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Chapter 10: Humanoid Kinematics & Dynamics

Introduction​

Humanoid robots require sophisticated kinematic and dynamic models to achieve stable and efficient locomotion. This chapter covers the mathematical foundations and practical implementation of humanoid robot kinematics and dynamics.

10.1 Forward Kinematics​

10.1.1 Joint Coordinate Systems​

import numpy as np
from scipy.spatial.transform import Rotation

class HumanoidKinematics:
    def __init__(self):
        # Joint configuration for humanoid robot
        self.joint_names = [
            'left_hip_yaw', 'left_hip_roll', 'left_hip_pitch',
            'left_knee_pitch', 'left_ankle_pitch', 'left_ankle_roll',
            'right_hip_yaw', 'right_hip_roll', 'right_hip_pitch',
            'right_knee_pitch', 'right_ankle_pitch', 'right_ankle_roll',
            'left_shoulder_pitch', 'left_shoulder_roll', 'left_shoulder_yaw',
            'left_elbow_pitch', 'left_wrist_pitch', 'left_wrist_roll',
            'right_shoulder_pitch', 'right_shoulder_roll', 'right_shoulder_yaw',
            'right_elbow_pitch', 'right_wrist_pitch', 'right_wrist_roll',
            'neck_pitch', 'neck_yaw', 'neck_roll'
        ]
        
        # DH parameters for each joint
        self.dh_parameters = self._initialize_dh_parameters()
        
    def _initialize_dh_parameters(self):
        """Initialize Denavit-Hartenberg parameters"""
        return {
            'left_hip_yaw': {'a': 0, 'alpha': 0, 'd': 0.1, 'offset': 0},
            'left_hip_roll': {'a': 0, 'alpha': np.pi/2, 'd': 0, 'offset': 0},
            'left_hip_pitch': {'a': 0.1, 'alpha': 0, 'd': 0, 'offset': 0},
            # ... more joints
        }
    
    def forward_kinematics(self, joint_angles):
        """Calculate forward kinematics for all joints"""
        positions = {}
        orientations = {}
        
        for joint_name in self.joint_names:
            if joint_name in joint_angles:
                pos, ori = self._compute_joint_transform(joint_name, joint_angles[joint_name])
                positions[joint_name] = pos
                orientations[joint_name] = ori
                
        return positions, orientations
    
    def _compute_joint_transform(self, joint_name, angle):
        """Compute transformation matrix for a joint"""
        dh = self.dh_parameters.get(joint_name)
        if not dh:
            return np.zeros(3), np.eye(3)
        
        # DH transformation
        theta = angle + dh['offset']
        a = dh['a']
        d = dh['d']
        alpha = dh['alpha']
        
        # Transformation matrix
        T = np.array([
            [np.cos(theta), -np.sin(theta)*np.cos(alpha), np.sin(theta)*np.sin(alpha), a*np.cos(theta)],
            [np.sin(theta), np.cos(theta)*np.cos(alpha), -np.cos(theta)*np.sin(alpha), a*np.sin(theta)],
            [0, np.sin(alpha), np.cos(alpha), d],
            [0, 0, 0, 1]
        ])
        
        position = T[:3, 3]
        orientation = T[:3, :3]
        
        return position, orientation

10.1.2 End-Effector Positioning​

class EndEffectorKinematics:
    def __init__(self, kinematics_solver):
        self.kinematics = kinematics_solver
        
    def get_foot_positions(self, joint_angles):
        """Get left and right foot positions"""
        positions, _ = self.kinematics.forward_kinematics(joint_angles)
        
        left_foot = positions.get('left_ankle_roll', np.zeros(3))
        right_foot = positions.get('right_ankle_roll', np.zeros(3))
        
        return left_foot, right_foot
    
    def get_hand_positions(self, joint_angles):
        """Get left and right hand positions"""
        positions, _ = self.kinematics.forward_kinematics(joint_angles)
        
        left_hand = positions.get('left_wrist_roll', np.zeros(3))
        right_hand = positions.get('right_wrist_roll', np.zeros(3))
        
        return left_hand, right_hand
    
    def get_center_of_mass(self, joint_angles, link_masses):
        """Calculate center of mass of the robot"""
        positions, _ = self.kinematics.forward_kinematics(joint_angles)
        
        total_mass = sum(link_masses.values())
        com = np.zeros(3)
        
        for link_name, mass in link_masses.items():
            if link_name in positions:
                com += mass * positions[link_name]
                
        return com / total_mass

10.2 Inverse Kinematics​

10.2.1 Jacobian-Based IK Solver​

class InverseKinematics:
    def __init__(self, kinematics_solver):
        self.kinematics = kinematics_solver
        self.joint_limits = self._initialize_joint_limits()
        
    def _initialize_joint_limits(self):
        """Initialize joint limits for humanoid robot"""
        return {
            'left_hip_yaw': (-np.pi, np.pi),
            'left_hip_roll': (-np.pi/4, np.pi/4),
            'left_hip_pitch': (-np.pi/2, np.pi/2),
            'left_knee_pitch': (0, np.pi),
            'left_ankle_pitch': (-np.pi/2, np.pi/2),
            'left_ankle_roll': (-np.pi/4, np.pi/4),
            # ... similar for right leg and arms
        }
    
    def inverse_kinematics(self, target_position, target_orientation, 
                          initial_angles, max_iterations=100):
        """Solve inverse kinematics using Jacobian method"""
        joint_angles = initial_angles.copy()
        
        for iteration in range(max_iterations):
            # Forward kinematics
            current_pos, current_ori = self.kinematics.forward_kinematics(joint_angles)
            
            # Position error
            pos_error = target_position - current_pos
            ori_error = self._orientation_error(target_orientation, current_ori)
            
            error = np.concatenate([pos_error, ori_error])
            
            if np.linalg.norm(error) < 1e-6:
                break
            
            # Compute Jacobian
            jacobian = self._compute_jacobian(joint_angles)
            
            # Damped least squares solution
            damping = 0.01
            delta_angles = np.linalg.solve(
                jacobian.T @ jacobian + damping * np.eye(jacobian.shape[1]),
                jacobian.T @ error
            )
            
            # Update joint angles
            joint_angles += delta_angles
            
            # Apply joint limits
            joint_angles = self._apply_joint_limits(joint_angles)
            
        return joint_angles
    
    def _compute_jacobian(self, joint_angles):
        """Compute Jacobian matrix"""
        epsilon = 1e-6
        jacobian = np.zeros((6, len(self.kinematics.joint_names)))
        
        for i, joint_name in enumerate(self.kinematics.joint_names):
            # Perturb joint angle
            angles_plus = joint_angles.copy()
            angles_plus[joint_name] += epsilon
            
            # Forward kinematics
            pos_plus, ori_plus = self.kinematics.forward_kinematics(angles_plus)
            pos, ori = self.kinematics.forward_kinematics(joint_angles)
            
            # Numerical derivative
            jacobian[:3, i] = (pos_plus - pos) / epsilon
            jacobian[3:, i] = self._orientation_derivative(ori_plus, ori) / epsilon
            
        return jacobian
    
    def _orientation_error(self, target, current):
        """Calculate orientation error"""
        # Convert to rotation matrices
        R_target = Rotation.from_euler('xyz', target).as_matrix()
        R_current = Rotation.from_euler('xyz', current).as_matrix()
        
        # Orientation error
        error_matrix = 0.5 * (R_target @ R_current.T - R_current @ R_target.T)
        
        # Extract vector from skew-symmetric matrix
        error_vector = np.array([
            error_matrix[2, 1],
            error_matrix[0, 2],
            error_matrix[1, 0]
        ])
        
        return error_vector

10.3 Dynamics​

10.3.1 Rigid Body Dynamics​

class HumanoidDynamics:
    def __init__(self):
        self.link_masses = self._initialize_link_masses()
        self.link_inertias = self._initialize_link_inertias()
        self.gravity = np.array([0, 0, -9.81])
        
    def _initialize_link_masses(self):
        """Initialize link masses"""
        return {
            'torso': 15.0,
            'left_thigh': 5.0,
            'left_shin': 3.0,
            'left_foot': 1.0,
            'right_thigh': 5.0,
            'right_shin': 3.0,
            'right_foot': 1.0,
            'left_upper_arm': 2.0,
            'left_forearm': 1.5,
            'right_upper_arm': 2.0,
            'right_forearm': 1.5,
            'head': 3.0
        }
    
    def _initialize_link_inertias(self):
        """Initialize link inertia tensors"""
        return {
            'torso': np.diag([0.5, 0.3, 0.2]),
            'left_thigh': np.diag([0.1, 0.1, 0.02]),
            'left_shin': np.diag([0.05, 0.05, 0.01]),
            # ... more links
        }
    
    def compute_dynamics(self, joint_angles, joint_velocities, joint_accelerations):
        """Compute joint torques using recursive Newton-Euler algorithm"""
        # Forward pass: compute velocities and accelerations
        link_velocities, link_accelerations = self._forward_pass(
            joint_angles, joint_velocities, joint_accelerations
        )
        
        # Backward pass: compute forces and torques
        joint_torques = self._backward_pass(
            link_velocities, link_accelerations
        )
        
        return joint_torques
    
    def _forward_pass(self, q, qd, qdd):
        """Forward pass of Newton-Euler algorithm"""
        velocities = {}
        accelerations = {}
        
        # Base link (assume fixed)
        velocities['base'] = np.zeros(6)  # [vx, vy, vz, wx, wy, wz]
        accelerations['base'] = np.array([0, 0, -9.81, 0, 0, 0])
        
        # Propagate through kinematic chain
        for i, joint_name in enumerate(q.keys()):
            # Transform to next link
            v_next = self._propagate_velocity(
                velocities['base'], qd[joint_name]
            )
            a_next = self._propagate_acceleration(
                accelerations['base'], qd[joint_name], qdd[joint_name]
            )
            
            velocities[joint_name] = v_next
            accelerations[joint_name] = a_next
            
        return velocities, accelerations
    
    def _backward_pass(self, velocities, accelerations):
        """Backward pass of Newton-Euler algorithm"""
        torques = {}
        
        # Initialize forces at end-effectors
        forces = {}
        for link_name in self.link_masses.keys():
            forces[link_name] = np.zeros(3)
        
        # Propagate forces backward
        for joint_name in reversed(list(velocities.keys())):
            # Compute joint torque
            if joint_name in self.link_inertias:
                I = self.link_inertias[joint_name]
                m = self.link_masses[joint_name]
                
                # Coriolis and centrifugal forces
                coriolis = self._compute_coriolis(velocities[joint_name], I)
                
                # Gravity compensation
                gravity_torque = self._compute_gravity_compensation(joint_name)
                
                # Total torque
                torques[joint_name] = coriolis + gravity_torque
                
        return torques

10.3.2 Contact Dynamics​

class ContactDynamics:
    def __init__(self):
        self.contact_points = ['left_foot', 'right_foot']
        self.friction_coefficient = 0.8
        self.ground_stiffness = 10000.0
        self.ground_damping = 100.0
        
    def compute_contact_forces(self, positions, velocities):
        """Compute contact forces with ground"""
        forces = {}
        
        for contact_point in self.contact_points:
            if contact_point in positions:
                pos = positions[contact_point]
                vel = velocities.get(contact_point, np.zeros(3))
                
                # Ground contact force
                if pos[2] < 0:  # Below ground
                    # Normal force
                    normal_force = -self.ground_stiffness * pos[2] - self.ground_damping * vel[2]
                    
                    # Friction force
                    tangential_vel = vel[:2]
                    if np.linalg.norm(tangential_vel) > 1e-6:
                        friction_direction = -tangential_vel / np.linalg.norm(tangential_vel)
                        friction_force = min(
                            self.friction_coefficient * normal_force,
                            np.linalg.norm(tangential_vel) * self.ground_damping
                        ) * friction_direction
                    else:
                        friction_force = np.zeros(2)
                    
                    # Total contact force
                    forces[contact_point] = np.array([
                        friction_force[0],
                        friction_force[1],
                        max(0, normal_force)
                    ])
                else:
                    forces[contact_point] = np.zeros(3)
                    
        return forces
    
    def check_stability(self, com_position, contact_forces):
        """Check if robot is statically stable"""
        # Compute center of pressure
        cop = np.zeros(2)
        total_normal_force = 0
        
        for contact_point, force in contact_forces.items():
            if force[2] > 0:  # Contact with ground
                cop += force[2] * force[:2]
                total_normal_force += force[2]
        
        if total_normal_force > 0:
            cop /= total_normal_force
        
        # Check stability margin
        stability_margin = np.linalg.norm(com_position[:2] - cop)
        
        return stability_margin < 0.1, stability_margin

10.4 Balance Control​

10.4.1 Zero Moment Point (ZMP)​

class ZMPController:
    def __init__(self):
        self.zmp_preview_window = 1.0  # seconds
        self.control_frequency = 100.0  # Hz
        
    def compute_zmp(self, com_position, com_acceleration, total_mass):
        """Compute Zero Moment Point"""
        # ZMP formula
        zmp_x = com_position[0] - com_position[2] * com_acceleration[0] / 9.81
        zmp_y = com_position[1] - com_position[2] * com_acceleration[1] / 9.81
        
        return np.array([zmp_x, zmp_y])
    
    def preview_control(self, reference_zmp, current_com, horizon=1.0):
        """Model predictive control for ZMP tracking"""
        # State space model
        dt = 1.0 / self.control_frequency
        A = np.array([
            [1, dt, dt**2/2],
            [0, 1, dt],
            [0, 0, 1]
        ])
        B = np.array([dt**3/6, dt**2/2, dt])
        
        # Preview control
        N = int(horizon * self.control_frequency)
        Q = np.eye(3 * N)
        R = 1.0
        
        # Solve optimization problem
        # This is a simplified version - in practice use QP solver
        control_sequence = self._solve_mpc(
            A, B, current_com, reference_zmp, N, Q, R
        )
        
        return control_sequence[0]  # First control input
    
    def _solve_mpc(self, A, B, x0, zmp_ref, N, Q, R):
        """Solve MPC optimization problem"""
        # Simplified MPC solution
        # In practice, use proper QP solver like cvxpy or qpOASES
        n_states = 3
        n_controls = 1
        
        # Build prediction matrices
        F = np.zeros((N * n_states, n_states))
        H = np.zeros((N * n_states, N * n_controls))
        
        x = x0.copy()
        for i in range(N):
            F[i*n_states:(i+1)*n_states, :] = np.linalg.matrix_power(A, i+1)
            
            for j in range(i+1):
                H[i*n_states:(i+1)*n_states, j*n_controls:(j+1)*n_controls] = \
                    np.linalg.matrix_power(A, i-j) @ B.reshape(-1, 1)
        
        # Cost matrices
        zmp_selection = np.array([1, 0, -9.81])  # Select ZMP from state
        C = np.kron(np.eye(N), zmp_selection)
        
        # Solve (simplified)
        P = H.T @ C.T @ Q @ C @ H + R * np.eye(N)
        q = H.T @ C.T @ Q @ (zmp_ref.flatten() - C @ F @ x0)
        
        # Solve linear system
        u_opt = np.linalg.solve(P, q)
        
        return u_opt

10.4.2 Balance Recovery​

class BalanceRecovery:
    def __init__(self, zmp_controller):
        self.zmp_controller = zmp_controller
        self.recovery_strategies = ['ankle_strategy', 'hip_strategy', 'step_strategy']
        
    def detect_fall(self, com_position, com_velocity, zmp_position):
        """Detect if robot is falling"""
        # Check if ZMP is outside support polygon
        support_polygon = self._get_support_polygon()
        
        outside_support = not self._point_in_polygon(zmp_position, support_polygon)
        high_velocity = np.linalg.norm(com_velocity) > 0.5
        large_tilt = abs(com_position[2]) > 0.1
        
        return outside_support or high_velocity or large_tilt
    
    def select_recovery_strategy(self, fall_state):
        """Select appropriate recovery strategy"""
        if fall_state['tilt_angle'] < 0.1:
            return 'ankle_strategy'
        elif fall_state['tilt_angle'] < 0.3:
            return 'hip_strategy'
        else:
            return 'step_strategy'
    
    def execute_ankle_strategy(self, current_angles, target_zmp):
        """Execute ankle strategy for balance recovery"""
        # Adjust ankle angles to shift ZMP
        ankle_adjustment = self._compute_ankle_adjustment(target_zmp)
        
        new_angles = current_angles.copy()
        new_angles['left_ankle_pitch'] += ankle_adjustment[0]
        new_angles['left_ankle_roll'] += ankle_adjustment[1]
        new_angles['right_ankle_pitch'] += ankle_adjustment[0]
        new_angles['right_ankle_roll'] += ankle_adjustment[1]
        
        return new_angles
    
    def execute_step_strategy(self, com_position, com_velocity):
        """Execute stepping strategy for balance recovery"""
        # Compute capture point
        capture_point = com_position[:2] + np.sqrt(com_position[2] / 9.81) * com_velocity[:2]
        
        # Plan foot placement
        foot_placement = self._plan_foot_placement(capture_point)
        
        return foot_placement
    
    def _get_support_polygon(self):
        """Get current support polygon"""
        # Simplified support polygon (feet positions)
        return np.array([
            [-0.1, -0.05],  # Left foot corners
            [-0.1, 0.05],
            [0.1, 0.05],   # Right foot corners
            [0.1, -0.05]
        ])
    
    def _point_in_polygon(self, point, polygon):
        """Check if point is inside polygon"""
        # Ray casting algorithm
        x, y = point
        n = len(polygon)
        inside = False
        
        j = n - 1
        for i in range(n):
            xi, yi = polygon[i]
            xj, yj = polygon[j]
            
            if ((yi > y) != (yj > y)) and (x < (xj - xi) * (y - yi) / (yj - yi) + xi):
                inside = not inside
                
            j = i
            
        return inside

10.5 Gait Planning​

10.5.1 Trajectory Generation​

class GaitPlanner:
    def __init__(self):
        self.step_duration = 0.8  # seconds
        self.step_height = 0.05   # meters
        self.step_length = 0.3    # meters
        
    def generate_foot_trajectory(self, start_pos, end_pos, phase):
        """Generate foot trajectory using cubic spline"""
        # Parameterize trajectory
        t = phase  # 0 to 1
        
        # Cubic spline coefficients
        a0 = start_pos
        a1 = np.zeros(3)
        a2 = 3 * (end_pos - start_pos) - 2 * self.step_height * np.array([0, 0, 1])
        a3 = -2 * (end_pos - start_pos) + self.step_height * np.array([0, 0, 1])
        
        # Trajectory
        position = a0 + a1 * t + a2 * t**2 + a3 * t**3
        
        # Add height variation
        position[2] += 4 * self.step_height * t * (1 - t)
        
        return position
    
    def generate_com_trajectory(self, foot_positions, duration):
        """Generate center of mass trajectory"""
        time_points = np.linspace(0, duration, int(duration * 100))
        com_trajectory = []
        
        for t in time_points:
            # Interpolate between foot positions
            phase = (t % self.step_duration) / self.step_duration
            
            if phase < 0.5:  # Single support
                com = foot_positions[0] + 0.1 * np.array([0, 0, 1])
            else:  # Double support
                com = (foot_positions[0] + foot_positions[1]) / 2 + 0.05 * np.array([0, 0, 1])
                
            com_trajectory.append(com)
            
        return np.array(com_trajectory)
    
    def plan_walking_sequence(self, start_position, target_position, num_steps):
        """Plan complete walking sequence"""
        foot_steps = []
        
        # Generate footstep positions
        for i in range(num_steps):
            if i % 2 == 0:  # Left foot
                foot_pos = start_position + np.array([
                    (i // 2 + 1) * self.step_length,
                    0.1,  # Lateral offset
                    0
                ])
            else:  # Right foot
                foot_pos = start_position + np.array([
                    (i // 2 + 1) * self.step_length,
                    -0.1,  # Lateral offset
                    0
                ])
                
            foot_steps.append(foot_pos)
            
        return foot_steps

10.6 ROS 2 Integration​

10.6.1 Kinematics Node​

#!/usr/bin/env python3

import rclpy
from rclpy.node import Node
from sensor_msgs.msg import JointState
from geometry_msgs.msg import PoseStamped, Point
from std_msgs.msg import Float64MultiArray

class HumanoidKinematicsNode(Node):
    def __init__(self):
        super().__init__('humanoid_kinematics_node')
        
        # Initialize kinematics
        self.kinematics = HumanoidKinematics()
        self.ik_solver = InverseKinematics(self.kinematics)
        
        # Publishers
        self.left_foot_pub = self.create_publisher(
            PoseStamped, 'left_foot_pose', 10
        )
        self.right_foot_pub = self.create_publisher(
            PoseStamped, 'right_foot_pose', 10
        )
        self.com_pub = self.create_publisher(
            Point, 'center_of_mass', 10
        )
        
        # Subscribers
        self.joint_sub = self.create_subscription(
            JointState, 'joint_states', self.joint_callback, 10
        )
        self.target_sub = self.create_subscription(
            PoseStamped, 'target_pose', self.target_callback, 10
        )
        
        self.get_logger().info('Humanoid Kinematics Node started')
    
    def joint_callback(self, msg):
        """Process joint states"""
        # Convert joint states to dictionary
        joint_angles = {}
        for i, name in enumerate(msg.name):
            joint_angles[name] = msg.position[i]
        
        # Forward kinematics
        positions, orientations = self.kinematics.forward_kinematics(joint_angles)
        
        # Publish foot positions
        self._publish_foot_positions(positions, orientations)
        
        # Publish center of mass
        self._publish_com(positions)
    
    def target_callback(self, msg):
        """Process target pose for inverse kinematics"""
        target_position = np.array([
            msg.pose.position.x,
            msg.pose.position.y,
            msg.pose.position.z
        ])
        
        target_orientation = np.array([
            msg.pose.orientation.x,
            msg.pose.orientation.y,
            msg.pose.orientation.z
        ])
        
        # Solve inverse kinematics
        current_angles = self._get_current_joint_angles()
        solution = self.ik_solver.inverse_kinematics(
            target_position, target_orientation, current_angles
        )
        
        # Publish joint commands
        self._publish_joint_commands(solution)
    
    def _publish_foot_positions(self, positions, orientations):
        """Publish foot positions"""
        # Left foot
        left_foot_msg = PoseStamped()
        left_foot_msg.header.stamp = self.get_clock().now().to_msg()
        left_foot_msg.header.frame_id = 'base_link'
        
        if 'left_ankle_roll' in positions:
            pos = positions['left_ankle_roll']
            left_foot_msg.pose.position.x = pos[0]
            left_foot_msg.pose.position.y = pos[1]
            left_foot_msg.pose.position.z = pos[2]
            
        self.left_foot_pub.publish(left_foot_msg)
        
        # Right foot
        right_foot_msg = PoseStamped()
        right_foot_msg.header.stamp = self.get_clock().now().to_msg()
        right_foot_msg.header.frame_id = 'base_link'
        
        if 'right_ankle_roll' in positions:
            pos = positions['right_ankle_roll']
            right_foot_msg.pose.position.x = pos[0]
            right_foot_msg.pose.position.y = pos[1]
            right_foot_msg.pose.position.z = pos[2]
            
        self.right_foot_pub.publish(right_foot_msg)
    
    def _publish_com(self, positions):
        """Publish center of mass"""
        com_msg = Point()
        
        # Simple COM calculation (would need link masses in practice)
        com = np.zeros(3)
        count = 0
        for pos in positions.values():
            com += pos
            count += 1
            
        if count > 0:
            com /= count
            
        com_msg.x = com[0]
        com_msg.y = com[1]
        com_msg.z = com[2]
        
        self.com_pub.publish(com_msg)
    
    def _get_current_joint_angles(self):
        """Get current joint angles (simplified)"""
        # In practice, this would read from current joint states
        return {name: 0.0 for name in self.kinematics.joint_names}
    
    def _publish_joint_commands(self, joint_angles):
        """Publish joint commands"""
        msg = Float64MultiArray()
        msg.data = [joint_angles.get(name, 0.0) for name in self.kinematics.joint_names]
        
        # Create publisher if needed
        if not hasattr(self, 'joint_cmd_pub'):
            self.joint_cmd_pub = self.create_publisher(
                Float64MultiArray, 'joint_commands', 10
            )
            
        self.joint_cmd_pub.publish(msg)

def main(args=None):
    rclpy.init(args=args)
    node = HumanoidKinematicsNode()
    
    try:
        rclpy.spin(node)
    except KeyboardInterrupt:
        pass
    finally:
        node.destroy_node()
        rclpy.shutdown()

if __name__ == '__main__':
    main()

Summary​

This chapter covered the fundamental concepts and practical implementation of humanoid robot kinematics and dynamics:

  • Forward Kinematics: Computing end-effector positions from joint angles
  • Inverse Kinematics: Solving joint angles for desired positions
  • Dynamics: Computing forces and torques for motion
  • Balance Control: ZMP-based stability and recovery strategies
  • Gait Planning: Generating walking trajectories
  • ROS 2 Integration: Real-time kinematics computation

The next chapter will explore human-robot interaction systems for creating natural and intuitive interfaces with humanoid robots.

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