Chapter 2: Basics of Humanoid Robotics

Learning Objectives

By the end of this chapter, you will be able to:

Understand the mechanical structure and design principles of humanoid robots
Explain forward and inverse kinematics for robotic manipulators
Identify different types of actuators and their trade-offs
Describe sensor integration for perception and proprioception
Understand basic control architectures for humanoid systems
Recognize the challenges unique to bipedal locomotion

2.1 Why Humanoid Robots?

Humanoid robots are designed to mimic the human form—typically featuring a torso, two arms, two legs, and a head. But why build robots that look like humans?

Advantages of Human-Like Form

1. Human-Designed Environments Our world is built for humans: doorways, stairs, chairs, tools, and vehicles all assume a human body shape. A humanoid robot can navigate these spaces and use these tools without requiring environmental modifications.

2. Natural Human Interaction Humans are more comfortable interacting with entities that resemble them. Humanoid robots can use gestures, facial expressions, and body language for intuitive communication.

3. Versatility & Generalization A humanoid form factor enables a single platform to perform diverse tasks—from opening doors to operating machinery to assisting with daily activities—without specialized end-effectors.

4. Leveraging Human Data Humanoid robots can learn from human demonstrations more directly. Motion capture data, teleoperation, and imitation learning become more straightforward when the robot has a similar kinematic structure.

Notable Humanoid Robots

ASIMO (Honda): Early pioneer, demonstrated walking, running, stair climbing
Atlas (Boston Dynamics): Athletic capabilities, parkour, backflips
Optimus (Tesla): Designed for factory and household tasks
Figure 01 (Figure AI): General-purpose humanoid for commercial deployment
Digit (Agility Robotics): Warehouse logistics and delivery
NAO (SoftBank): Educational and research platform
Pepper (SoftBank): Social interaction and customer service

2.2 Mechanical Structure & Degrees of Freedom

2.2.1 Kinematic Chain

A humanoid robot is composed of rigid links connected by joints. This forms a kinematic chain where motion at one joint affects the position of all subsequent links.

Key Terminology:

Link: Rigid body segment (e.g., forearm, thigh)
Joint: Connection allowing relative motion between links
Degree of Freedom (DOF): Independent motion capability (e.g., shoulder rotation)
End-effector: Terminal link (e.g., hand, foot)

2.2.2 Typical DOF Distribution

A full humanoid robot typically has 30-50+ DOF:

Head (2-3 DOF):

Yaw (side-to-side)
Pitch (up-down)
Roll (tilt) [optional]

Torso (1-3 DOF):

Waist yaw
Spine pitch/roll [optional]

Arms (7 DOF each = 14 total):

Shoulder: 3 DOF (pitch, roll, yaw)
Elbow: 1 DOF (flexion/extension)
Wrist: 3 DOF (pitch, roll, yaw)
Hand: 1-20+ DOF depending on complexity

Legs (6 DOF each = 12 total):

Hip: 3 DOF (pitch, roll, yaw)
Knee: 1 DOF (flexion/extension)
Ankle: 2 DOF (pitch, roll)

Total: ~30-40 DOF for basic humanoid, 50+ with articulated hands

2.2.3 Joint Types

Revolute Joint (Hinge)

Rotates around a single axis
Most common in humanoid robots
Examples: elbow, knee, finger joints

Prismatic Joint (Slider)

Translates along a single axis
Less common in humanoids
Examples: telescoping limbs, spinal extension

Spherical Joint (Ball-and-Socket)

3 DOF rotation around a point
Mechanically complex to implement
Often approximated with 3 revolute joints (shoulder, hip)

2.3 Kinematics: The Mathematics of Motion

Kinematics describes the geometry of motion without considering forces. For humanoid robots, we need to solve two fundamental problems:

2.3.1 Forward Kinematics (FK)

Problem: Given joint angles θ₁, θ₂, ..., θₙ, what is the position and orientation of the end-effector?

Solution: Use transformation matrices to propagate from base to end-effector.

Denavit-Hartenberg (DH) Convention: A systematic way to assign coordinate frames to each joint. Each link is described by 4 parameters:

a: Link length
α: Link twist
d: Link offset
θ: Joint angle (variable for revolute joints)

Transformation Matrix:

T = [cos(θ)  -sin(θ)cos(α)   sin(θ)sin(α)  a·cos(θ)]
    [sin(θ)   cos(θ)cos(α)  -cos(θ)sin(α)  a·sin(θ)]
    [0        sin(α)         cos(α)         d       ]
    [0        0              0              1       ]

Chain Rule: Multiply transformations from base to end-effector:

T_end = T₀₁ · T₁₂ · T₂₃ · ... · Tₙ₋₁,ₙ

Example: 2-Link Planar Arm

Given:

Link 1 length: L₁ = 1.0 m, angle θ₁ = 45°
Link 2 length: L₂ = 0.8 m, angle θ₂ = 30°

End-effector position:

x = L₁·cos(θ₁) + L₂·cos(θ₁ + θ₂)
y = L₁·sin(θ₁) + L₂·sin(θ₁ + θ₂)

x = 1.0·cos(45°) + 0.8·cos(75°) ≈ 0.914 m
y = 1.0·sin(45°) + 0.8·sin(75°) ≈ 1.480 m

2.3.2 Inverse Kinematics (IK)

Problem: Given desired end-effector position/orientation, what joint angles achieve it?

Challenge: Often has multiple solutions (or no solution if position unreachable).

Approaches:

1. Analytical (Closed-Form)

Geometric reasoning to derive equations
Fast computation
Only feasible for simple kinematic chains (≤6 DOF)

Example for 2-Link Planar Arm:

θ₂ = ±arccos((x² + y² - L₁² - L₂²) / (2·L₁·L₂))
θ₁ = arctan2(y, x) - arctan2(L₂·sin(θ₂), L₁ + L₂·cos(θ₂))

2. Numerical (Iterative)

Jacobian-based methods: Linearize and iterate
Optimization: Minimize ||f(θ) - target||²
Handles redundant systems (more DOF than needed)
May get stuck in local minima

3. Learning-Based

Train neural networks to predict joint angles
Handles complex constraints
Requires large datasets

2.3.3 Jacobian Matrix

The Jacobian J relates joint velocities to end-effector velocities:

ẋ = J(θ) · θ̇

Where:

ẋ: End-effector velocity (linear + angular)
θ̇: Joint velocities
J(θ): Jacobian matrix (6×n for n joints)

Applications:

Velocity control
Singularity detection (det(J) = 0)
Force control (τ = Jᵀ · F)

2.4 Actuation Systems

Actuators convert energy into motion. Choosing the right actuator involves trade-offs between power, speed, precision, size, and cost.

2.4.1 Electric Motors

Brushed DC Motors

Simple control, low cost
Wear due to brushes
Used in hobbyist robots

Brushless DC Motors (BLDC)

Higher efficiency, longer life
Requires electronic commutation (ESC)
Common in drones and small humanoids

Servo Motors

Integrated position feedback and control
Easy to use (PWM signal)
Limited torque for humanoid applications

Stepper Motors

Precise positioning without feedback
Lower speed and efficiency
Used in CNC and 3D printers

2.4.2 Hydraulic Actuators

Advantages:

High power-to-weight ratio
Large force output
Used in Atlas (Boston Dynamics)

Disadvantages:

Complex system (pump, valves, reservoir)
Messy (oil leaks)
Difficult to control precisely
Noisy operation

2.4.3 Pneumatic Actuators

Advantages:

Safe (compliant, no electrical hazard)
Fast response
Low cost

Disadvantages:

Compressibility makes control difficult
Requires air supply
Limited force compared to hydraulics
Noisy

2.4.4 Series Elastic Actuators (SEA)

Design: Spring in series between motor and load

Advantages:

Compliant, safe for human interaction
Force sensing via spring deflection
Energy storage for dynamic motions
Used in many research humanoids

Disadvantages:

Lower bandwidth (spring introduces lag)
More complex mechanical design

2.4.5 Comparison Table

Actuator Type	Power Density	Speed	Precision	Cost	Typical Use
Brushed DC	Low	Medium	Low	Low	Toys, hobbyist
BLDC	Medium	High	Medium	Medium	Drones, arms
Hydraulic	Very High	Medium	Low	High	Heavy-duty (Atlas)
Pneumatic	Medium	High	Low	Low	Soft robots
SEA	Medium	Low	High	High	Research humanoids

2.5 Sensor Integration

Humanoid robots require diverse sensors for perception and state estimation.

2.5.1 Proprioceptive Sensors (Internal State)

Joint Encoders

Measure joint angles
Types: Incremental, absolute
Resolution: 12-bit to 20-bit

Inertial Measurement Unit (IMU)

3-axis accelerometer: linear acceleration
3-axis gyroscope: angular velocity
3-axis magnetometer (optional): absolute heading
Used for balance and orientation estimation

Force/Torque Sensors

Measure interaction forces
Locations: feet (ground contact), wrists (manipulation), joints (load)
Enables compliant control and safety monitoring

Current Sensors

Monitor motor currents
Indirectly measure torque (τ ≈ k · I)
Detect collisions and overloads

2.5.2 Exteroceptive Sensors (External Environment)

Cameras

RGB: Color images for object recognition
Depth: Stereo, structured light, or time-of-flight
Typical placement: head (binocular vision), hands (manipulation)

LIDAR

2D or 3D point clouds
Long range (10-100m)
Robust to lighting conditions
Used for mapping and obstacle avoidance

Ultrasonic Sensors

Short-range proximity detection (0.1-5m)
Low cost, simple interface
Used for basic obstacle avoidance

Tactile Sensors

Pressure/touch detection
Important for manipulation (e.g., object grasping)
Complex and expensive to integrate

2.5.3 Sensor Fusion

Combining multiple sensors improves robustness:

Kalman Filter

Optimal fusion under Gaussian noise
Used for pose estimation (IMU + encoders)

Particle Filter

Handles non-Gaussian distributions
Used for localization (cameras + LIDAR)

Complementary Filter

Simple, lightweight fusion
Common for IMU orientation estimation

2.6 Control Architectures

2.6.1 Hierarchical Control Structure

Humanoid robots typically use a layered control architecture:

┌─────────────────────────────────────┐
│   High-Level Planning               │  <- Task/behavior planning
│   (seconds to minutes)              │
└──────────────┬──────────────────────┘
               ↓
┌─────────────────────────────────────┐
│   Mid-Level Control                 │  <- Trajectory generation
│   (100-500 Hz)                      │     Inverse kinematics
└──────────────┬──────────────────────┘
               ↓
┌─────────────────────────────────────┐
│   Low-Level Control                 │  <- Motor control (PID)
│   (1-10 kHz)                        │     Force/torque control
└─────────────────────────────────────┘

2.6.2 PID Control

Proportional-Integral-Derivative control is the workhorse of robotics:

u(t) = Kₚ·e(t) + Kᵢ·∫e(t)dt + Kd·de(t)/dt

Where:

e(t) = reference - actual (error)
Kₚ: Proportional gain (immediate response)
Kᵢ: Integral gain (eliminate steady-state error)
Kd: Derivative gain (damping, reduce overshoot)

Tuning:

Start with Kₚ only, increase until oscillations
Add Kd to dampen oscillations
Add Kᵢ to eliminate steady-state error
Iterate and test

2.6.3 Impedance Control

For safe human-robot interaction, control both position and force:

F = K·(x_desired - x_actual) + B·(ẋ_desired - ẋ_actual)

Where:

K: Stiffness (spring constant)
B: Damping (resistance to velocity)

Benefits:

Compliant behavior
Safe contact with humans and environment
Natural-feeling interactions

2.6.4 Model Predictive Control (MPC)

Optimize control over a future horizon:

Predict future states using a model
Optimize control inputs to minimize cost
Apply first control input
Repeat (receding horizon)

Advantages:

Handles constraints (joint limits, torque limits)
Anticipates future states
Optimal trajectories

Disadvantages:

Computationally expensive
Requires accurate model
Tuning can be complex

2.7 Bipedal Locomotion

Walking on two legs is one of the hardest problems in humanoid robotics.

2.7.1 Static vs. Dynamic Walking

Static Walking

Center of Mass (CoM) always above support polygon
Slow, stable
Easy to control
Energy inefficient

Dynamic Walking

CoM can momentarily leave support polygon
Faster, more natural
Requires active balance control
Energy efficient (passive dynamics)

2.7.2 Zero Moment Point (ZMP)

The ZMP is the point on the ground where the net moment from gravity and inertia is zero.

Stability Condition: ZMP must stay inside the support polygon (foot contact area).

ZMP-Based Walking:

Plan CoM trajectory such that ZMP stays inside support polygon
Compute inverse kinematics for leg joints
Execute trajectory with feedback control

Limitations:

Conservative (always statically stable)
Cannot handle dynamic maneuvers (running, jumping)

2.7.3 Gait Patterns

Walk Cycle Phases:

Double Support: Both feet on ground
Single Support: One foot on ground, other swinging
Swing Phase: Foot moves forward
Heel Strike: Foot touches down

Key Parameters:

Step length: Distance between foot placements
Step width: Lateral spacing
Step height: Swing foot clearance
Step time: Duration of one step

2.7.4 Balance & Stabilization

Ankle Strategy

Small perturbations
Torque at ankles to adjust CoM
Fast, minimal motion

Hip Strategy

Larger perturbations
Torque at hips to move CoM
Slower, more motion

Stepping Strategy

Very large perturbations
Take a step to regain balance
Last resort, most effective

2.8 Challenges in Humanoid Robotics

1. Complexity

30-50 DOF to coordinate simultaneously
Coupled dynamics (moving one limb affects whole body)
Real-time computation requirements

2. Energy Efficiency

Humanoid walking is energy-intensive
Battery life limits autonomy (typically 1-2 hours)
Optimization of gait and control crucial

3. Robustness

Uneven terrain, obstacles, disturbances
Slip detection and recovery
Fall detection and mitigation

4. Manipulation

Dexterous grasping requires complex hands
Coordination of arm, wrist, and fingers
Force control to avoid damaging objects

5. Perception

Real-time processing of visual and depth data
Occlusions and lighting variations
Integration with control (closed-loop)

6. Cost

High-quality actuators and sensors are expensive
Custom mechanical design and fabrication
Research platforms cost $100K-$1M+

2.9 Design Example: Simplified Humanoid Arm

Let's design a basic 3-DOF arm for manipulation tasks.

Specifications

Reach: 0.8 m
Payload: 2 kg
Precision: ±5 mm

Kinematic Design

Joint 1 (Shoulder Pitch): 0° to 180°
Joint 2 (Elbow Pitch): 0° to 150°
Joint 3 (Wrist Roll): -90° to 90°

Link Lengths

Upper arm: L₁ = 0.4 m
Forearm: L₂ = 0.4 m

Actuator Selection

Joints 1-2: BLDC motors with 50:1 gearbox (high torque)
Joint 3: Small servo motor (low torque, wrist rotation)

Sensors

Encoders: 14-bit absolute (0.02° resolution)
6-axis IMU in forearm (orientation feedback)
Force/torque sensor at wrist (manipulation tasks)

Control

Low-level: PID joint control at 1 kHz
Mid-level: Inverse kinematics at 100 Hz
High-level: Task planner (pick-and-place, handover)

2.10 Chapter Summary

Humanoid robots are designed with human-like form to navigate our environments and interact naturally. They consist of rigid links connected by joints, typically totaling 30-50 degrees of freedom across head, torso, arms, and legs.

Forward kinematics computes end-effector position from joint angles using transformation matrices, while inverse kinematics solves for joint angles given a desired position. The Jacobian matrix relates joint velocities to end-effector velocities.

Actuation options include electric motors (BLDC, servos), hydraulic actuators (high power), pneumatic systems (compliant), and series elastic actuators (safe interaction). Each has trade-offs in power, speed, precision, and cost.

Sensors are divided into proprioceptive (encoders, IMUs, force sensors) for internal state and exteroceptive (cameras, LIDAR) for environment perception. Sensor fusion techniques like Kalman filters combine data for robust estimation.

Control architectures are hierarchical: high-level planning, mid-level trajectory generation, and low-level motor control. PID control is standard, with impedance control for safe interaction and MPC for optimal trajectories.

Bipedal locomotion is challenging, requiring balance control through ZMP planning or dynamic approaches. Walking involves gait patterns, balance strategies, and robustness to disturbances.

Key challenges include complexity, energy efficiency, robustness, dexterous manipulation, real-time perception, and high costs. Despite these challenges, humanoid robotics continues to advance, driven by applications in manufacturing, healthcare, and service domains.

Key Takeaways

✅ Humanoid robots navigate human environments using human-like morphology
✅ Forward kinematics: joint angles → end-effector position
✅ Inverse kinematics: desired position → joint angles (multiple solutions)
✅ Actuators: BLDC motors common, hydraulics for power, SEA for safety
✅ Sensors: Encoders + IMU (proprioception), cameras + LIDAR (exteroception)
✅ Control: Hierarchical (planning → trajectories → motor control)
✅ Bipedal walking: ZMP-based (stable) or dynamic (efficient)
✅ Challenges: complexity, energy, robustness, manipulation, perception, cost

Learning Objectives​

2.1 Why Humanoid Robots?​

Advantages of Human-Like Form​

Notable Humanoid Robots​

2.2 Mechanical Structure & Degrees of Freedom​

2.2.1 Kinematic Chain​

2.2.2 Typical DOF Distribution​

2.2.3 Joint Types​

2.3 Kinematics: The Mathematics of Motion​

2.3.1 Forward Kinematics (FK)​

2.3.2 Inverse Kinematics (IK)​

2.3.3 Jacobian Matrix​

2.4 Actuation Systems​

2.4.1 Electric Motors​

2.4.2 Hydraulic Actuators​

2.4.3 Pneumatic Actuators​

2.4.4 Series Elastic Actuators (SEA)​

2.4.5 Comparison Table​

2.5 Sensor Integration​

2.5.1 Proprioceptive Sensors (Internal State)​

2.5.2 Exteroceptive Sensors (External Environment)​

2.5.3 Sensor Fusion​

2.6 Control Architectures​

2.6.1 Hierarchical Control Structure​

2.6.2 PID Control​

2.6.3 Impedance Control​

2.6.4 Model Predictive Control (MPC)​

2.7 Bipedal Locomotion​

2.7.1 Static vs. Dynamic Walking​

2.7.2 Zero Moment Point (ZMP)​

2.7.3 Gait Patterns​

2.7.4 Balance & Stabilization​

2.8 Challenges in Humanoid Robotics​

1. Complexity​

2. Energy Efficiency​

3. Robustness​

4. Manipulation​

5. Perception​

6. Cost​

2.9 Design Example: Simplified Humanoid Arm​

Specifications​

Kinematic Design​

Link Lengths​

Actuator Selection​

Sensors​

Control​

2.10 Chapter Summary​

Key Takeaways​

Further Reading​

Learning Objectives

2.1 Why Humanoid Robots?

Advantages of Human-Like Form

Notable Humanoid Robots

2.2 Mechanical Structure & Degrees of Freedom

2.2.1 Kinematic Chain

2.2.2 Typical DOF Distribution

2.2.3 Joint Types

2.3 Kinematics: The Mathematics of Motion

2.3.1 Forward Kinematics (FK)

2.3.2 Inverse Kinematics (IK)

2.3.3 Jacobian Matrix

2.4 Actuation Systems

2.4.1 Electric Motors

2.4.2 Hydraulic Actuators

2.4.3 Pneumatic Actuators

2.4.4 Series Elastic Actuators (SEA)

2.4.5 Comparison Table

2.5 Sensor Integration

2.5.1 Proprioceptive Sensors (Internal State)

2.5.2 Exteroceptive Sensors (External Environment)

2.5.3 Sensor Fusion

2.6 Control Architectures

2.6.1 Hierarchical Control Structure

2.6.2 PID Control

2.6.3 Impedance Control

2.6.4 Model Predictive Control (MPC)

2.7 Bipedal Locomotion

2.7.1 Static vs. Dynamic Walking

2.7.2 Zero Moment Point (ZMP)

2.7.3 Gait Patterns

2.7.4 Balance & Stabilization

2.8 Challenges in Humanoid Robotics

1. Complexity

2. Energy Efficiency

3. Robustness

4. Manipulation

5. Perception

6. Cost

2.9 Design Example: Simplified Humanoid Arm

Specifications

Kinematic Design

Link Lengths

Actuator Selection

Sensors

Control

2.10 Chapter Summary

Key Takeaways

Further Reading