Optimization of Leg Mechanism and PD Control for Efficient Quadruped Robot Locomotion

Subject Areas : Journal of Simulation and Analysis of Novel Technologies in Mechanical Engineering

Mobin Salehi ¹ , Arash Emami ² , Mojtaba Norozi ³

1 -
2 -
3 -

Received: 2024-10-14 Accepted : 2024-12-30 Published : 2025-03-01

Keywords: Quadruped, Genetic algorithm, Dynamics-independent control, Trotting gait,

Abstract :

This paper presents the design and control of a quadruped robot. One of the primary challenges in building quadruped robots is the need for high torque density actuators and an efficient control algorithm. To address these challenges, this work focuses on optimizing the transmission torque ratio of the 4-bar linkage used in the robot's legs, using a genetic algorithm. The optimization is achieved by deriving the kinematic equations of the robot’s legs and introducing a novel objective function tailored to the robot’s application. To evaluate the impact of the optimization, the full dynamics of the robot are derived and validated through variations in total mechanical energy. A kinematics-based controller, suitable for real-time applications, is proposed, and its performance is tested in various scenarios to assess its effectiveness. The controller is applied to robots with two different linkage lengths, one optimized for maximum and the other for minimum torque requirements. The results show that the optimization reduces the required torque by nearly 42% when comparing the maximum to the minimum case.

References:


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Full-Text:



Journal of Simulation and Analysis of Novel Technologies in Mechanical Engineering 16 (3) (2024) 0041~0060 DOI 10.71939/jsme.2024. 1186910

 

Research article

Optimization of leg mechanism and design of a PD control

for a quadruped robot

 

Mobin Salehi1*, Arash Emami2 and Mojtaba Norozi3

 

1Mechanical Engineering Department, Sharif University of Technology, Tehran, Tehran, Iran

2Mechanical Engineering Department, Islamic Azad University of Central Tehran Branch, Tehran, Tehran, Iran

3Industrial Engineering Department, Malek Ashtar University of Technology, Tehran, Tehran, Iran

 

*[e-mail]

 

(Manuscript Received --- 14 Oct. 2024; Revised --- 25 Dec. 2024; Accepted --- 30 Dec. 2024)

Abstract

This paper presents the design and control of a quadruped robot. One of the primary challenges in building quadruped robots is the need for high torque density actuators and an efficient control algorithm. To address these challenges, this work focuses on optimizing the transmission torque ratio of the 4-bar linkage used in the robot's legs, using a genetic algorithm. The optimization is achieved by deriving the kinematic equations of the robot’s legs and introducing a novel objective function tailored to the robot’s application. To evaluate the impact of the optimization, the full dynamics of the robot are derived and validated through variations in total mechanical energy. A kinematics-based controller, suitable for real-time applications, is proposed, and its performance is tested in various scenarios to assess its effectiveness. The controller is applied to robots with two different linkage lengths, one optimized for maximum and the other for minimum torque requirements. The results show that the optimization reduces the required torque by nearly 42% when comparing the maximum to the minimum case.

Keywords: Quadruped robots, Genetic algorithm, Model-independent controller, Four-bar linkage, Mechanism optimization.

1- Introduction

Legged robots, with their unique abilities, can traverse uneven terrain more easily compared to wheeled robots. Due to the configuration of these robots, their locomotion can adapt to the environment [1, 2], allowing them to handle tasks in unknown or unpredictable conditions. Researchers have also explored combining wheeled and quadruped robots to leverage the advantages of both systems [3].

Among legged robots, quadrupeds are a major area of research because their design offers greater stability compared to bipedal robots. The design of quadruped robots is often inspired by nature, with researchers trying to mimic the locomotion of animals like dogs [4], cats [5], turtles [6], and cheetahs [7]. These robots can be deployed in environments that reduce the risk of human injury while improving overall performance. Some of their applications include inspection, search and rescue, delivery, monitoring, and more.

Quadruped robots can be classified into different categories based on their actuators, topology, configuration, and more [8]. Depending on the robot’s application, the type of actuators may vary. The most common actuators used in such robots are electrical, pneumatic, and hydraulic. Hydraulic actuators are typically used for tasks that involve carrying heavy loads. Robots like BigDog, LS3, and WildCat are examples of those utilizing hydraulic actuators [9, 10, 11]. Electrical actuators, which are gaining popularity in recent research, have been increasingly adopted. For instance, MIT's new electrical actuator design for impedance control in quadruped robots has led many researchers to switch to electrical actuators. Mini Cheetah is a well-known example of a lightweight quadruped robot using these actuators [12,13]. Another prominent robot is ANYmal from ETH Zurich, which is recognized for its ability to operate in unknown environments and is often used for industrial inspection. Lastly, pneumatic actuators are less commonly used due to their lower control accuracy. However, some researchers favor them for their lightweight properties [14].

One of the critical components of a quadruped robot that has a tremendous influence on the robot's locomotion stability and adaptability is robots’ legs [15]. There are various types of leg designs that can categorize quadruped robots into different groups. Some examples include rigid legs, articulated legs, parallel mechanisms, and spring-loaded legs, among others [16-18]. Some researchers are exploring flexible materials to enhance robot degrees of freedom and ensure safer interaction. One innovative approach involves using honeycomb-structured flexible materials for robot legs, combined with pneumatic actuators. This design effectively reduces the robot's total weight while maintaining functional integrity [19]. Additionally, leg structures incorporating tensegrity mechanisms have been explored in research [20]. The study primarily focused on enhancing the payload capacity of the tensegrity mechanism to make it viable for use in quadruped robots.

Quadruped robots can also generate different gaits through various combinations of leg movements, which contribute to both stability and locomotion speed. Common gaits include walking, trotting, pacing, and more [21]. Recently a study investigated the relationship between different gaits and stability by employing mathematical models rooted in spiral theory [22]. Similarly, the main body of the robot (torso) can be divided into two categories: rigid and flexible [23, 24]. In flexible torsos, the robot can achieve higher speeds, but designing an efficient control algorithm for such torsos is more challenging compared to rigid ones.

Prior to motion control, path planning has been a critical area of investigation in robotics research. For instance, two staged optimization approach was explored in a study to enhance the robot's performance in densely clustered environments [25]. The control algorithms for quadruped robots can be divided into model-independent and model-based methods. Generally speaking, each leg of the robot has three degrees of freedom: two for pitching and one for rolling [26]. Some model-independent methods use concepts like Central Pattern Generators (CPGs) to generate periodic motions [27, 28]. These types of oscillators require fewer feedback inputs compared to model-based controllers. For more robust control, researchers use model-based controllers. Due to the complex dynamics of the system, a simplified kinematic model that approximates the robot’s behavior is often employed in the control algorithm. The spring-loaded inverted pendulum (SLIP) model is commonly used as it approximates the behavior of such systems well [29]. This simplified model can be applied in methods like Model Predictive Control (MPC) to design control signals based on the predicted future response of the system [30, 31]. In contrast, some approaches rely on more accurate dynamic modeling, such as Whole Body Dynamics [26]. Also, a virtual method has been presented that models the ground reaction as a virtual spring. The task of the controller is to control the reaction virtual forces via actuators through the Jacobian matrix [33].

An accurate design plays a significant role in the stability, agility, and performance of quadruped robots [26], and therefore must be carefully considered. In this article, we focus on the design of quadruped robots equipped with electrical actuators and a four-bar linkage mechanism. To achieve optimal performance in force transmission, a heuristic optimization approach has been applied to the link lengths. Subsequently, the kinematic and kinetic equations of the robot have been derived to design a PD controller based on the dynamic model. Finally, the results of the controller, both with and without optimization, have been compared to evaluate the effectiveness of the design. This article employs a genetic algorithm to optimize the lengths of a quadruped robot's links based on a proposed cost function and constraints. The optimization reduces the required torque for forward motion, enabling the use of lightweight actuators in the robot's main torso, improving performance and reducing costs. Additionally, a model-independent controller is presented for real-time applications. The article systematically derives and explains the robot's kinematics and dynamics, emphasizing efficiency in design and control.

In the first section, the 4-bar linkages of the robot's leg are optimized through the kinematic equations and transmission ratio. In the next section, the entire kinematics of the leg is expressed, and the robot's desired path, considering the kinematic equations, is developed using polynomial equations. Subsequently, the dynamics of the robot are modeled and verified. After this section, the control algorithm is presented based on two phases of the robot's dynamics. The results section demonstrates the control performance of the proposed algorithm and compares the results of the length optimization. Finally, the findings are summarized, and future work is suggested.

2- Four bar linkage

The robot's conceptual design of robot is illustrated in Fig. 1. Each leg of the robot consists of two joints for pitching (the hip and knee joints) and one joint for rolling (the thigh joint), resulting in three degrees of freedom (DOF) to control the robot's movement. To reduce the inertia of the robot's leg, no actuators are placed directly on the leg itself. Instead, all motors are located on the torso of the robot, necessitating a transmission mechanism for the knee joints. Two common methods for transmitting knee torque are timing belts and four-bar linkages [34, 35]. The concept of the robot's leg for pitching is depicted in Fig. 2.

 

Fig. 1 The conceptual design of quadruped robot.

The four-bar linkage shown in Fig. 2 functions like a gear with a unique gear ratio. Consequently, the lengths of the four-bar links are optimized to achieve maximum torque at the output.

 

Fig. 2 Robot’s leg considering 4 bar mechanism and showing the simplified version as a 2 link.

The dynamics of the robot's leg is divided into two parts. In the swing phase, the actuators function to step forward without any contact with the ground. In contrast, during the stance phase, the robot remains in contact with the ground. The main objective of the optimization is to reduce the torque density of the actuators. This can be achieved by selecting appropriate link lengths to enhance force transmission efficiency. Therefore, the four-bar linkage mechanism needs to be analyzed.

Figure 3 illustrates the four-bar linkage used in the robot's leg.

Fig. 3 The 4-bar mechanism of leg.

Since the four-bar linkage is a one-degree-of-freedom (DOF) system, only the angle ​ is considered as the input angle generated from the desired angle in the robot's leg, which is given by . Based on the angles of the other links can be expressed as follows:

In Eqs. (1) and (2), the and  equals to:

	(1)
	(2)

which  is calculated as below:

	(3)
	(4)

In a conventional four-bar mechanism, the angle  and the distance  between the first and last joint (as shown in Fig. 3) are fixed. However, due to the rotation of the robot's actuator, these values can change, as illustrated in Fig. 2.

The angle  is given by:

(5)

which  and  are the distance between robot’s actuators according to Fig. 2. The distance  is calculated as:

(6)

As all of the angles in the four-bar linkage are defined, the transmission angle, which relates the output to the input torque, can be formulated. The force diagram of the four-bar linkage is shown in Fig. 4.

 

Fig. 4 The force diagram of four bar linkage.

As shown in Fig. 4, the output torque to the input torque is given by:

(7)

The angles between the linkages are denoted by and , as depicted in Fig. 4. Since the robot's leg has low inertia during the stance phase, we can consider it a stationary phase, making Eq. (8) criteria applicable in this context.

The main objective of the optimization is to determine the lengths that maximize the transmission ratio. This can be achieved by selecting appropriate lengths. To find the optimal lengths, a genetic algorithm is employed, as it is well-suited for handling the numerous constraints present in the mechanism through principles of natural selection. The chosen cost function is defined as follows:

(8)

The first part of Eq. (9) calculates the absolute value of the transmission ratio for different input angles, while the second part checks the feasibility of these angles and lengths within the four-bar mechanism. This is defined as follows:

(9)

which function  is derived from Freudenstein's Equation [36] and is expressed as follows:

(10)

In Eq. (10), if the configuration is valid, the combination yields a positive value that contributes to increasing the cost. Conversely, if the possibility condition is violated, the response is penalized with a negative cost.

 

Fig. 5 The cost function of optimizing lengths over the iteration.

To maximize the cost function, the offspring population (60 percent of the parent population) is generated through single, double, and uniform crossovers. Additionally, the mutant population (90 percent of the total population) is created by applying a normal probability function to the parents. The roulette wheel selection method is employed for choosing parents, and the termination condition is based on reaching the maximum number of iterations. Also, there are 150 chromosomes in the population, and each chromosome contains 3 genes, which represent the lengths of the four-bar mechanism. Before conducting the optimization, the intervals for angles  and ​ were selected to ensure that the robot could operate within a desired workspace for a specific geometry, as it is not designed to reach arbitrary angles. The range for ​ was discretized into 10 steps spanning from −110 deg to −140deg, while ​ was defined across 10 steps ranging from 20deg to 70deg. There is one extreme case that can lead to a high value of the transmission ratio, which occurs when links 2 and 3 are aligned. The genetic algorithm tends to reach this configuration in each run; however, due to the singularity associated with this alignment, it is considered a violation and is thus avoided. The resulting lengths of the links are  and  respectively. Fig. 5 illustrates the convergence plot over the iterations.

To compare these results with other values, the minimum of the cost function is derived by inverting the first part of the formula. Table 1 presents three cases of the robot's leg, two of which are generated from the genetic algorithm, while one represents a feasible solution for the robot's configuration.

Table 1: The results of GA optimization on robot’s links.

(11)

 

In the next section, the kinematic model of the robot's leg is developed, and the desired trajectory for the robot's leg is presented.

3- Kinematic analysis

In the previous section, the kinematic equations of the four-bar linkage were developed. To determine the desired angles of the links, the inverse and forward kinematics of the two-bar linkage (as shown on the right side of Fig. 2) must be formulated. The Denavit-Hartenberg (DH) parameters for the series section are shown in Table 2.

Table 2: The DH parameters of series links.  denotes link’s number.

Response	(cm)	(cm)	(cm)	(cm)	(cm)
Ordinary response	6	15	6	15	15
Maximum ratio	18.22	19.77	18.22	15	15
Minimum Ratio	12.88	8.83	12.88	15	15

 

The position of the leg's end effector for rolling in the ​ coordinate system is given below:

0			0
0			0

The velocity of end effector, according to Eq. (12) is given by:

(12)

which matrix  can be represented by:

(13)

By inverting Eq. (14), the joint angular velocities of the robot can be calculated for a given desired trajectory.

(14)

The desired angles of the two links can be obtained by integrating the desired angular velocities in Eq. (15).

Now, the desired trajectory of the robot’s leg can be developed. Generally, the robot’s movement consists of two phases. Here, the desired trajectory for the swing phase is developed. Polynomial functions are used to define the desired trajectory based on the problem's conditions [37]. While other methods, such as trigonometric and Bezier functions, can also represent the desired trajectory, polynomial functions are chosen because they meet the specific requirements for velocity and position constraints.

In order to reduce the order of the polynomials, the swing phase trajectory is divided into two parts: the acceleration phase and the deceleration phase [37]. The conditions for these two phases follow the same concept, which requires that the initial and final positions, velocities, and accelerations must be continuous. As an example, the conditions for the x-direction in the acceleration phase can be written as follows [37]:

 

(15)

The subscripts  and  refer to the swing and stance phases, respectively. The value following the motion phase indicates the time, where  represents the initial time of the phase, and  represents the final time. The parameter α represents the acceleration during the acceleration phase and  declares the step length. Given the six specified conditions (continuity of position, velocity, and acceleration), the polynomial has an order of 5. During the robot's stance phase, the controller's primary task is to propel the robot forward. Following the stance phase, the robot transitions into the swing phase. Utilizing (16) conditions as an example for the acceleration path of the leg endpoint in the x-direction (the deceleration phase and y-direction can be similarly derived), the robot’s virtual path in the stance phase defines the desired swing phase trajectory. By fixing the torso and assuming inverse leg movement, terms such as ​​ can be computed.

The desired polynomials, based on the leg's conditions, can be expressed as follows:

 

(16)

 

Based on the mentioned conditions, the constant terms in Eqs. (17) and 18) can be computed. These constants depend on the virtual path of the stance phase, which is determined through the control algorithm in real-time operation. In Fig. 6, the desired trajectory is depicted for a maximum height of 8 cm, an initial position () of 10 cm in the x-direction, and a swing time of 0.7 seconds. This trajectory ensures smooth transitions during the swing phase while adhering to the specified conditions in Eq. (16).

 

Fig. 6 The swing phase desired trajectory based on the polynomials function.

The derivative of the polynomial function in Eqs. (17) and (18) with respect to time gives the desired velocity trajectory, which can then be used in the kinematic equations (Equation (15)) to determine the desired joint angles for the robot's leg. Once the desired angles for the two main links are found, the desired angle for the 4-bar mechanism can also be evaluated according to past section. Fig. 7 illustrates the robot's desired joint angles derived from the given trajectory, showing how the robot’s leg moves according to the predefined motion path.

 

 

Fig. 7 The angle of links according to desired trajectory during the motion.

In Fig. 7, the link lengths correspond to the ordinary case presented in Table 1. In the following section, the dynamics of the robot will be enhanced to simulate the robot's response to the actuator forces.

4-Dynamics equation

Several gaits are inspired by nature, each differing in the number of phases and combinations of leg movements [38]. In this project, the trotting gait has been selected to move the robot. The sequence of leg movements is shown in Table 3. The trotting gait can reach speeds where inertia must be considered in the dynamic equations.

Table 3: Leg movement in trotting phase. Each cycle contains two phases.

	(17)
	(18)

leg	1	2	1	2	1	2	1	2
Left front
Left rear
Right front
Right rear

 

To calculate the robot's dynamics, the Lagrange method is applied. Based on the robot’s gait, the dynamics are divided into four parts within each cycle. In the first part, two diagonal legs are in the swing phase, while the other two diagonal legs are in the stance phase. Once the swing legs make contact with the ground, the legs switch, and the same pattern is repeated for the other diagonal legs. Thus, each cycle consists of a swing-stance phase for the diagonal legs (four legs) and an impact phase for the swing legs (two legs). The stance phase introduces constraints on the robot’s legs, as they can be treated like joints, assuming sufficient friction. Consequently, the Lagrange equation can be expressed as follows [39]:

In Eq. (19),  represents the Lagrangian, which is the difference between the kinetic energy  and potential energy  ().  denotes the virtual forces, while  and ​ are the generalized coordinates and velocities, respectively.  represents the Lagrange multipliers. The matrix  arises from the constraints and can be written as follows:

(19)

To simulate the robot's response, the augmented Lagrange method is used, which can be expressed as follows:

(20)

In Eq. (21),  represents the number of generalized coordinates, while  denotes the number of constraints.

To verify the correctness of derived equations, the time derivative of the total mechanical energy is examined. In the absence of virtual forces, if the derivative of the total mechanical energy equals zero, it can be concluded that the equations are correct. The time derivative of mechanical energy is expressed as follows:

(21)

where  represents the total mechanical energy, which is the sum of kinetic and potential energy (). Due to the high number of constraints in the system, there may be intervals where the constraint equations are not fully satisfied. Therefore, to ensure the validity of the derived equations, the following condition is checked:

(22)

Each leg contains 3 degrees of freedom (DOF), which can be represented by 5 generalized coordinates and 2 constraints. The constraints mentioned here arise from the 4-bar linkage. For example, the right front leg has the generalized coordinates . Therefore, the robot's legs collectively have a total of 15 generalized coordinates, represented as follows:

(23)

which  represents the leg number.

If any leg is in the stance phase, additional constraints are applied to it: the velocity of the end effector in the ground plane, due to friction, is equal to zero.

The torso of the robot contains 6 degrees of freedom (DOF) in space, represented as . The  coordinates are expressed in the global coordinate system, while  represent body-fixed rotations. The first rotation, , is around the vertical (gravity) axis,  is around the transverse direction, and the last rotation, , is around the longitudinal axis. The angular and linear velocity of the main body can be expressed as follows:

 

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