditions and high real-time requirements, it cannot meet the high precision control requirements. Therefore, this study combines PID with a feedforward controller to form a feedforward feedback controller, as shown in Figure 2.

In Figure 2, G_f(s) and G_p(s) are the transfer functions of the feedforward controller and PID. To make the LLE meet the trajectory tracking of different scenes, G_f(s) needs to compensate for the target trajectory R(s) to minimize E(s). The error transfer function is represented as follows:

$$G_e(s)={\displaystyle \frac{E(s)}{R(s)}}={\displaystyle \frac{a{s}^3+b{s}^2+cs+d-G_f(s)}{a{s}^3+b{s}^2+cs+d+G_p(s)}}$$

(11)

where G_e(s) is the error transfer function, describing the relationship between R(s) and E(s). By designing G_f(s) to make G_e(s) zero, the full compensation of the system can be achieved. Therefore, G_f(s) can be obtained from Equation (11):

$$G_f\left(s\right)=a{s}^3+b{s}^2+cs+d$$

(12)

Using the first difference approximation method to discretize Equation (12),

$$\begin{array}{l}{u}_f\left(k\right)=k_1\left[r\left(k\right)-3r\left(k-1\right)+3r\left(k-2\right)-r\left(k-3\right)\right]+\hfill \\ k_2\left[r\left(k\right)-2r\left(k-1\right)+r\left(k-2\right)\right]+k_3\left[r\left(k\right)-r\left(k-1\right)\right]+k_{4}r\left(k\right)\hfill \end{array}$$

(13)

where u_f(k) is the output of the feedforward controller for the kth time. k₁ = a/T³, k₂ = b/T², k₃ = c/T, k₄ = d. T is the system sampling period. Although the values of k₁, k₂, k_3, and k₄ can be calculated from the process parameters (a, b, c, d) and sampling period, the acquisition of process parameters relies on prior knowledge, such as the establishment of mathematical models. And the parameters obtained cannot fully reflect the reality. In contrast, the optimizer does not require prior knowledge. Through iterative optimization processes on LLE prototypes, the optimizer can automatically search for parameters that are suitable for real systems. Furthermore, it can jointly optimize the parameters of both the PID and feedforward controllers, resulting in a better overall controller tracking performance. Therefore, the optimizer is introduced in this study to obtain the values of k₁, k₂, k_3, and k₄. The FPID controller is obtained by combining Equations (9) and (13):

$$u(k)=u_p(k)+u_f(k)$$

(14)

where u(k) is the output of the FPID for the kth time.

3.2. DBO Optimizer

The DBO simulated the survival behaviors of dung beetles in nature through mathematical modeling, including ball–rolling, dancing, breeding, foraging, and stealing, and constructed a search framework based on these behaviors. In ball-rolling behavior, dung beetles use celestial navigation to update their position, which is also influenced by light intensity and natural factors. When dung beetles encounter an obstacle, they are redirected by dancing behavior, which is simulated by the tangent function. In breeding behavior, female dung beetles will choose safe locations to lay eggs. This process is simulated by defining the spawning area and dynamically adjusting its boundaries based on the number of iterations. In foraging behavior, some mature dung beetles emerge from the ground in search of food. Its position update is simulated by defining the optimal feeding area and dynamically adjusting its boundaries based on the number of iterations. In stealing behavior, some dung beetles will steal the dung balls of other dung beetles as their food source. This behavior combines the current position, global optimal position, local optimal position, and random vector to simulate. The DBO constantly updates the individual position of dung beetles through the above behaviors to search for the optimal solution in the defined space.

3.3. DBO–FPID Control Strategy

Previous studies have obtained appropriate PID parameters through popular PSO. However, PSO needs to set more initial parameters, including two learning factors, the inertia coefficient, and the lower and upper limits of speed. The optimization effect of PSO is sensitive to these parameter settings. In addition, these fixed parameters lead to a tendency to get trapped in local optima and result in slow convergence speed in the later stages of iteration. The above issues make it difficult to achieve satisfactory results in LLE parameter tuning. Therefore, this study adopts the DBO to obtain all parameters of FPID. According to Equation (14), the DBO requires tuning the parameters k_p, k_i, and k_d of the PID and the parameters k₁, k₂, k₃, and k₄ of the feedforward controller to minimize the tracking error between the target trajectory and the actual trajectory. The objective function of the DBO is designed as follows:

$$J={\displaystyle \sum _{j=1}^k\left|e(j)\right|}$$

(15)

where |e(j)| represents the absolute value of the j-th tracking error; k represents the total number of runs; and J is the sum of the absolute value of the error in the number of runs. Since there are seven parameters to be obtained, simultaneous tuning will increase the complexity of optimization. This study adopts the stepwise optimization strategy. It is worth noting that, if the feedforward controller is tuned first and then the PID controller is added for tuning, it will result in a small PID gain and reduce the stability of the controller. Therefore, this study first tunes the PID to ensure basic stability, and then adds the feedforward controller for tuning to improve control accuracy. The procedure is as follows: Firstly, the LLE is controlled by PID according to Equation (9), and the DBO is used to obtain the optimal parameter solution k_p, k_i, and k_d of the PID. Based on the tuned PID, the feedforward controller is added to form FPID. FPID is used to control the LLE according to Equation (14), and the DBO is used to obtain four parameters of FPID, including k₁, k₂, k₃, and k₄. After completing the tuning of the seven parameters, the proposed controller is obtained. The flowchart of DBO–FPID is shown in Figure 3.

4. Simulation Experiment

After completing the design of the control strategy, simulation experiments will be carried out in this section. In Section 4.1, the gait data of different scenes will be obtained, which will be used as the target movement trajectory of the simulation experiment; Section 4.2 will set up the simulation and then show the simulation results.

4.1. Gait Data Collection

The attitude sensors are inertial measurement units, with an accuracy of 0.2 degrees. The subject obtained gait data in the four scenes shown in Figure 4. Two attitude sensors are fixed on the back of the left thigh and left calf with straps. They communicate with the upper computer through the Modbus protocol with a sampling interval of 120 ms. Figure 4a is scene 1, where the subject walks at a constant speed on the horizontal ground. Figure 4b is scene 2, where the subject needs to climb stairs (0.15 m per step). Figure 4c is scene 3. The subject walks along the red arrow, which in turn is horizontal ground walking, climbing stairs, horizontal ground walking, descending stairs, and horizontal ground walking. Figure 4d is scene 4, where the subject gradually transitions from slow walking to fast walking.

4.2. Simulation Settings and Results

The simulation experiments were carried out on MATLAB2021b, using the LLE transfer function model obtained in Equation (8). The target trajectory is based on the test data of Section 4.1. Since piecewise cubic Hermite interpolation can generate smooth curves and retain the features of the original data, the data are interpolated by it to make the sampling frequency change to 100 Hz. To comprehensively evaluate the tracking performance, four performance indexes including IAE, MEAE, MAE, and RMSE were proposed. IAE reflects the accumulation of absolute errors, MEAE focuses on the average level of tracking performance, MAE evaluates the worst tracking situation, and RMSE focuses on the size and distribution of errors. The smaller the values of the above indexes, the better the tracking performance. The DBO and PSO are set with the same population size and number of iterations, which are 30 and 100. The two learning factors of PSO are set to 1, and the inertia coefficient is set to 0.5. The upper limit and lower limit of speed are set to 20 and −20. The constraint range for FPID and PID parameters is set to between −250 and 250. The controller parameters obtained in scene 1 are shown in

Table 1. DBO and PSO obtain parameters by tuning FPID and PID in scene 1.

Parameters	Hip				Knee
	DBO–FPID	DBO–PID	PSO–FPID	PSO–PID	DBO–FPID	DBO–PID	PSO–FPID	PSO–PID
kp	8.0381	8.0381	8.23722	8.23722	2.3221	2.3221	9.814255	9.814255
ki	1.07255	1.07255	1.68581	1.68581	0.024061	0.024061	0.105555	0.105555
kd	50.112	50.112	83.4124	83.4124	6.0659	6.0659	137.1816	137.1816
k1	125.022		−85.62981		54.2868		−26.6664
k2	69.40883		156.3278		20.7605		64.5435
k3	5.815122		5.226777		2.30838		32.838
k4	0.108704		0.2433991		0.0476948		0.0533645

. The trajectory tracking performance indexes in scene 1 are shown in

Table 2. Trajectory tracking performance indexes of hip and knee joint in scenes 1.

Indexes	Hip				Knee
	DBO–FPID	DBO–PID	PSO–FPID	PSO–PID	DBO–FPID	DBO–PID	PSO–FPID	PSO–PID
IAE (°)	58.2124	222.5931	461.1682	547.1105	131.5939	801.2457	2225.4	2472.6
MEAE (°)	0.0582	0.2224	0.4607	0.5466	0.1315	0.8004	2.2232	2.4701
MAE (°)	0.3766	1.4608	1.5968	2.5298	0.9952	5.7764	6.2693	9.2723
RMSE (°)	0.0820	0.3378	0.5374	0.6971	0.1943	1.1497	2.6133	3.0370

. The corresponding hip and knee joint trajectory tracking simulation results are shown in Figure 5.

As can be seen from Table 2, the IAE, MEAE, MAE, and RMSE of DBO–FPID and PSO–FPID are lower than those of DBO–PID and PSO–PID, respectively, indicating that FPID improves the control accuracy over PID when using the same optimizer. The four indexes of DBO–FPID and DBO–PID are also lower than PSO–FPID and PSO–PID, respectively, indicating that the DBO provides a better optimization result than PSO when optimizing the same controller. Furthermore, among the four control strategies, the values of the IAE, MEAE, MAE, and RMSE of DBO–FPID are 58.2124, 0.0582, 0.3766, and 0.0820, respectively. This shows the smallest four indexes compared to DBO–PID, PSO–FPID, and PSO–PID, which verifies the effectiveness of combining FPID with the DBO proposed in this study. Figure 5 shows the trajectory tracking diagram and tracking error diagram of the hip and knee joints in scene 1. It can be seen that DBO–FPID has the smallest tracking error. To further verify the robustness of DBO–FPID, the controller parameters obtained in Table 1 are used for trajectory tracking in scenes 2, 3, and 4. The trajectory tracking performance indexes are shown in

Table 3. Trajectory tracking performance indexes of hip and knee joint in scenes 2, 3, and 4.

Indexes	Hip				Knee
	DBO–FPID	DBO–PID	PSO–FPID	PSO–PID	DBO–FPID	DBO–PID	PSO–FPID	PSO–PID
IAE (°)
Scene 2	384.1358	814.3416	1170.4	1534.7	207.0941	1070.6	2837.1	3430.5
Scene 3	123.7093	404.9149	626.8465	892.2566	204.3195	1118.9	2715.9	3594.4
Scene 4	338.5276	887.4463	993.4773	1681.7	389.4159	1762.6	2800.8	3735.8
MEAE (°)
Scene 2	0.3838	0.8135	1.1692	1.5332	0.2069	1.0695	2.8343	3.4271
Scene 3	0.1236	0.4045	0.6262	0.8914	0.2041	1.1178	2.7132	3.5908
Scene 4	0.3382	0.8866	0.9925	1.6800	0.3890	1.7608	2.7980	3.7320
MAE (°)
Scene 2	5.3048	9.7206	9.2546	13.7469	1.4831	8.1599	10.4331	13.9228
Scene 3	1.0152	3.1406	3.7607	7.0526	1.7605	10.8578	12.4174	15.6317
Scene 4	6.8780	10.1990	8.8998	12.5339	3.2037	10.2793	12.5177	15.7884
RMSE (°)
Scene 2	0.8560	1.7227	2.0339	2.7306	0.3023	1.7040	3.5310	4.3846
Scene 3	0.1895	0.6838	0.8614	1.3557	0.3187	1.8387	3.3156	4.6806
Scene 4	0.8312	1.7722	1.6249	2.7269	0.6384	2.6138	3.6101	4.8806

. The corresponding hip and knee joint trajectory tracking simulation results are shown in Figure 6 and Figure 7.

As can be seen from Table 3, the four performance indexes of DBO–FPID and PSO–FPID are smaller than those of DBO–PID and PSO–PID, respectively, indicating that FPID has stronger robustness than PID when tracking different target trajectories. The four performance indexes of DBO–FPID and DBO–PID are lower than those of PSO–FPID and PSO–PID, respectively, indicating that the controller optimized by the DBO is more robust than that of PSO. Furthermore, DBO–FPID has the smallest performance indexes in the trajectory tracking process of scenes 2, 3, and 4, verifying that the proposed control strategy has the best robustness in different scenes. In addition, it can be seen from Figure 6 and Figure 7 that DBO–FPID has the smallest tracking error in the trajectory tracking process of hip and knee joints in different scenes.

5. Joint Module Experiments

After the verification of the above simulation experiments, joint module experiments will be carried out in this section to further verify the practicability of the proposed control strategy. The joint module experiment includes the unloaded motor test and hip load test, using the hip joint trajectories from scene 1 to scene 4, measured in Section 4.1, as the target trajectory.

Figure 8a shows the unloaded motor test. Controller parameter tuning is performed in the hip joint trajectory of scene 1, and the obtained parameters are also used for hip trajectory tracking tests in scenes 2, 3, and 4. The rated torque and speed of the DC motor are 21 Nm and 40 rpm. It is powered by a 48 V lithium battery with a capacity of 7.8 Ah. The main control board is equipped with an STM32F427IIH6 chip, which drives the DC motor through the CAN protocol. Figure 8b is the hip load test. The load test uses the same controller parameters as the unloaded motor test. The weight of the LLE in the experiments is approximately 15 kg, which can effectively reflect the practicality of the controller in the load state. The number of iterations for PID parameter tuning is set to 4, with a constraint range from 0.01 to 100. For FPID, the number of iterations is set to 6, and the constraint range is set between –100 and 100. The remaining parameters are set to be consistent with the simulation experiments.

Table 4. Trajectory tracking performance indexes of the unloaded motor and hip joint.

Indexes	Unloaded Motor				Hip
	DBO–FPID	DBO–PID	PSO–FPID	PSO–PID	DBO–FPID	DBO–PID	PSO–FPID	PSO–PID
IAE (°)
Scene 1	861.7772	1007.1	1015.9	1192.1	2341.2	2570.5	2789.3	2933.8
Scene 2	1291.2	1485.1	1465.8	1742.0	3278.3	3608.2	3813.1	3992.3
Scene 3	964.5609	1181.8	1135.3	1400.4	2750.4	3119.3	3256.7	3493.3
Scene 4	1462.0	1821.5	1728.0	2155.2	3556.2	3867.2	3996.5	4289.5
MEAE (°)
Scene 1	0.8609	1.0061	1.0149	1.1909	2.3388	2.5679	2.7865	2.9308
Scene 2	1.2899	1.4837	1.4644	1.7402	3.2750	3.6046	3.8093	3.9883
Scene 3	0.9636	1.1806	1.1342	1.3990	2.7476	3.1162	3.2535	3.4898
Scene 4	1.4605	1.8196	1.7262	2.1530	3.5527	3.8634	3.9925	4.2852
MAE (°)
Scene 1	3.8383	4.1705	4.7346	5.1705	9.0412	9.5008	10.3127	10.8686
Scene 2	6.1421	7.0115	7.1811	7.9666	19.5760	20.0768	20.0768	20.0768
Scene 3	5.0142	5.8058	6.2402	7.3933	12.8158	13.6784	14.8158	14.9534
Scene 4	6.1624	7.4472	6.7636	8.5789	18.6408	20.2123	20.2123	20.6408
RMSE (°)
Scene 1	1.1586	1.2822	1.3652	1.5201	2.9430	3.2003	3.4895	3.6510
Scene 2	1.7620	2.0222	2.0898	2.3668	5.0957	5.3124	5.5868	5.6105
Scene 3	1.3570	1.5926	1.6456	1.9210	3.8778	4.1414	4.4274	4.5987
Scene 4	1.9461	2.2895	2.2507	2.7342	4.9633	5.3544	5.5185	5.8429

shows the trajectory tracking performance indexes of the unloaded motor and hip joint.

As shown in Table 4, the results are consistent with the simulation experiments. DBO–FPID and PSO–FPID have smaller indexes than DBO–PID and PSO–PID, respectively. DBO–FPID and DBO–PID have smaller indexes than PSO–FPID and PSO–PID, respectively. Notably, DBO–FPID still has the smallest indexes in all four scenes in unloaded and loaded tests. In addition, the MEAE of DBO–FPID is less than 1.5 degrees in unloaded tests and less than 3.6 degrees in load tests, requiring only a few iterations. Therefore, DBO–FPID has practical potential and can provide more effective closed-loop responses for the LLE.

6. Conclusions

The LLE control strategy of DBO–FPID is proposed in this paper. Its effectiveness and practicability are verified by simulation experiments and joint module experiments. It is found that DBO–FPID is more competitive in the process of trajectory tracking, making it suitable for different scenes such as climbing stairs, variable speed walking, and switching between different terrains. The proposed control strategy is universal and applicable to different LLEs. It is not necessary to design the parameters of PID and feedforward controller through system identification or mathematical modeling. Instead, using the DBO can automatically obtain appropriate controller parameters on actual LLE prototypes or simulation systems. There are also some limitations in this study, and external disturbances are not fully considered. Future studies will focus on the influence of these disturbances on the LLE system and explore effective suppression strategies to ensure the robust suppression of external disturbances.