In my research on legged robotics, I have focused on the development and analysis of quadruped robot dogs, which offer significant advantages in mobility and adaptability across unstructured environments. This paper presents a comprehensive study on dynamic walking gait planning and simulation for a quadruped robot dog, inspired by biological principles and aimed at achieving stable, high-speed locomotion. I will delve into the mechanical structure, gait analysis, planning methodologies, and simulation results, emphasizing the trot gait—a diagonal synchronous pattern that enables efficient movement. Throughout this work, the term “robot dog” will be frequently used to highlight its application as a pet or service robot, with implications for disaster response, exploration, and entertainment. The integration of mathematical modeling, control theory, and simulation tools like Matlab allows for a rigorous validation of the proposed gait strategies, ensuring that the robot dog can navigate rough terrains with reliability and grace.
The walking mechanism of a quadruped robot dog is fundamental to its performance. I analyzed a typical robot dog model, such as the i-Cybie, which features four legs, each with multiple joints for articulated movement. Each leg comprises four joints: the hip joint, shoulder joint, knee joint, and ankle joint. Among these, the hip, shoulder, and knee joints are active, driven by DC motors to enable planar and spatial motions, while the ankle joint is passive, adjusting to ground contact for better balance and force distribution. This design mimics the biological structure of canine limbs, allowing for complex maneuvers like walking, trotting, and jumping. To facilitate analysis, I developed a simplified joint model, as shown in the schematic below, which represents the robot dog’s leg as a series of linked segments with rotational degrees of freedom. This model is crucial for deriving kinematics and dynamics equations that govern the robot dog’s motion.
In this joint structure, I define coordinate frames for each segment to compute forward and inverse kinematics. For instance, the position of the foot tip relative to the body can be expressed using homogeneous transformation matrices. Let the joint angles be denoted as $\theta_1$ for the shoulder, $\theta_2$ for the hip, and $\theta_3$ for the knee, with link lengths $l_1$, $l_2$, and $l_3$. The foot position $(x, y, z)$ in the body frame is given by:
$$ x = l_1 \cos(\theta_1) + l_2 \cos(\theta_1 + \theta_2) + l_3 \cos(\theta_1 + \theta_2 + \theta_3) $$
$$ y = l_1 \sin(\theta_1) + l_2 \sin(\theta_1 + \theta_2) + l_3 \sin(\theta_1 + \theta_2 + \theta_3) $$
$$ z = 0 \text{ (for planar motion)} $$
For 3D motion, additional transformations account for the shoulder’s lateral swing. This kinematic model enables precise control of the robot dog’s leg movements during gait cycles. To summarize the joint parameters, I present the following table:
| Joint | Type | Motion Plane | Actuation | Range (degrees) |
|---|---|---|---|---|
| Shoulder | Active | Lateral (inner/outer) | DC Motor | -30 to 30 |
| Hip | Active | Sagittal (forward/backward) | DC Motor | -45 to 45 |
| Knee | Active | Sagittal (forward/backward) | DC Motor | -60 to 60 |
| Ankle | Passive | Adaptive | Spring-loaded | -15 to 15 |
Gait planning for a robot dog involves coordinating leg movements to achieve stable locomotion. A gait is defined as a sequence of leg lifts and placements, characterized by phases: the stance phase (supporting the body) and the swing phase (moving forward). The gait cycle is the time taken for all legs to complete one full sequence, and the duty factor $\beta$ is a key parameter representing the fraction of time a leg is in the stance phase. For a quadruped robot dog, $\beta$ determines the gait type: when $\beta > 0.5$, the robot dog walks slowly with three legs always on the ground; when $\beta = 0.5$, it trots with two legs in stance at any time; and when $\beta < 0.5$, it runs or gallops with periods of no ground contact. The stability of the robot dog depends on keeping the center of gravity (CoG) within the support polygon formed by the stance legs. I analyze this using the zero-moment point (ZMP) criterion, where the net moment about the CoG should be zero to prevent tipping. For a trot gait with $\beta = 0.5$, the CoG must lie on the diagonal line connecting the two stance legs, ensuring dynamic balance. The duty factor is calculated as:
$$ \beta_i = \frac{T_{\text{stance}, i}}{T_{\text{cycle}, i}} $$
where $T_{\text{stance}, i}$ is the stance time for leg $i$, and $T_{\text{cycle}, i}$ is the total cycle time. In my planning, I set $\beta = 0.5$ for the trot gait, enabling the robot dog to move swiftly while maintaining stability through coordinated diagonal leg pairs.
The trot gait, or diagonal synchronous gait, is a dynamic walking pattern where diagonal legs move in unison. This gait is efficient for high-speed locomotion in a robot dog, as it reduces energy consumption and minimizes lateral oscillations. I analyzed the foot state and trajectory for this gait, as illustrated in gait diagrams. The legs are grouped into two pairs: Group A (left front and right rear) and Group B (right front and left rear). During the gait cycle, Group A and Group B alternate between stance and swing phases. The foot state diagram shows the timing of lift and placement, while the trajectory diagram depicts the path of each foot relative to the body. For the robot dog, the foot trajectory during the swing phase is planned as a parabolic curve to avoid obstacles and ensure smooth landing. The coordinates of the foot tip during swing can be parameterized as:
$$ x(t) = x_0 + v_x t $$
$$ z(t) = z_0 + A \sin\left(\frac{\pi t}{T_{\text{swing}}}\right) $$
where $x_0$ and $z_0$ are initial positions, $v_x$ is the forward velocity, $A$ is the step height, and $T_{\text{swing}}$ is the swing duration. This planning ensures that the robot dog’s feet clear the ground and land with minimal impact. To detail the trot gait parameters, I use the following table:
| Gait Phase | Duration (s) | Leg Group Action | CoG Position |
|---|---|---|---|
| Initial Stance | 0.0-0.5 | Group A stance, Group B swing | On diagonal A |
| Transition | 0.5-1.0 | Group B stance, Group A swing | On diagonal B |
| Mid-Swing | 0.25-0.75 | Feet at maximum height | Moving forward |
| Landing | 0.5 and 1.0 | Feet touch down | Stabilized |
In my first-person perspective as a researcher, I planned the trot gait by defining joint angle sequences for each leg. Considering the right side legs as an example, the joint actions are coordinated across four stages per cycle. Let the joint angles be $\phi_1$ for the right shoulder, $\phi_2$ for the right hip, $\phi_3$ for the right knee, and $\phi_4$ for the right ankle (though passive, it adapts). The angle changes $\Delta \phi$ are set based on a predefined increment $\theta = 15^\circ$ for simplicity. The stages are as follows: Stage 1—right shoulder swings back, right hip swings forward, right knee swings forward, and right ankle remains neutral; Stage 2—reverse of Stage 1; Stage 3—right shoulder swings forward, right hip neutral, right knee swings back, right ankle adapts; Stage 4—reverse of Stage 3. This sequence generates the diagonal motion, with the left side mirroring these actions. The joint angle profiles over time $t$ can be expressed as piecewise functions. For instance, for the right shoulder in a cycle period $T$:
$$ \phi_1(t) = \begin{cases}
\theta & \text{for } 0 \leq t < T/4 \\
-\theta & \text{for } T/4 \leq t < T/2 \\
-\theta & \text{for } T/2 \leq t < 3T/4 \\
\theta & \text{for } 3T/4 \leq t < T
\end{cases} $$
Similar equations apply to other joints, ensuring synchronous movement. This planning approach simplifies control by reducing the degrees of freedom, as the ankle joint follows passively based on ground reaction forces. The robot dog’s gait stability is enhanced by this coordinated pattern, which I verified through simulation.
To validate the trot gait planning, I conducted simulations using Matlab, focusing on the robot dog’s CoG trajectory and joint dynamics. The simulation model incorporated the kinematic equations and duty factor $\beta = 0.5$, with a step length $\lambda = 40$ mm and cycle time $T = 2$ s, resulting in an average speed of 20 mm/s. I computed the CoG displacement in the x (lateral), y (forward), and z (vertical) axes over multiple gait cycles. The results show that the CoG in the x-axis oscillates but nets zero displacement over a cycle due to symmetric leg movements, confirming lateral stability. The z-axis displacement is minimal (less than 0.4 mm), indicating smooth vertical motion without significant bouncing. The y-axis displacement increases linearly, demonstrating consistent forward progression. The CoG coordinates $(X_{cog}, Y_{cog}, Z_{cog})$ are derived from the foot positions and body geometry. Assuming a symmetric robot dog body with mass $m$ and leg masses negligible, the CoG is at the body center, and its motion is given by:
$$ X_{cog}(t) = \frac{1}{4} \sum_{i=1}^{4} x_i(t) $$
$$ Y_{cog}(t) = \frac{1}{4} \sum_{i=1}^{4} y_i(t) $$
$$ Z_{cog}(t) = \frac{1}{4} \sum_{i=1}^{4} z_i(t) $$
where $(x_i, y_i, z_i)$ are the foot positions of leg $i$. The simulation output graphs illustrate these trajectories, aligning with the planned gait. For instance, the x-axis trajectory shows periodic oscillations that cancel out over time, ensuring the robot dog does not drift sideways. This is crucial for a robot dog navigating narrow paths. I also simulated joint torques using inverse dynamics to assess energy efficiency. The torque $\tau_j$ for joint $j$ is calculated using the Lagrangian formulation:
$$ \tau_j = \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_j} \right) – \frac{\partial L}{\partial q_j} $$
where $L$ is the Lagrangian (kinetic minus potential energy), and $q_j$ are generalized coordinates (joint angles). The results indicate peak torques during leg liftoff and landing, within motor limits. To summarize simulation parameters, I present this table:
| Simulation Parameter | Value | Description |
|---|---|---|
| Step Length ($\lambda$) | 40 mm | Forward distance per step |
| Cycle Time ($T$) | 2 s | Duration of one gait cycle |
| Duty Factor ($\beta$) | 0.5 | Stance time fraction |
| Swing Height ($A$) | 10 mm | Maximum foot clearance |
| CoG Oscillation (x) | ±5 mm | Lateral displacement amplitude |
| CoG Oscillation (z) | ±0.2 mm | Vertical displacement amplitude |
| Average Speed | 20 mm/s | Forward locomotion speed |
The simulation confirmed that the trot gait planning is effective for the robot dog. I implemented a joint controller using FPGA-based hardware to generate PWM signals for the DC motors, executing the planned angle sequences. The robot dog exhibited stable trotting with quick leg swings and accurate foot placements, no noticeable sway or jerks, validating the gait design. This practical verification underscores the robustness of the $\beta = 0.5$ parameter and the diagonal synchronization. Moreover, the gait algorithm ensures continuity between cycles, as the robot dog returns to its initial posture at each cycle end, enabling seamless transitions. This feature is vital for a robot dog operating in dynamic environments where gait adaptation may be required. Compared to traditional static gaits, this dynamic approach allows higher speeds and better energy management, key for prolonged operations of a robot dog in search-and-rescue missions.
In conclusion, my analysis and planning of the trot gait for a quadruped robot dog demonstrate a viable method for achieving dynamic walking. Through structural modeling, gait theory, and simulation, I showed that diagonal synchronous movement with a duty factor of 0.5 provides stability and efficiency. The Matlab simulations and hardware controller tests confirm the correctness of the approach, with the robot dog maintaining balance and linear progression. This work contributes to the field of legged robotics by offering a detailed gait planning framework that can be extended to other patterns like pace or gallop for a robot dog. Future directions include incorporating sensors for adaptive gait control on uneven terrain and optimizing energy consumption through machine learning. The robot dog, as a versatile platform, holds promise for various applications, and this research lays a foundation for advanced locomotion strategies in real-world scenarios.
Throughout this paper, I emphasized the term “robot dog” to reflect its role as an intelligent, mobile agent. The integration of mathematical models, such as the kinematic equations and duty factor formulas, with simulation tools like Matlab, enables precise gait design. The use of tables to summarize parameters and formulas to describe motion dynamics enhances clarity and reproducibility. This comprehensive study not only advances the understanding of quadruped locomotion but also provides practical insights for developers aiming to build agile and stable robot dogs. As robotics technology evolves, such gait planning methodologies will be crucial for creating robot dogs that can seamlessly interact with humans and navigate complex environments, ultimately expanding the horizons of autonomous systems.