Figure 2 shows the mechanical structure of the quadruped robot model. The robot model comprises a front fork, rear frame, and four legs with a pantograph mechanism. The legs are connected to the front fork and rear frame. The front fork and the rear frame are perpendicularly connected with a trunk joint serving as a steering joint. The trunk joint rotation makes the front legs rotate in the axis of the trunk joint, as shown in Figure 2B. The inclined front fork, adjusted by α , provides negative feedback from tilt to steering, resulting in self-stabilization, similar to the bicycle model described in (Åström et al., 2005).

The leg structure has a pantograph mechanism with four DoF, as shown in Figure 2C. The hip joint is an actuated rotary joint, which swings the leg in the anterior and posterior directions. The knee joint is an actuated rotary joint, which flexes and extends the leg with a pantograph mechanism. The two prismatic joints at the hip and the pantograph are passive spring dampers acting as suspensions.

Drawing inspiration from the bicycle balance control, as described in Equation 2, we designed a balance control steering the trunk DoF angle η , as follows: η ̄ = − a θ r − ζ (5) τ = − k p η − η ̄ − k d η ̇ (6) where η ̄ denotes the target angle of the trunk joints, a is the weight of the feedback gain and θ r is the roll angle of the rear frame, similar to the variable θ in Equation 4; ζ denotes the bias that controls the moving direction. The generated torque τ at the trunk joint is controlled by proportional-derivative (PD) control in response to the target angle η ̄ and angular velocity η ̇ of the trunk joint; k p and k d are the proportional and derivative gains, and η ̇ is the angular velocity of the trunk angle. The steering control is assumed to generate a centrifugal force that contributes to model balance.

The leg actuators are controlled by PD control and make the foot tip draw the target foot trajectory, as shown in Figure 3. The trajectory was designed to emulate the cheetah’s high-speed locomotion pattern (Zhang et al., 2022). Key characteristics include the initiation of backward leg movement during the swing phase and the considerable clearance. The frequency is controlled by phase oscillators implemented in each leg. The oscillator phase ϕ i determines the target angles of the hip joint ψ ̄ i hip and the knee joint ψ ̄ i hip as shown in Equations 7–9, ψ ̄ i hip = ψ i hip,c + ψ i hip,a ⁡ sin ϕ i (7) ψ ̄ i knee = ψ i knee,c − ψ i knee,a ⁡ tanh ρ ϕ i − π 2 0 ≤ ϕ < π ψ i knee,c + ψ i knee,a ⁡ tanh ρ ϕ i − 3 π 2 π ≤ ϕ < 2 π , (8) ϕ ̇ i = ω , (9) where ψ i hip,c is the offset angle of the hip joint and ψ i hip,a is the amplitude of the hip joint; ψ i knee,c is the offset angle of the knee joint and ψ i knee,a is the amplitude of the knee joint, and ρ is a positive constant; ω denotes the angular frequency of the oscillator ϕ i . Here, the index i denotes the leg identifier (left fore: LF, right fore: RF, left hind: LH, and right hind: RH). The initiation of backward leg movement during swing phase is designed to set the offset angle of the hip joint ψ i hip,c , and the clearance from the ground is configured to determine the amplitude of the knee joint ψ i knee,a .

Quadruped animals can move in various gaits (Alexander, 1984) unlike bicycles. The experiment investigated the gait variations of the robot model. The gait pattern is determined by the phase relationships between oscillators. We evaluated the locomotion performance for various gait patterns, which determine the left-right phase relationship Δ ϕ L R and the fore-hind phase relationship ϕ F H derived as shown in Equations 10, 11, Δ ϕ L R = ϕ L F − ϕ R F = ϕ L H − ϕ R H , (10) Δ ϕ F H = ϕ L F − ϕ L H = ϕ R F − ϕ R H . (11) Here, when Δ ϕ L R = 0 [rad], the left fore leg and the right fore leg are inphase, ϕ L F = ϕ R F . The hind legs are also the same. When Δ ϕ F H = π [rad], the legs of the ipsilateral side are antiphase, ϕ L F = ϕ L H + π . We investigated the combination of Δ ϕ L R from 0 to π [rad] and Δ ϕ F H from 0 to π [rad]. The other experimental conditions are as follows: the phase angular velocity ω = 28 [rad/s], the angle bias ζ = 0 [deg], the experimental duration 15 [s], and ten trials for each phase relationship. In each trial, the initial phase ϕ L F was set incrementally from 0 to 1.8 π in steps of 0.2 π . To evaluate the performance, we measured the locomotion sustainability. The gait pattern is regarded as sustainable when the running speed is 2.0 [m/s] or more at the end of the experiment.

The simulation results are shown in Figure 4. The success rate represents the proportion of sustainable gait shown within ten trials for each parameter set. There are two parameter areas showing a higher success rate, although the robot cannot achieve stable running within most of the parameter set. The first area, Δ ϕ L R close to π [rad] and Δ ϕ F H close to 0.3 π [rad], is a four-beat gait pattern similar to walking gait shown in slow locomotion of quadruped animal (Alexander, 1984). The second area, Δ ϕ L R close to π and Δ ϕ F H close to π is a two-beat gait pattern similar to trotting gait shown in intermediate-speed locomotion of quadruped animal. These results demonstrate that the robot model can achieve stable movement when following specific animal gait patterns. On the other hand, the robot cannot run at Δ ϕ L R close to zero, similar to galloping gait, which is the fastest gait of quadruped animals. This instability can be attributed to the characteristics of galloping, which involves a phase where both the left and right legs are off the ground. These characteristics are incompatible with the bicycle-inspired balance control, which assumes continuous grounding. Based on the results, we adopted the trot gait parameter set ( Δ ϕ L R = Δ ϕ F H = π ) for the following experiments.

We conducted disturbance experiments that added the external force to the running robot model. The experimental duration is 7 [s], and the external force is 20 [N] in the x direction applied to the center of gravity of the rear frame from 4.0 to 4.5 [s]. The other experimental conditions are as follows: the phase angular velocity ω = 28 [rad/s], the experimental duration 15 [s], the initial phase and phase relationship parameters ( ϕ L F , Δ ϕ L R , Δ ϕ F H ) = ( 0 , π , π ) [rad] that generate a trotting gait.

We observed the resulting behavior, as shown in Figure 5 and Supplementary Video S1. Figure 5A shows the time evolution of the roll angle θ r and the steering angle η . Due to the disturbance, the roll angle tilted to 0.23 [rad] at 4.5 [s] and recovered to the neutral angle at around 6.0 [s]. The steering angle was controlled in response to the roll angle following Equations 5 and 6. The time evolution of the velocity, as shown in Figure 5B, also presents the impact of disturbance that decreases the speed temporarily.

Figure 5C shows the relationship between the roll angle θ r and the angular velocity θ ̇ r . The color of the lines indicates the time zones: black for before the disturbance, red for during, and blue for after. The trajectory after the disturbance converges to the same range as the trajectory before the disturbance. It indicated that the robot’s posture has recovered, and the robot shows robustness against the external force.

Figure 5D illustrates the robot’s running trajectory on the x y plane. The movement direction aligns with the steering angle as the disturbance is applied. These results show that the proposed robot model can recover its posture from the external forces by steering the trunk joint in response to the roll angle and steering the trunk joint changes the direction of movement.

Next, we investigated the relationship between the oscillator angular velocity ω , related to the locomotion speed, and the robustness against external force. The experimental parameters are the oscillator angular velocity ω ranging from 20 to 38 [rad/s] and the external forces ranging from 0 to 45 [N]. For each parameter set, ten trials were conducted with the initial phase and phase relationship parameters set to ( ϕ L F , Δ ϕ L R , Δ ϕ F H ) = ( π , π , π ) . To evaluate robustness across various postures, an external force was applied for 0.5 s, with the application timing shifted in increments of 0.1 of the gait cycle period ( 2 π / ω [s]), beginning at 4 [s]. To evaluate performance, we used locomotion sustainability, as described in Section 4.1, as the metric.

The simulation results are shown in Figure 6. The success rate represents the proportion of sustainable gaits demonstrated after the disturbance, based on ten trials for each parameter set. There is a tendency that higher ω provide robustness against the greater external forces. When ω = 34 [rad/s], the model can withstand a disturbance of 45 [N] in some trials. The reduced robustness against disturbances observed at ω = 36 [rad/s] is likely due to insufficient steering feedback gain a for high-frequency locomotion. This property is consistent with the bicycle balance control, in which restoring force increases with speed, as described in Section 2.

The steering handle is used to change the direction of movement of the bicycle, as well as to control balance. We conducted turning experiments to investigate the maneuverability of the proposed model. The turning behavior is generated by controlling the roll angel bias ζ as described in Equation 5. The experimental duration is 11 [s]. At 4 [s], ζ is set to 8.6 [deg]. When the robot achieves the target yaw angle, ζ is set to zero. The target yaw angle is π / 2 [rad]. The other experimental conditions are as follows: The oscillator angular velocity ω = 28 [rad/s], the initial oscillator phase ( ϕ L F , ϕ R F , ϕ L H , ϕ R H ) = ( 0 , π , π , 0 ) [rad].

We observed the turning behavior, as shown in Figure 7 and Supplementary Video S1. Figure 7A shows the time evolution of the roll angle θ r , the trunk joint angle η , and the roll angle bias ζ . The angle bias ζ shifts to 8.6 [deg] from 4.0 until 7.7 [s], as shown in the green area. The trunk joint angle was controlled to achieve the angle bias ζ and make the roll angle tilt. During turning, the running speed slightly decreases, as shown in Figure 7B. Figure 7C shows the time evolution of the yaw angle of the robot, i.e., the direction of movements. The robot started to change the direction of movement at 4.0 [s] and reached the yaw angle of π / 2 [rad] at 7.7 [s]. Subsequently, the yaw angle remained stable at π / 2 [rad]. These movements are observed from the trajectory path, as shown in Figure 7D. The color of the lines indicates the time zone: black for before the turning, green for during, and blue for after. These results showed that the robot model changes the moving direction by simply changing the roll angle bias ζ .

Next, we investigated the relationship between the turning performance and the parameter sets of the oscillator angular velocity ω and the roll angle bias ζ . The parameter sets are ω ranging from 18 to 36 and the maximum value of ζ ranging from 4 to 14. In each trial, the roll angle bias ζ was set to change linearly, as shown in Figure 8. The turning behavior was evaluated during 7–11 [s] where ζ reaches its maximum value. The turning radius is calculated from the ratio of the mean speed to the rotational velocity of the rear frame over the evaluation period. To evaluate performance, we used velocity, turn radius, and average roll angle. The experimental duration is 14 [s].

The simulation results are shown in Figure 9. Each color map shows velocity, turn radius, and average roll angle, respectively. The black area represents the trials in which falls occurred. The combination of lower ω and higher ζ tends to result in falls due to insufficient speed for sharp steering.

Figure 9A shows the velocity. The higher ω tends to run at a higher velocity. Figure 9B shows the turn radius. The higher ω tends to be a larger turn radius. It can be assumed that the higher velocity generates greater centrifugal forces to resist sharp turning. Besides, the larger angle bias ζ tends to be a smaller turn radius. Figure 9C shows that the angle bias ζ causes the roll angle to tilt, exhibiting a behavior similar to lean-in turning, which helps maintain high running speed, as observed in bicycle turning. The angle bias ζ affects the turn radius and not velocity, as shown in Figures 9A,B. The maximum rotational angular velocity reached 22.8 [deg/s], and the roll angle magnitude was 8.7 [deg] at the parameter set of ω = 22 [rad/s] and ζ max = 11 [deg]. The results show that the proposed model changes the moving direction by simply changing ζ and performs lean-in turning, maintaining a high running speed, as shown in bicycle turning.