2.1 System overview

The proposed implementation consists of two stages: offline centralized training and online two-rate execution, as illustrated in Fig 1. During offline training, the shared leg-level actor and centralized critic are optimized in 32 parallel Isaac Sim environments under the centralized-training–decentralized-execution paradigm. Terrain properties, robot parameters, and external disturbances are randomized during training. After convergence, only the trained shared actor parameters are exported to the embedded controller; the centralized critic is not required during online execution.

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Fig 1. Detailed implementation pipeline of the proposed CPG-conditioned leg-level multi-agent controller.

The upper panel shows offline centralized training in Isaac Sim, during which the shared actor and centralized critic are optimized using PPO. After training, only the shared actor parameters are deployed on the physical robot. The lower panel shows the online two-rate control architecture. Sensor acquisition, state construction, CPG integration, inverse kinematics, safety projection, command filtering, and servo actuation operate at 200 Hz, whereas the shared actor is evaluated independently for the four leg agents at 50 Hz. The leg-level frequency and swing-height proposals are aggregated and clipped before being fed back to the CPG. The nominal and residual joint targets are then fused, projected to hardware-safe limits, filtered, and transmitted to the servo controller, thereby closing the sensing–control loop.

https://doi.org/10.1371/journal.pone.0356505.g001

During online execution, proprioceptive measurements are acquired and synchronized at 200 Hz. The IMU orientation and angular velocity, joint states, and foot-contact signals are filtered to construct the global state , the four local observations , and the shared coordination message . The low-level branch integrates the four coupled Hopf oscillators using contact and body-attitude feedback, maps the oscillator states to desired Cartesian foot trajectories, and applies inverse kinematics to obtain the nominal joint target .

The high-level multi-agent policy is evaluated every fourth low-level cycle, corresponding to 50 Hz. At each policy update, the same shared actor is evaluated once for each of the LF, RF, LH, and RH leg agents. Each agent outputs two local joint residuals and two gait-adaptation proposals, and , for the step frequency and swing height. The four frequency and swing-height proposals are averaged and clipped to hardware-safe intervals before being fed back to the CPG. The latest residual actions and common gait parameters are held constant between two consecutive policy updates, while the CPG and servo loops continue to operate at 200 Hz.

At every low-level cycle, the nominal joint target and the most recently available multi-agent residual command are combined as

(1)

where is the nominal joint target generated by the CPG and inverse-kinematics modules, is the concatenated residual command generated by the four leg agents, and applies joint-position and command-rate limits before the command is passed to the hardware interface. A first-order filter is subsequently applied to suppress servo-command chattering.

2.2 CPG-based gait generation module

Each leg is associated with one nonlinear oscillator with amplitude and phase , where . We use a Hopf-type formulation:

(2)

(3)

where is the amplitude convergence gain, is the nominal oscillation radius, is the step frequency, is the coupling weight between oscillators i and j, is the desired phase difference, is the measured foot-contact state, and is the desired contact state inferred from the current gait phase. The feedback term

(4)

uses foot contact and IMU pitch/roll to slightly advance or delay the oscillator when terrain or body attitude deviates from the nominal gait. The desired contact signal is computed from the oscillator phase as

(5)

where denotes the nominal stance part of the gait cycle. This definition makes the contact feedback term in Eq. (4) interpretable as a phase correction driven by the mismatch between expected and measured contact.

For the trot gait used in the experiments, the desired phase vector is

(6)

for the ordered leg set , so that diagonal legs move in phase. The phase-offset matrix is defined by , and in the implementation we use symmetric all-to-all coupling with for and . The nominal oscillator parameters used in our experiments are , , and base step frequency f₀ = 1.6 Hz.

The oscillator states are mapped to Cartesian foot trajectories as

(7)

with fixed by the robot geometry. Here is the stride amplitude and is the swing-foot clearance. In the nominal setting, m and h₀ = 0.025 m. The desired foot positions are converted to hip and knee targets by planar inverse kinematics for each leg, giving in Eq. (1).

2.3 CPG parameter tuning and terrain-adaptive adjustment

The CPG parameters were selected before reinforcement learning using a three-stage tuning procedure. First, oscillator-internal parameters (, , , and ) were chosen to ensure fast convergence to the desired trot limit cycle and stable diagonal-leg phase locking. Second, kinematic parameters (, h₀, and f₀) were swept on flat ground to obtain a nominal gait that matched the target speed without joint saturation. Third, the feedback gains (, , , , and ) were increased gradually under slope, obstacle, and push disturbances, and the smallest gains that reduced contact mismatch and body-attitude drift without introducing command oscillation were retained. The final parameter values used in the experiments are summarized in Table 1.

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Table 1. Controller parameters used in the experiments.

https://doi.org/10.1371/journal.pone.0356505.t001

Terrain adaptation is implemented continuously rather than by manually switching parameter sets for different terrains. Contact and IMU deviations perturb the oscillator through Eqs. (2)–(4), while the multi-agent residual policy adjusts the common step frequency and swing height through and . The resulting frequency and clearance are clipped to hardware-safe intervals in Eq. (14). Thus, ramps primarily induce phase and frequency corrections through body pitch and contact timing, whereas obstacles and stairs mainly increase the learned swing-height correction when premature contact or missed clearance is detected.

A one-at-a-time sensitivity check was used to verify that the controller was not dependent on a single narrow parameter setting. The most sensitive parameters were f₀, , and h₀, because they directly determine forward speed, reachable footholds, and obstacle clearance. Moderate changes in and preserved the trot pattern as long as the oscillator remained phase locked. Feedback gains above the selected values improved recovery in some disturbed trials but increased command chattering on the physical robot. For this reason, conservative gains were used and only the residual policy was allowed to make bounded online adjustments.

2.4 Leg-level multi-agent coordination architecture

We explicitly model the four legs as four cooperating agents,

(8)

Each agent controls one leg and outputs residual corrections for the hip and knee joints of that leg. To balance modularity and coordination, we use centralized training with decentralized execution and shared actor parameters across all agents. During training, each agent has access to a global state for value estimation, whereas at execution time each agent acts from its local observation augmented with a compact shared coordination signal.

For leg agent i, the local observation is defined as

(9)

where and are the hip and knee states of leg i, and are the corresponding oscillator states, is the measured contact state, is the desired contact state from the gait generator, and are body-attitude measurements from the IMU, is the previous action of the same agent, is the target forward speed, and is a shared coordination message broadcast through the global state bus.

The shared coordination message is

(10)

where is the common step-frequency command, is the common swing-height command, and indicates the desired contact schedule of each leg. This message allows each agent to remain aware of the global gait pattern without requiring a fully centralized action output.

Each agent outputs a local residual action

(11)

where and are residual corrections for the hip and knee joints of leg i, is a local contribution to step-frequency adaptation, and is a local contribution to swing-height adaptation. The joint residual vector applied to the full robot is then

(12)

The common gait parameters are obtained by averaging the leg-level proposals,

(13)

so that

(14)

This aggregation preserves a coherent whole-body gait while allowing each leg to contribute to adaptation. In the implementation, joint residuals are bounded by rad, frequency proposals by Hz, and swing-height proposals by m before the final safety projection in Eq. (1).

The local observation in Eq. (9) contains leg-specific variables (, , , , , and ), low-dimensional body-stability variables (, , , and ), the previous local action, the commanded speed, and the compact coordination message . This design intentionally avoids giving each decentralized actor the full 8-joint robot state at runtime, which would increase redundancy and make the policy more sensitive to missing or noisy channels. At the same time, it does not omit the global information needed for gait coherence: the shared message carries the common phase schedule, frequency, and swing-height commands, while the IMU terms provide whole-body balance information. The action space is residual rather than absolute because the CPG already provides a dynamically consistent nominal gait; the policy therefore learns bounded corrections for local joint targets and bounded proposals for global gait parameters. During training, the centralized critic receives the global state in Eq. (15), which contains the full robot posture, contacts, oscillator states, and previous actions. This CTDE split reduces partial-observability problems during learning while retaining decentralized execution on the embedded hardware. The main missing information is exteroceptive terrain geometry; the present controller therefore targets proprioceptive terrain adaptation and not vision-guided foothold planning.

2.5 Multi-agent network architecture and training

The shared actor network is implemented as a multilayer perceptron with two hidden layers of 256 and 128 units and ReLU activations. The same actor parameters are shared across the four leg agents. A centralized critic is trained from the global state

(15)

which is the same global observation used for full-system value estimation. We therefore adopt a centralized-training/decentralized-execution paradigm: the critic evaluates whole-body coordination during training, while each leg executes from its own local observation and shared message at runtime.

The actor outputs the mean of a Gaussian policy over the 4-dimensional action of each leg agent together with a learned log-standard-deviation vector. The critic outputs a scalar state value. Optimization is performed using a PPO-style clipped objective with clip ratio 0.2, discount factor , and generalized-advantage parameter . The policy was optimized with Adam using an initial learning rate of , linear learning-rate decay, rollout length of 2048 policy steps per update, minibatch size of 4096 samples, 10 optimization epochs per update, entropy coefficient 0.01, value-loss coefficient 0.5, and gradient-norm clipping at 0.5. Training used 32 parallel simulated environments and was stopped after simulator steps or when the evaluation success rate changed by less than 2% over ten consecutive evaluation windows. These implementation settings are summarized here to make the learning procedure reproducible.

Deployment complexity. The local observation in Eq. (9) has 23 scalar elements and the action in Eq. (11) has four elements. For the 23–256–128–4 actor, the total number of trainable parameters is

(16)

where the last four parameters are the learned log-standard deviations. This corresponds to approximately 155 KiB when stored in 32-bit floating-point format. One deterministic actor evaluation requires 39,168 multiply–accumulate operations in the three fully connected layers. Evaluating the shared actor for four legs at 50 Hz therefore requires approximately multiply–accumulate operations per second, excluding the comparatively small fixed-cost CPG and inverse-kinematics operations. Sharing one actor across four legs reduces actor storage by 75% relative to deploying four independent actors of the same architecture. The centralized critic, PPO optimizer, rollout buffers, and simulation environments are used only during training and are not deployed on the physical robot. These counts quantify model size and arithmetic complexity; processor-specific latency and electrical power depend on the embedded software and hardware implementation.

To encourage both local leg effectiveness and global locomotion quality, we use a shared team reward supplemented with a coordination regularizer. The total reward is

(17)

where each term is computed at the control step as follows:

(18)

(19)

(20)

(21)

(22)

(23)

(24)

Here is the measured forward body velocity, is the target velocity, is the foot height, is the horizontal foot velocity, is the estimated joint effort, and avoids division by zero. We used m/s and rad. For the low-cost servo platform, direct torque sensing was unavailable; therefore, was estimated from the calibrated servo command-tracking error and used consistently for both the reward penalty and the CoT estimate. The terminal reward is applied when the robot falls, exceeds the safety tilt threshold, or terminates the traversal prematurely; otherwise .

The coordination penalty is defined as

(25)

This term penalizes disagreement between realized and desired contact schedules and discourages excessive divergence among the leg agents in their frequency and swing-height corrections. The reward weights were chosen by first normalizing each reward component to a comparable numerical range on nominal flat-ground rollouts and then increasing the velocity and posture weights until the controller tracked the commanded speed without excessive body attitude variation. The penalties on power, slip, smoothness, and coordination were kept smaller to regularize the learned behavior without overriding traversal success. In practice, each agent receives the same team reward, which promotes cooperation rather than competitive local behavior.

Training is conducted in Isaac Sim with 20 s episodes and domain randomization over terrain and robot parameters. During training, ground friction is randomized in [0.5,1.0], base mass is perturbed by , slope is randomized within , and lateral push disturbances of 2–4 N are applied for 0.15 s at random times. A simple curriculum is used: obstacle height is gradually increased from 5 mm to 20 mm once the success rate over the previous evaluation window exceeds 85%. The multi-agent policy is evaluated every 20 ms, corresponding to the 50 Hz coordination layer.

2.6 Core advantages and low-cost platform integration

The low-cost merit of the proposed method is produced by the algorithmic decomposition rather than by the robot price alone. The CPG supplies a stable nominal gait using a small set of oscillator states; consequently, the learned policy solves only a bounded residual-correction problem. Parameter sharing allows all four leg agents to reuse the same actor, while CTDE removes the centralized critic from online execution. The controller also avoids an online rigid-body dynamics solver or receding-horizon optimizer: its runtime operations consist of fixed-dimensional CPG integration, planar inverse kinematics, shared-actor inference at 50 Hz, residual addition, and simple command projection at 200 Hz.

The sensing burden is similarly limited. Runtime observations are obtained from the IMU, joint states, and foot-contact signals already available on the platform. The method does not depend on an external motion-capture system, a terrain-reconstruction network, or direct joint-torque measurements. The monocular camera installed on the PuppyPi platform was not used by the reported controller. Table 2 summarizes the online implementation characteristics that support the low-cost claim.

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Table 2. Online implementation characteristics of the proposed controller. Training-side resources such as the centralized critic, PPO optimizer, rollout buffers, and parallel simulation environments are excluded because they are not deployed on the robot.

https://doi.org/10.1371/journal.pone.0356505.t002

Actuation commands generated by the CPG and multi-agent coordination modules are mapped to the low-cost actuators through a hardware abstraction layer. The final command is clipped to the joint range of each servo and then rate-limited before transmission to the motor controller. A first-order filter,

(26)

is used to suppress command chattering. Safety logic also stops the gait and resets the robot to a neutral posture if sustained saturation or excessive tilt is detected. The hardware abstraction and bounded residual interface allow the same learned actor to be transferred between servo-based platforms by changing joint calibration, limits, and command scaling rather than redesigning the policy architecture.

2.7 Integrated control loop

The online implementation is executed as a two-rate closed loop. Let k denote the index of the 200-Hz low-level cycle. Because the policy layer operates at 50 Hz, the shared actor is evaluated when . The most recently computed policy outputs are applied by zero-order hold during the remaining low-level cycles. The implementation proceeds as follows:

1. Sensor acquisition and preprocessing: Read the IMU orientation and angular velocity, joint angles and velocities, and foot-contact measurements at 200 Hz. Filter and temporally synchronize the measurements to suppress high-frequency sensing noise.
2. State construction: Construct the global state , the four local observations , and the shared coordination message from the filtered measurements, oscillator states, previous actions, and target forward speed.
3. Multi-agent policy update: If , evaluate the shared actor independently for the LF, RF, LH, and RH leg agents to obtain the local actions defined in Eq. (11). Concatenate the local joint residuals according to Eq. (12).
4. Gait-parameter aggregation: Average the four frequency and swing-height proposals using Eq. (13), and clip the common gait parameters according to Eq. (14). The resulting , , and are held constant until the next 50-Hz policy update.
5. CPG update: Integrate the coupled oscillator dynamics in Eqs. (2)–(3) using the current contact feedback, IMU attitude feedback, and common gait parameters. The updated oscillator states determine the desired contact schedule and gait phase of each leg.
6. Trajectory generation and inverse kinematics: Map the oscillator states to desired Cartesian foot trajectories using Eq. (7), and calculate the nominal hip and knee targets through inverse kinematics.
7. Command fusion and hardware projection: Combine the nominal and residual joint commands according to Eq. (1). Apply the joint-position limits and command-rate limits before passing the command to the hardware interface.
8. Command filtering and actuation: Apply the first-order command filter in Eq. (26), and transmit the filtered joint targets to the low-level servo controller.
9. Safety supervision: Continuously monitor body tilt, sustained joint saturation, and command-rate violations. If a safety condition is detected, stop the gait command and return the robot to its predefined neutral posture.

The resulting robot motion produces the measurements used in the next control cycle, thereby closing the sensing–control loop. This implementation preserves the interpretability and rhythmic stability of the 200-Hz CPG controller while allowing the 50-Hz leg-level policy to adapt the joint targets, step frequency, and swing height to terrain variations and asymmetric contact events without sacrificing whole-body gait coherence.

3.1 Experimental setup

To evaluate the proposed framework, we conducted experiments in both simulation and real-world environments using a low-cost quadruped robot platform. The evaluation focuses on four representative terrain scenarios: ramp, stairs, obstacles, and uneven terrain. These scenarios were selected because they represent typical locomotion challenges for low-cost quadrupeds, including slope adaptation, discrete foothold placement, disturbance rejection, and robustness on continuously varying surfaces.

Fig 2 presents representative snapshots of the simulation environments and corresponding real-world experiments. The same control architecture is deployed in both simulation and hardware experiments. The low-level CPG and servo loop runs at 200 Hz, while the high-level coordination policy runs at 50 Hz.

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Fig 2. Overview of the experimental environments.

The upper panel shows the simulation environments used for training and evaluation across ramp, stairs, obstacles, and uneven-terrain scenarios. The lower panel shows representative real-world experiments in the corresponding terrains.

https://doi.org/10.1371/journal.pone.0356505.g002

In all scenarios, the robot is commanded to move forward at a constant target speed. Each controller–terrain pair is evaluated over 20 independent trials. A trial is considered successful if the robot completes the assigned traversal without falling. A failure is declared when the robot loses balance, undergoes unrecoverable body collapse, or terminates locomotion prematurely.

Physical hardware setup. The physical platform used for experimental validation was the Hiwonder PuppyPi quadruped robot. The robot is a low-cost aluminum-alloy quadruped platform with 8 degrees of freedom, powered by a 7.4 V 2200 mAh LiPo battery and equipped with an onboard Raspberry Pi controller, a dedicated expansion board, and a monocular camera. Although the camera was physically present, camera images were not used by the reported controller. Online feedback was limited to IMU measurements, joint-angle feedback, and foot-contact signals; no external motion-capture system or direct torque sensors were required. This hardware configuration is suitable for evaluating whether the proposed controller can maintain robust locomotion under the sensing, computation, and actuation constraints typical of affordable quadruped platforms.

3.2 Compared controllers

We compare four controllers. All controllers use the same joint limits, command-rate limits, low-level servo interface, safety termination conditions, target speed, and evaluation terrains, so that the comparison isolates the controller architecture rather than hardware settings.

• CPG-only: this baseline uses Eqs. (2)–(7), the same inverse kinematics, and the same safety projection, but sets , , and . It therefore tests the nominal oscillator and feedback controller without learned residual adaptation.
• Multi-Agent-only: this baseline keeps the leg-level CTDE policy and shared actor, but removes the CPG prior by replacing with a neutral standing pose. The agents directly output bounded joint increments for each leg. This evaluates whether leg-level multi-agent coordination alone can generate effective locomotion without the structured oscillator prior.
• RL-only: this baseline uses a single centralized PPO actor that maps the global state to an 8-dimensional joint command correction. It uses the same reward and training randomization, but removes the leg-level agent decomposition, the shared coordination message, and the CPG gait prior. This evaluates the effect of replacing the proposed modular residual structure with a monolithic learning controller.
• Integrated: the proposed controller combines the CPG gait generator, the shared-parameter leg-level residual policy, frequency and swing-height aggregation, and the hardware-aware command projection.

3.3 Evaluation metrics and statistical analysis

To ensure consistency between the evaluation design and the reported results, the following metrics are used throughout the experiments:

• Pitch Mean and Pitch Std: mean and standard deviation of body pitch angle measured by the IMU;
• Roll Mean and Roll Std: mean and standard deviation of body roll angle measured by the IMU;
• Success Rate: percentage of successful trials for each controller–terrain pair;
• Cost of Transport (CoT): a dimensionless energy-efficiency metric estimated from actuation effort during traversal.

The CoT is computed as

(27)

where m is the robot mass, g is gravitational acceleration, d is the forward distance traveled during the trial, is the control interval, and is the estimated effort of joint k. Because the physical platform does not provide direct torque sensing, is obtained from the calibrated servo command-tracking error. The same estimator is used for all controllers, making the CoT values comparable across methods even though they should be interpreted as effort-based estimates rather than direct electrical-energy measurements.

These metrics jointly characterize body-attitude stability, traversal robustness, and energy efficiency. Lower pitch/roll mean and standard deviation indicate better body stabilization, higher success rate indicates stronger terrain adaptability, and lower CoT indicates more efficient locomotion. For continuous stability metrics, we report means and standard deviations over repeated trials and use Welch’s t-tests for pairwise comparisons between the integrated controller and the best-performing baseline in each scenario. For success rate, Wilson 95% confidence intervals and Fisher’s exact tests are used when integer success/failure counts are available. Multiple pairwise tests are interpreted with Holm–Bonferroni correction. Statistical results are used to qualify the strength of the conclusions rather than to claim universal superiority across all operating conditions.

4.1 Overall performance

The physical-robot results are summarized in Figs 3–Fig 4, and Table 3. Fig 3 reports the continuous body-attitude metrics as the mean with one standard deviation over 20 independent trials, while traversal success is shown both as a percentage and as the underlying number of successful trials out of 20. Across the four scenarios, the integrated controller provides the most balanced performance, with comparatively small pitch and roll deviations, high traversal success, and low estimated CoT.

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Table 3. Physical-robot performance of the four controllers across the obstacle, ramp, stair, and uneven-terrain scenarios. Each controller–terrain pair was evaluated over 20 independent trials. Continuous metrics are reported as the mean and standard deviation over the repeated trials. Traversal success is reported as the number of successful trials out of 20, with the corresponding percentage in parentheses.

https://doi.org/10.1371/journal.pone.0356505.t003

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Fig 3. Physical-robot comparison of the four controllers across the obstacle, ramp, stair, and uneven-terrain scenarios.

Panels (a) and (b) show the mean pitch and roll angles, respectively, with error bars representing one standard deviation over 20 independent trials. Panel (c) shows traversal success; the labels above the bars report the number of successful trials out of 20, and the bar heights show the corresponding percentages. Panel (d) reports the effort-based cost-of-transport estimate. Lower pitch, roll, and CoT values and higher success values indicate better performance.

https://doi.org/10.1371/journal.pone.0356505.g003

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Fig 4. Additional physical-robot analysis synchronized with the revised 20-trial success results.

Panels (a) and (b) show the pitch and roll means with one standard deviation over 20 independent trials. Panel (c) presents the scenario-level composite score obtained by globally min–max normalizing pitch mean, pitch standard deviation, roll mean, roll standard deviation, traversal success, and CoT, and then assigning equal weight to the six normalized metrics. Panel (d) reports the arithmetic mean of the four scenario scores for each controller. All scores range from 0 to 1, and higher values indicate a more favorable overall combination of stability, success, and estimated effort. The composite score is descriptive and should be interpreted together with the original metrics and statistical comparisons.

https://doi.org/10.1371/journal.pone.0356505.g004

To provide a transparent descriptive summary across the six reported metrics, we compute a composite score using global min–max normalization over all 16 controller–terrain combinations. For the lower-is-better metrics (pitch mean, pitch standard deviation, roll mean, roll standard deviation, and CoT), the normalized score is

whereas traversal success is normalized as

Here, denotes metric m for controller c in scenario s, and the minimum and maximum are calculated globally for each metric. The six normalized values are equally averaged to obtain the scenario-level composite score. The controller-level score is the arithmetic mean of its four scenario scores. This composite index is used only as a descriptive visualization and does not replace the individual metrics or the statistical tests.

The pitch and roll results indicate that the integrated controller reduces both the average body-angle deviation and the trial-to-trial variability in most tested cases. After recalculation using the corrected 20-trial traversal-success values, its normalized composite scores are 0.95, 1.00, 0.92, and 0.88 for the obstacle, ramp, stair, and uneven-terrain scenarios, respectively, with a controller-level average of 0.94. The corresponding average scores of CPG-only, Multi-Agent-only, and RL-only are 0.39, 0.29, and 0.50. These values provide a compact descriptive summary of the reported metrics; because the score uses global min–max normalization and equal metric weights, the individual measurements and statistical tests remain the primary basis for interpretation.

4.2 Statistical comparison

Table 4 reports Welch-test comparisons between the integrated controller and the best baseline for pitch and roll in each scenario, using the reported trial mean and standard deviation. The analysis shows that the roll-angle reduction is consistently favorable, while the pitch-angle reduction is statistically strong on the ramp scenario and more trend-like in the other scenarios. Therefore, the revised conclusions are phrased in terms of a favorable overall trade-off rather than uniform statistically significant improvement for every metric.

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Table 4. Statistical comparison between the integrated controller and the best baseline for body-attitude metrics. Values are Welch-test p-values computed from the reported mean, standard deviation, and 20 repeated trials.

https://doi.org/10.1371/journal.pone.0356505.t004

4.3 Ablation interpretation

The three baselines also serve as component ablations of the integrated controller. CPG-only removes learned residual adaptation and the multi-agent policy; Multi-Agent-only removes the structured CPG prior; and RL-only removes both the leg-level decomposition and the oscillator prior in favor of a monolithic learned policy. The performance gap between these baselines and the integrated controller suggests that the best results are obtained when rhythmic priors, leg-level residual adaptation, and hardware-aware projection are used together. However, the present data do not fully isolate the coordination regularizer from the rest of the multi-agent architecture. For this reason, we describe as a stabilizing regularization term and avoid claiming that it is independently necessary unless an additional “Integrated without ” experiment is added.

4.4 Scenario-wise results

In the obstacle scenario, the integrated controller achieves the best overall performance, with lower body-angle deviation and higher success rate than the baselines. This suggests stronger disturbance rejection when the robot encounters scattered local protrusions.

In the ramp scenario, the integrated controller also produces the lowest pitch and roll deviations and the highest success rate. This indicates better slope adaptation and more stable body posture during ascent and descent.

In the stair scenario, the integrated controller maintains the highest success rate together with low body-attitude variation. This result suggests improved foot clearance and more reliable inter-leg coordination during step negotiation.

In the uneven-terrain scenario, the same overall trend is preserved. The integrated controller again shows lower body-angle deviation, higher success rate, and lower CoT than the other controllers. This indicates that the framework remains effective not only for discrete terrain events, but also for continuously varying surfaces.

4.5 Controller comparison

The baseline controllers exhibit different strengths but also clear limitations. The CPG-only controller preserves regular rhythmic motion and relatively low energy consumption, but its adaptability is limited on more challenging terrain. The RL-only controller provides adaptive capacity, but its performance is less consistent across stability, success rate, and efficiency. The Multi-Agent-only controller improves coordination flexibility, but without the structured gait prior it does not achieve the same overall balance as the integrated framework.

By comparison, the integrated controller provides the most consistent trade-off among the reported metrics in the tested scenarios, which is particularly important for practical deployment on low-cost quadruped platforms. The statistical analysis indicates that some improvements, especially roll stabilization, are stronger than others; therefore, the method should be viewed as improving the overall balance of stability, success rate, and effort rather than dominating every metric in every terrain.

4.6 Sim-to-real consistency

The simulation and real-world results show a consistent overall trend. In both domains, the integrated controller provides the most favorable trade-off among body-attitude stability, traversal success rate, and energy efficiency. This consistency suggests that the proposed combination of CPG-based gait generation, leg-level multi-agent coordination, and hardware-aware command projection transfers reasonably well from Isaac Sim to the physical low-cost platform.

At the same time, some discrepancy between simulation and hardware is expected. The physical robot is affected by actuator backlash, battery-voltage variation, contact uncertainty, and sensing noise that can only be approximated in simulation. The preserved controller ranking suggests that the proposed framework transfers reasonably well to the PuppyPi platform, but broader deployment claims should be verified on additional robots, speeds, and terrain geometries.