Leg Phase detection

Author: Teng-Hu Cheng
email: [e-mail]
Date: Jan. 2025
Website: https://ncrl.lab.nycu.edu.tw/member/

Sensor Fusion and Probabilistic Methods in Quadruped Robot Gait Phase Switching

Quadruped robots rely on sensor fusion and probabilistic models to accurately determine the switching between swing and stance phases for each leg. Since real-world terrains and robot dynamics are often uncertain, robust detection and prediction of gait phases are critical for stable locomotion, adaptive terrain response, and efficient movement.

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1. Swing and Stance Phase Detection

Each leg of a quadruped robot alternates between two key phases:

• Stance phase: The foot is in contact with the ground, supporting the robot’s weight.
• Swing phase: The foot is lifted and moves forward to the next step.

Challenges in Phase Switching:

1. Uneven terrain: Different terrains introduce variations in expected ground reaction forces (GRF).
2. Foot slippage: Detecting false contact (e.g., foot dragging or slipping).
3. External disturbances: Unexpected forces (wind, impacts) can alter the expected gait pattern.
4. Sensor noise and drift: Individual sensors are prone to measurement errors, making sensor fusion essential.

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2. Sensor Fusion for Gait Phase Estimation

To determine the current phase, quadruped robots fuse data from multiple sensors:

(1) Ground Reaction Force (GRF) Sensors

• Measures contact force at the foot.
• If GRF > threshold, the leg is likely in stance; otherwise, it's in swing.
• Useful but can fail on soft surfaces (e.g., sand, grass).

(2) IMU (Inertial Measurement Unit)

• Measures angular velocity (gyroscope) and acceleration (accelerometer).
• Helps detect foot impacts, velocity changes, and slip events.
• Used to estimate body acceleration and tilt, which correlates with phase transition.

(3) Joint Encoders & Motor Current Feedback

• Measures joint angles and torques.
• Detects if the leg is moving (swing) or resisting external forces (stance).
• Higher torque → likely stance phase.

(4) Vision & Depth Sensors (Optional)

• LiDAR, stereo cameras, or depth sensors estimate the terrain height.
• Helps anticipate foot placement and predict when the leg should transition to stance.

Sensor Fusion Process:

Sensor fusion combines the above sources to filter out noise and improve phase estimation using:

1. Extended Kalman Filter (EKF)

   • Fuses IMU + GRF + joint encoders.
   • Estimates leg states (stance/swing) probabilistically.

2. Bayesian Inference

   • Uses probability models to infer stance/swing likelihood based on noisy sensor data.

3. Machine Learning / Neural Networks

   • Uses historical gait data to predict optimal switching points.

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3. Probabilistic Models for Phase Switching

Since sensors are noisy and terrain conditions vary, probabilistic models help predict the correct phase transition.

(1) Hidden Markov Model (HMM) for Phase Transitions

• The stance/swing phase is treated as a hidden state
  $S_t$.
• Observations
  $O_t$ come from fused sensor data.
• The transition probability
  $P(S_t|S_{t-1})$ depends on:
  • Foot velocity.
  • Contact force.
  • Terrain estimation.

HMM State Transition Matrix Example:

$$P(S_t|S_{t-1})=\left[\begin{array}{cc}P(\text{stance}|\text{stance})& P(\text{swing}|\text{stance})\\ P(\text{stance}|\text{swing})& P(\text{swing}|\text{swing})\end{array}\right]$$

(2) Gaussian Mixture Models (GMM)

• Clusters gait sensor data into stance and swing phases.
• Uses likelihood estimation to classify the current phase.

(3) Reinforcement Learning for Gait Adaptation

• State
  $s_t$ → Sensor measurements.
• Action
  $a_t$ → Decide swing/stance transition.
• The model learns optimal transitions from real-world terrain variations.

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4. Example: Bayesian Decision Rule for Phase Switching

To determine whether a foot should switch to stance or swing, we compute:

$$P(\text{stance}|O_t)=\frac{P(O_t|\text{stance})P(\text{stance})}{P(O_t)}$$

Where:

• $P(O_t|\text{stance})$ is the likelihood of observing current sensor data given stance phase.
• $P(\text{stance})$ is the prior probability of stance (e.g., if robot is slow, stance probability is higher).
• $P(O_t)$ is the normalization factor.

Decision Rule:

• If
  $P(\text{stance}|O_t)>0.5$, switch to stance.
• Else, stay in swing phase.

Conclusion

• Input: the observation, Prior Probability
• Output: the Probability

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5. Implementation in MATLAB/Python

Here’s a MATLAB example using Extended Kalman Filter (EKF) for gait estimation:

function phase = gait_phase_estimator(GRF, IMU, JointTorque)
    % Define prior probabilities
    P_stance = 0.6; 
    P_swing = 0.4;
    
    % Likelihood estimation
    L_stance = normpdf(GRF, 80, 15) * normpdf(IMU, 0, 0.1) * normpdf(JointTorque, 5, 2);
    L_swing = normpdf(GRF, 20, 10) * normpdf(IMU, 2, 0.5) * normpdf(JointTorque, 2, 1);

    % Compute posterior probabilities
    P_stance_given_O = (L_stance * P_stance) / (L_stance + L_swing);
    P_swing_given_O = (L_swing * P_swing) / (L_stance + L_swing);

    % Decision rule
    if P_stance_given_O > P_swing_given_O
        phase = 'Stance';
    else
        phase = 'Swing';
    end
end

% Example usage
GRF = 85; IMU = 0.1; JointTorque = 6;
phase = gait_phase_estimator(GRF, IMU, JointTorque);
disp(['Predicted phase: ', phase]);

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6. Summary

✅ Sensor Fusion: Combines GRF, IMU, Joint Sensors, and Vision for accurate phase detection.
✅ Probability Models: Uses HMM, Bayesian Inference, and GMM to infer gait transitions.
✅ Adaptive Control: Machine Learning or Reinforcement Learning helps adjust gait based on terrain.
✅ Practical Use: Implemented in Boston Dynamics' Spot, ANYmal, and MIT Cheetah robots.

With sensor fusion and probabilistic models, quadruped robots achieve adaptive, stable, and efficient gait control on complex terrains. 🚀🐾