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Lesson 11.2: Humanoid Robot Kinematics and Dynamics

Learning Objectives​

After completing this lesson, you will be able to:

  • Understand forward and inverse kinematics for humanoid robots
  • Apply mathematical models for humanoid robot dynamics
  • Explain the relationship between joint control and robot stability
  • Analyze the dynamic behavior of multi-link humanoid systems
  • Implement basic kinematic solutions for humanoid robots

Introduction​

Humanoid robot kinematics and dynamics form the mathematical foundation for understanding and controlling these complex systems. Kinematics deals with the geometric relationships between joints and end-effectors without considering forces, while dynamics encompasses the forces and torques that cause motion. This lesson explores both areas, focusing on their application to humanoid robots with their unique multi-link, multi-degree-of-freedom structures.

Forward Kinematics for Humanoid Robots​

Forward kinematics is the process of determining the position and orientation of the end-effector (hand, foot, head) given the joint angles. For humanoid robots, this involves complex multi-chain kinematics due to the presence of multiple limbs.

Mathematical Representation​

The forward kinematics of a humanoid robot can be represented using homogeneous transformation matrices. For a joint chain with n joints:

T = T1(θ1) * T2(θ2) * ... * Tn(θn)

Where T is the final transformation matrix representing the end-effector pose relative to the base frame, and T_i(θ_i) represents the transformation due to joint i with angle θ_i.

Humanoid-Specific Considerations​

Multi-Chain Kinematics

  • Humanoid robots have multiple kinematic chains (arms, legs, spine, head)
  • Each chain has its own forward kinematics solution
  • The chains are interconnected through the torso, requiring coordination

Degrees of Freedom

  • Humanoid robots typically have 30+ degrees of freedom
  • The human body has approximately 206 joints, but humanoid robots simplify this
  • Common configurations: 12-18 DOF for legs, 6-8 DOF for arms, 2-3 DOF for spine/head

Redundancy

  • Humanoid robots often have redundant DOF (more DOF than required for a task)
  • This redundancy allows for multiple solutions to reach the same pose
  • Optimization criteria needed to select the best solution

Denavit-Hartenberg (DH) Parameters​

The DH convention provides a systematic method for defining coordinate frames for each joint in a humanoid robot:

T(i-1,i) = [
    [cos(θi), -sin(θi)*cos(αi),  sin(θi)*sin(αi), ai*cos(θi)],
    [sin(θi),  cos(θi)*cos(αi), -cos(θi)*sin(αi), ai*sin(θi)],
    [0,        sin(αi),          cos(αi),          di        ],
    [0,        0,                0,                1         ]
]

Where:

  • θi: joint angle (variable for revolute joints)
  • αi: twist angle (constant)
  • ai: link length (constant)
  • di: link offset (constant)

Inverse Kinematics for Humanoid Robots​

Inverse kinematics (IK) determines the joint angles required to achieve a desired end-effector pose. This is more complex than forward kinematics and often has multiple solutions or no solutions.

Analytical vs. Numerical Methods​

Analytical Methods

  • Closed-form solutions for simple kinematic chains
  • Fast computation but limited to specific configurations
  • Not generally applicable to complex humanoid structures

Numerical Methods

  • Iterative approaches like Jacobian-based methods
  • Applicable to arbitrary kinematic structures
  • Computationally more expensive but more general

Jacobian-Based Methods​

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

v = J(θ) * θ̇

Where:

  • v: end-effector velocity vector (linear and angular)
  • J(θ): Jacobian matrix
  • θ̇: joint velocity vector

To solve for joint velocities given desired end-effector velocity:

θ̇ = J⁻¹(θ) * v

For redundant systems, the pseudoinverse is used:

θ̇ = J⁺(θ) * v

Humanoid-Specific IK Challenges​

Multiple End-Effectors

  • Need to coordinate multiple IK solutions simultaneously
  • Arms and legs may need to work together
  • Balance constraints must be considered

Joint Limits

  • Solutions must respect physical joint limits
  • May require optimization to avoid joint limit violations
  • Self-collision avoidance is critical

Real-time Requirements

  • IK solutions needed at high frequency (200Hz+)
  • Efficient algorithms required for real-time control
  • Approximation methods often used for speed

Robot Dynamics​

Robot dynamics describes the relationship between forces/torques applied to the robot and resulting motion. For humanoid robots, this is essential for stable control and dynamic behaviors.

Lagrangian Formulation​

The dynamic equations of motion can be derived using the Lagrangian approach:

τ = M(q)q̈ + C(q, q̇)q̇ + G(q) + JᵀF

Where:

  • τ: joint torque vector
  • M(q): inertia matrix
  • C(q, q̇): Coriolis and centrifugal forces
  • G(q): gravitational forces
  • JᵀF: external forces transformed to joint space
  • q, q̇, q̈: joint positions, velocities, and accelerations

Inertia Matrix (M(q))​

The inertia matrix represents how the robot's mass is distributed relative to its joints:

M(q) = Σ(mi * Jiᵀ * Ji + Ii * Ri * Jiᵀ * Riᵀ)

Where:

  • mi: mass of link i
  • Ji: Jacobian of link i's center of mass
  • Ii: inertia tensor of link i
  • Ri: rotation matrix of link i

Coriolis and Centrifugal Forces (C(q, q̇))​

These forces arise from the motion of the robot's links:

C(q, q̇)q̇ = Σ(ċi * q̇i + ½ * Σ(∂Ji/∂qj * q̇i * q̇j))

Gravitational Forces (G(q))​

Gravitational forces depend on the robot's configuration:

G(q) = Σ(mi * g * ∂hi/∂q)

Where:

  • g: gravitational acceleration vector
  • hi: height of link i's center of mass

Joint Control and Stability​

Control Architecture​

Humanoid robots typically use hierarchical control structures:

High-Level Controller

  • Generates desired trajectories and behaviors
  • Handles balance and gait planning
  • Manages interaction with environment

Mid-Level Controller

  • Implements impedance control
  • Manages contact transitions
  • Coordinates multiple limbs

Low-Level Controller

  • Joint-level PID control
  • Motor command execution
  • Real-time feedback control

Balance Control​

Balance control is critical for humanoid robots due to their inherently unstable nature:

Zero Moment Point (ZMP)

  • Point where the net moment of ground reaction forces is zero
  • For stable walking, ZMP must remain within the support polygon
  • Used as a stability criterion in gait planning

Capture Point

  • Location where the robot can come to rest given its current state
  • Useful for balance recovery strategies
  • Computed from current position and velocity

Mathematical Models for Balance​

The linear inverted pendulum model (LIPM) is commonly used for humanoid balance:

ẍ = ω²(x - x₀)

Where:

  • x: center of mass position
  • x₀: reference point (typically ZMP)
  • ω: natural frequency (sqrt(g/h), h = height)

Mathematical Modeling Approaches​

Multi-Body Dynamics​

Humanoid robots can be modeled as interconnected rigid bodies:

Recursive Newton-Euler Algorithm

  • Efficient computation of inverse dynamics
  • O(n) complexity for n DOF
  • Suitable for real-time control

Lagrangian Method

  • Systematic derivation of equations of motion
  • Good for analysis and simulation
  • More complex for real-time control

Contact Modeling​

When humanoid robots interact with the environment, contact forces must be modeled:

Rigid Contact Model

  • No penetration allowed
  • Friction cone constraints
  • Impulse-based collision handling

Soft Contact Model

  • Allows slight penetration
  • Spring-damper based forces
  • More stable but less accurate

Practical Implementation Considerations​

Real-time Computation​

  • Kinematic and dynamic calculations must be performed at high frequency
  • Approximation methods may be necessary for real-time performance
  • Parallel computation can help with complex calculations

Sensor Integration​

  • Joint encoders provide position feedback
  • IMUs provide orientation and acceleration data
  • Force/torque sensors provide contact information
  • Vision systems provide environmental data

Control Stability​

  • Proper tuning of control parameters is essential
  • Robust control methods handle model uncertainties
  • Adaptive control adjusts to changing conditions

Summary​

Humanoid robot kinematics and dynamics form the mathematical foundation for their control and operation. Forward kinematics determines end-effector positions from joint angles, while inverse kinematics finds joint angles for desired end-effector poses. Robot dynamics describes the relationship between forces and motion, essential for stable control. For humanoid robots, these concepts become particularly complex due to multiple kinematic chains, redundancy, and the need for balance and stability. Proper mathematical modeling and real-time implementation are critical for successful humanoid robot operation.

Further Reading​

  • Spong, M.W., et al. "Robot Modeling and Control" - Comprehensive treatment of robot kinematics and dynamics
  • Siciliano, B. & Khatib, O. "Springer Handbook of Robotics" - Advanced topics in humanoid robotics
  • Featherstone, R. "Rigid Body Dynamics Algorithms" - Efficient computation of robot dynamics
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