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Module 03: Kinematics Fundamentals

Introduction​

Kinematics describes the geometry of motion without considering forces. For humanoid robots, kinematics answers: "Given joint angles, where is the hand?" (forward kinematics) and "What joint angles place the hand here?" (inverse kinematics).

Section 1: Coordinate Transformations​

1.1 Rotation Matrices​

Rotation Matrix: A 3×3 orthogonal matrix R\mathbf{R} with determinant +1 that transforms vectors between coordinate frames.

Properties:

  • RT=R−1\mathbf{R}^T = \mathbf{R}^{-1}
  • det⁡(R)=1\det(\mathbf{R}) = 1
  • Columns are orthonormal

1.2 Homogeneous Transformations​

T=[Rp0T1]\mathbf{T} = \begin{bmatrix} \mathbf{R} & \mathbf{p} \\ \mathbf{0}^T & 1 \end{bmatrix}

Section 2: Forward Kinematics​

2.1 DH Parameters​

Denavit-Hartenberg (DH) Parameters: A standard convention using four parameters (aa, α\alpha, dd, θ\theta) to describe the relationship between consecutive links.

 
def dh_transform(a, alpha, d, theta):
    """Compute DH transformation matrix."""
    ct, st = np.cos(theta), np.sin(theta)
    ca, sa = np.cos(alpha), np.sin(alpha)
    return np.array([
        [ct, -st*ca,  st*sa, a*ct],
        [st,  ct*ca, -ct*sa, a*st],
        [0,     sa,     ca,    d],
        [0,      0,      0,    1]
    ])

2.2 Chain Multiplication​

The end-effector pose is computed by chaining transforms:

T0n=T01⋅T12⋯Tn−1n\mathbf{T}_{0}^{n} = \mathbf{T}_{0}^{1} \cdot \mathbf{T}_{1}^{2} \cdots \mathbf{T}_{n-1}^{n}

Section 3: Inverse Kinematics​

3.1 Analytical Solutions​

For simple kinematic chains, closed-form solutions exist using geometric relationships.

3.2 Numerical Methods​

For complex robots, iterative methods are required:

def inverse_kinematics_jacobian(robot, target_pose, q_init, max_iter=100):
    """Solve IK using Jacobian pseudo-inverse."""
    q = q_init.copy()
    for _ in range(max_iter):
        current_pose = robot.forward_kinematics(q)
        error = pose_error(target_pose, current_pose)
```python
```python
        if np.linalg.norm(error) < 1e-6:
            break
        J = robot.jacobian(q)
        dq = np.linalg.pinv(J) @ error
        q += dq
    return q

Inverse kinematics may have multiple solutions, no solutions (unreachable targets), or infinite solutions (redundant manipulators).

Section 4: Jacobians and Singularities​

4.1 Velocity Kinematics​

x˙=J(q)q˙\dot{\mathbf{x}} = \mathbf{J}(\mathbf{q})\dot{\mathbf{q}}

4.2 Singularities​

When det⁡(J)=0\det(\mathbf{J}) = 0, the robot is in a singular configuration where certain directions of motion are impossible.

Summary​

Key takeaways:

  1. Forward kinematics maps joint angles to end-effector pose
  2. DH parameters provide a systematic parameterization
  3. Inverse kinematics finds joint angles for desired poses
  4. The Jacobian relates joint and Cartesian velocities

Key Concepts​

  • Forward Kinematics: Computing end-effector pose from joint angles
  • Inverse Kinematics: Computing joint angles from desired pose
  • Jacobian: Matrix relating joint and task space velocities
  • Singularity: Configuration where Jacobian loses rank

Further Reading​

  1. Murray, R., Li, Z., & Sastry, S. (1994). "A Mathematical Introduction to Robotic Manipulation"
  2. Spong, M., Hutchinson, S., & Vidyasagar, M. (2020). "Robot Modeling and Control"