Analysis of a Four-Legged Robot Kinematics during Rotational Movements of Its Body

Introduction. Currently, walking mobile robots are widely used to solve problems of observation and exploration of territories [1]. An example of one of the common designs of such robots is the four-legged robot Mini Cheetah (Fig. 1), developed at MIT Biomimetic Robotics Lab (https://biomimetics.mit.edu). These four-legged robots can move on various types of terrain, unlike wheeled or tracked robots.

The design of the walking robots under consideration includes a platform to which four legs are attached. Each of the legs is a three-link manipulator. One of the key aspects of a walking robot is the topology of its leg structure [2][3]. The topology of the legs refers to the composition and arrangement of the links, as well as the hinges of the limbs of a four-legged robot, which play an important role in providing its movement and stability. The development of an optimal topology for the leg structure of quadruped robots is one of the challenges in engineering. The solution to this problem not only helps to provide stable and efficient gait (locomotion) of the robot on various surfaces, but also makes its mechanical design more reliable and energy efficient.

Another aspect that needs to be considered is the drive system applied in quadruped robots [4]. Different types of drives are widely used in such robots, including pneumatic [5], hydraulic [6][7], and electric [2][6], each of which has its own advantages and disadvantages.

Four-legged robots have complex kinematics due to a large number of degrees of freedom. The robot considered in this paper has 18 degrees of freedom. To provide successful locomotion of four-legged robots, it is necessary to solve both direct and inverse problems of kinematics, which allows organizing dynamic control of the position of the legs of these robots. This control should provide the robot balancing, the ability to overcome various surfaces, and adaptation to various types of locomotion, including walking, running and jumping. The direct problem of walking robot kinematics is to determine the position and orientation of the robot platform based on the known rotation angles of the leg links. The inverse problem is to find the rotation angles of the leg links that provide a given position and orientation of the robot platform.

Paper [8] described in detail the application of the Denavit-Hartenberg method for modeling the forward and inverse kinematics of a walking robot. The study on the kinematic model of the robot was conducted with the aim of creating a gait algorithm.

The authors of [9] developed advanced kinematic and dynamic modeling methodologies based on the screw theory for quadruped robots. The proposed methods accounted for different gaits and leg mechanism topologies using simplified models of foot contact (e.g., a ball-and-socket model). Models were described for three separate phases of robot motion: standing, walking, and unsupported (flying) phase. The developed control strategies used the presented models for gait planning and jogging.

In recent years, scientists have been working on control and trajectory planning systems. The motion control method is the basis of a four-legged robot, directly affecting flexibility, stability and the ability to adapt to various support surfaces [3].

The authors of [10] specified the design of a regulator for the force control of a four-legged robot. The main purpose of the regulator was to equalize the forces acting on the robot during symmetrical gait, as well as to reduce disturbances and absorb impacts.

In [11], a new motion control strategy for a quadruped robot that enabled it to navigate efficiently on uneven terrains regardless of visual perception conditions was proposed. Paper [12] discussed the design of a motion controller for a quadruped robot that used torque optimization and performance control to account for the impact of unpredictable external forces such as surface irregularities.

Experimental studies devoted to the application of hierarchical controllers for motion control of quadruped robots were presented in [13], illustrated by the ANYmal robot, and in [14], exemplified by the StarlETH robot.

In [15], the authors discussed the development of a quadruped robot designed for construction and emergency response tasks. Their paper examined the statics, kinematics, and control system for the specific design of this robot.

Finally, in [16], a new central pattern generator model controller and gait switching tactic based on the Wilson-Cowan model were presented. This development aimed to create a smooth robot gait and reduce the adjustment time in the oscillating mechanical system.

Despite the development of the problem of modeling the kinematics of quadruped robots, in existing publications there was no complete kinematic model of the motion of robots whose leg design was similar to the Mini Cheetah robot (Fig. 1). Therefore, the objective of this work was to develop a kinematic model of a quadruped robot using the example of the Mini Cheetah and to develop proposals for planning the movements of the robot links that could provide the rotation of the robot around various axes.

To achieve the stated goal, we needed to formulate the laws of change of coordinates of the endpoints of the robot legs and angular coordinates of its body. It was also necessary to solve the inverse problem of robot kinematics to check the operability of the proposed motion plan. The results obtained can be used to develop robot control at the kinematic level.

Materials and Methods. System Definition. The object of the study is a robotic system consisting of a rigid body (robot platform) and four legs with three degrees of freedom for each of the legs. The links of the legs are connected to the robot platform by several hinges, which are driven by electric drives. Angle β_i is the angle of rotation of link O_iA_i around the longitudinal axis of the robot body and characterizes the rotation of the plane of the leg (containing points A_i, B_i, K_i) relative to the vertical plane of symmetry CXZ. And angles α_i, β_i characterize the rotation of links A_iB_i and B_iK_i in the plane of the leg. The hinges of the leg links at points O_i, A_i, B_i are controlled, i.e., the rotation of the leg links at angles φ_i, α_i, β_i is provided independently by the corresponding drives.

Fixed coordinate system XYZ and moving coordinates Cxyz are introduced, where point C is the geometric center of the robot platform (Fig. 2 a). The orientation of the robot platform is described by course angles ψ (rotation around axis z), roll θ (rotation around axis x) and pitch γ (rotation around axis y), and the position of the robot platform is determined by the coordinates of the geometric center x_C, y_C and z_C in the fixed coordinate system.

The distances along the longitudinal and transverse axes of the robot platform from the center of mass C and the attachment points of the legs to the robot platform O_i in the moving axes x_iy_iz_i are specified by ρ_xi and ρ_yi. The lengths of the links are designated as , (Fig. 2 b).

In projections onto the axes of the fixed coordinate system XYZ, the position vector of the endpoint of the leg K_i (or the point in contact with the surface) is determined from the formula:

(1)

where — position vector of the geometric center of the robot platform C; Г_ψγθ = Г_ψГ_θГ_γ — rotation matrix, which is calculated as the product of the elementary rotation matrices by the course, roll, and pitch angles. The rotation matrices included in equation (1), have the form:

(2)

Other radius vectors included in equation (1) are determined from the expressions:

(3)

here, the superscript “T” means the vector transpose operation.

After performing intermediate calculations on the right side of equation (1), taking into account expressions (2) and (3), we obtain projections of the position vector r_Ki for each leg.

Differentiating equation (1) with respect to time, we obtain a kinematic model in the form:

(4)

where — vector of angular generalized coordinates of the robot platform; — vector of angular generalized coordinates of the robot leg links; — vector of linear velocity of the endpoint of the i-th leg; — vector of linear velocity of the geometric center of the robot platform; — matrices of coefficients for the corresponding vectors of generalized velocities.

Research Results. To analyze the rotational motions of the robot by solving the equations of motion geometry (1) (or integrating the kinematic equations (4)), we set the initial conditions for the rotation angles of the leg links and the laws of change in the velocities of the endpoints of the legs v_Ki, the geometric center of the robot platform v_C and the vector of angular generalized coordinates q_ψθγ. Considering the rotational movements of the robot platform, we assume that the geometric center is stationary during the movement v_C = 0.

As the initial position of the leg links, we take the following angle values φ_i, α_i, β_i for each of the legs:

(5)

The given combination of angle values φ_i, α_i, β_i corresponds to the robot standing on semi-bent legs, which is presented in a simplified visualization (Fig. 3) made in Wolfram Mathematica. The following values of the robot geometric parameters were used in the visualization: l₀ = 0.15 m, l₁ = 0.45 m, l₂ = 0.45 m, ρ_x1 = ρ_x2 = –ρ_x3 = –ρ_x4 = 0.5 m, ρ_y1 = –ρ_y2 = –ρ_y3 = ρ_y4 = 0.25 m.

To perform periodic rotary movements of the robot platform, limited only by the roll angle, we set the laws for changing the orientation angles of the platform in the form:

(6)

where ψ₀ = 0°, θ₀ = 0°, γ₀ = 0° — initial values of orientation angles; A = 0.2 rad and Т = 20 s — constants characterizing the amplitude and period of oscillations by angle. Since it is optional to move the legs when the robot platform is rotated along the roll angle, the coordinates of the contact points of the legs with the support surface remain unchanged (v_Ki = 0):

, (7)

where x_Ki(0) and y_Ki(0) — initial coordinates of the contact points of the robot legs with the support surface.

Solving equations (1) with respect to (2) and (3), when substituting the program motion (6) and (7), we obtain the dependences of the rotation angles φ_i, α_i, β_i of the leg links (Fig. 4).

According to the results of the simulation presented in Figure 4, it can be noted that the rotation angles of the leg links are harmonic functions. As a consequence, it can be mentioned that the rotations of the robot platform around the longitudinal axis of symmetry of the robot (6) can be performed without tearing the contact points of the legs from the surface.

To perform periodic rotary movements of the robot platform, only in pitch angle, we set the laws for changing the orientation angles of the platform in the form:

(8)

When the robot platform rotates along the pitch angle, the transfer of the legs is optional, and the coordinates of the points of contact of the legs with the supporting surface remain unchanged.

Solving equations (1) taking into account (2) and (3), and substituting program movements (7) and (8), we obtain the dependences of the rotation angles φ_i, α_i, β_i of the leg links (Fig. 5).

According to the simulation results presented in Figure 5, it can be noted that the rotation angles of the leg links are harmonic functions. As a consequence, it can be mentioned that the rotations of the robot platform around the transverse axis of symmetry of the robot (8) can be performed without tearing the contact points of the legs from the surface.

To perform rotation around axis Z, we consider the laws of changing the orientation angles of the robot platform in the form:

(9)

where ω_z = 0.085 rad/s angular velocity of rotation of the robot platform.

To provide the rotation of the robot around the vertical axis, it is required to plan the movement of the legs. Let us consider the case when the trajectories of the contact points of the legs in projections onto the horizontal plane XY belong to a circle of radius ρ. In this case, we will assume that the axis, around which the rotation of the robot platform occurs, passes through the geometric center of the platform. Then, the expressions for the coordinates of the endpoints of the legs can be determined from the relations:

(10)

where — angle characterizing the position of the endpoint of the i-th leg at the initial moment of time; h_K = 0.3 m — maximum height to which the endpoints of the robot legs rise; T₁ = 6 s — time of one step; Δ₁₃, Δ₂₄ — auxiliary functions used to provide a piecewise assignment of the change in the coordinates of the endpoints of the legs:

(11)

Here, function (t)mod(T₁) returns the remainder of the integer division t by T₁. The piecewise character of the coordinate change assignment is associated with the consideration of gaits (elementary leg movements), in which a sequential transfer of the crossed legs is performed.

Solving equation (1) with respect to (2) and (3), when substituting the program movement (9)–(11), we obtain the dependences of the rotation angles φ_i, α_i, β_i of the leg links (Fig. 6).

The simulation results show that the dependences of the rotation angles of the leg links are periodic functions and, as a consequence, the considered rotational movements of the robot platform can take place without the occurrence of singular configurations. Thus, the proposed plan for transferring the legs (10), (11) during rotation around vertical axis Z is suitable for implementing the rotation of the robot at an arbitrary angle ψ.

Discussion and Conclusion. The constructed mathematical model of kinematics allows for the determination of the angles in the joints of the robot leg links, which must be provided to implement the desired movement of the robot platform. The obtained results of numerical modeling of the rotary movements of the robot body confirmed the operability of the developed kinematic model for assessing the implementation of the specified robot movements, taking into account the limitations and dimensions of the structure.

In cases where the robot platform performs a flat motion (at zero pitch and roll angles), the constructed mathematical model of kinematics coincides with the published results of other authors. The results obtained in this paper are a generalization and refinement of the equations that couple the rotation angles of the robot leg links with the position and orientation of the robot platform.

The application of the developed kinematic model (1), (4), as well as the proposed plan for transferring legs (10), (11) during rotation around the vertical axis, will allow constructing control at the kinematic level for the robot movement along curvilinear trajectories. This approach will significantly expand the robot functionality and increase its performance efficiency under various operating conditions.