Chapter 68A: Robot Kinematics¶

Prerequisites: Homogeneous Coordinates (Ch67) · Rotation Matrices (Ch05) · Geometric Algebra (Ch50)

Chapter Outline: From Degrees of Freedom to Pose Description → Representations of Rotation: Axis-angle, Euler Angles, and Quaternions → Core Description: Denavit-Hartenberg (DH) Parameterization → Forward Kinematics (FK): Concatenated Multiplication of Transformation Matrices → Algebraic Challenges of Inverse Kinematics (IK) → The Robot Jacobian Matrix ($J$) and Singular Poses → Differential Motion: From Velocity Space to Force Space → Applications: Industrial Manipulator Path Planning, Surgical Robots, and Humanoid Bipedal Locomotion

Extension: Robot kinematics is the "dynamic topology" of linear algebra; it transforms the physical constraints of mechanical structures into continuous motion on matrix manifolds. It proves that precise control of an end-effector is essentially the solving of a series of highly coupled non-linear matrix equations—the mathematical foundation for embodied intelligence.

In robotics, every joint movement corresponds to a coordinate transformation. The core task of Robot Kinematics is to establish a precise mathematical mapping between joint angles (internal variables) and the end-effector pose (external variables). By cascading Homogeneous Transformation Matrices, we can describe extremely complex mechanical chains like building blocks. This chapter introduces the motion geometry that powers industrial automation and precision manufacturing.

68A.1 Pose Description and DH Parameters¶

Definition 68A.1 (Pose Matrix)

The pose of a robot end-effector relative to the base frame is described by a $4 \times 4$ matrix $T$: $$T = \begin{pmatrix} R & \mathbf{p} \\ 0 & 1 \end{pmatrix}$$ where $R$ represents orientation (rotation) and $\mathbf{p}$ represents spatial position.

Technique: DH Parameterization

Using four parameters (link length $a$, link twist $\alpha$, link offset $d$, and joint angle $\theta$), the transformation between adjacent joints is uniquely represented by a matrix $A_i$.

68A.2 Forward and Inverse Kinematics¶

Theorem 68A.1 (Fundamental Kinematic Equations)

Forward Kinematics (FK): Uniquely determines the end-effector pose by computing the product $T = A_1 A_2 \cdots A_n$.
Inverse Kinematics (IK): Given a target $T$, solve for the joint vector $\mathbf{q} = (\theta_1, \ldots, \theta_n)$.
Property: FK is a direct mapping, while IK is an inverse mapping that often involves multiple solutions or none at all.

68A.3 The Robot Jacobian and Velocity¶

Definition 68A.2 (Robot Jacobian $J$)

Establishes the linear mapping between joint velocities $\dot{\mathbf{q}}$ and end-effector velocities $\mathbf{v}$: $$\mathbf{v} = J(\mathbf{q}) \dot{\mathbf{q}}$$ Singular Pose: A configuration where $\det(J) = 0$, causing the robot to lose mobility in certain directions.

Exercises¶

1. [Basics] Write the DH transformation matrix for a rotation $\alpha$ around the $x$-axis followed by a translation $a$ along the $x$-axis.

Solution

Construction: According to DH rules, this is a composite transform: $A = \operatorname{Rot}(x, \alpha) \operatorname{Trans}(x, a)$. Result: $\begin{pmatrix} 1 & 0 & 0 & a \\ 0 & \cos\alpha & -\sin\alpha & 0 \\ 0 & \sin\alpha & \cos\alpha & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}$.

2. [Calculation] For a 2-link planar arm with link lengths $L_1=L_2=1$ and angles $\theta_1=0, \theta_2=90^\circ$, find the end-effector coordinates.

Solution

Geometric Derivation: 1. First segment extends 1 unit along $x$. 2. Second segment rotates $90^\circ$ relative to the first, extending 1 unit along $y$. 3. Final coordinates: $x = 1+0=1, y=0+1=1$. Matrix Verification: $T = \operatorname{Rot}_z(0)\operatorname{Trans}_x(1) \cdot \operatorname{Rot}_z(90^\circ)\operatorname{Trans}_x(1)$ yields the same $(1, 1)$ result.

3. [Inverse Kinematics] Explain why a 6-DOF manipulator typically has 8 inverse solutions.

Solution

Algebraic Context: 1. IK involves solving a system of coupled trigonometric equations. 2. Each major configuration ("Shoulder," "Elbow," and "Wrist") has two symmetric possibilities (e.g., Left/Right, Elbow up/down). 3. $2 \times 2 \times 2 = 8$. Conclusion: This reflects the global multi-valuedness of non-linear mappings.

4. [Jacobian] What do the columns of the Jacobian matrix represent geometrically?

Solution

Conclusion: The $i$-th column represents the velocity vector at the end-effector generated if only the $i$-th joint moves. For revolute joints, it is the cross product of the joint axis and the vector from the joint to the tip.

5. [Singularity] Determine: If a manipulator is fully extended (straight line), is it in a singular pose?

Solution

Yes. Reasoning: In a straight configuration, the arm cannot move any further outward radially (the velocity component in that direction is always 0). Algebraically, the Jacobian loses rank, and a zero appears in its singular value spectrum.

6. [Application] Briefly state the role of "Pseudoinverse Control" in redundant robots.

Solution

If a robot has >6 DOF (redundant), the equation $\mathbf{v} = J\dot{\mathbf{q}}$ has infinitely many solutions. Using the Moore-Penrose Pseudoinverse $J^+$, we get $\dot{\mathbf{q}} = J^+ \mathbf{v}$. This guarantees the end-effector task is met while minimizing joint energy consumption (the norm of $\dot{\mathbf{q}}$).

7. [Quaternions] Why is Spherical Linear Interpolation (SLERP) preferred over linear interpolation of rotation matrices?

Solution

Reasoning: Linear combinations of rotation matrices are generally not orthogonal (causing distortion). SLERP moves along a geodesic on the unit 4-sphere, ensuring constant angular velocity and maintaining valid rotation properties throughout the transition (see Ch51).

8. [Calculation] A point $P^B$ is observed in frame $B$. If frame $B$ relative to $A$ is $T_B^A$, find $P^A$.

Solution

Formula: $P^A = T_B^A P^B$ (using homogeneous coordinates). This demonstrates how linear algebra unifies observations across different reference frames.

9. [Duality] What role does the transpose $J^T$ play in robotics?

Solution

Conclusion: Statics Mapping. The formula $\boldsymbol{\mu} = J^T \mathbf{f}$ maps the external force $\mathbf{f}$ at the end-effector to the required joint torques $\boldsymbol{\mu}$. This embodies the duality between velocity space and force space (Principle of Virtual Work).

10. [Application] Briefly state the meaning of the Image Jacobian in Visual Servoing.

Solution

The Image Jacobian relates camera motion to the movement of feature points in the image plane. By inverting this matrix, a robot can automatically calculate motor corrections based on image errors to achieve closed-loop target tracking.

Chapter Summary¶

Robot kinematics is the "embodiment" of linear algebra in physical space:

Chained Logic: Through matrix cascading, complex mechanical topologies are simplified into continuous operator compositions, establishing a universal algebraic paradigm for multi-link systems.
Differential View of Velocity: The Jacobian matrix proves that local linearization is the key to understanding global motion, providing a unified framework for velocity, torque, and singularity analysis.
Inverse Challenges: The multi-valuedness and non-linearity of inverse mappings reveal the inherent difficulty of moving from "Task Space" back to "Joint Space," driving continuous breakthroughs in geometric algebra and numerical optimization.