Part 2: The Skeleton of Robotics (Kinematics)

Welcome to Part 2 of "Robotics Zero to Hero." In Part 1 we learned to describe a single rigid body with $SE(3)$ . Now we chain those transforms together to form a robot's "skeleton." This is Kinematics — the geometry of motion without worrying about the forces that cause it.

The Math: Forward Kinematics

Forward Kinematics (FK) answers: given the joint values, where is the end-effector (the "hand" or "tip")? For a serial robot with $n$ joints, FK is a product of transforms:

T^{tip}_{base}(\mathbf q) = T_0^1(q_1)\, T_1^2(q_2)\cdots T_{n-1}^n(q_n)

where $q_i$ is an angle for a revolute joint or a displacement for a prismatic one. FK is always well-defined and unique — it is the easy direction. (Inverse kinematics, Part 3, is where the pain lives.)

There are two dominant ways to fill in those $T_{i-1}^i$ matrices, and they have genuinely different trade-offs.

Approach 1 — Denavit–Hartenberg (D-H) parameters

D-H describes each link with exactly four numbers, exploiting a clever convention that aligns successive joint axes:

$\theta_i$ — joint angle (rotation about $z_{i-1}$ )
$d_i$ — link offset (translation along $z_{i-1}$ )
$a_i$ — link length (translation along $x_i$ )
$\alpha_i$ — link twist (rotation about $x_i$ )

giving

T_{i-1}^i = \begin{bmatrix} \cos\theta_i & -\sin\theta_i \cos\alpha_i & \sin\theta_i \sin\alpha_i & a_i \cos\theta_i \\ \sin\theta_i & \cos\theta_i \cos\alpha_i & -\cos\theta_i \sin\alpha_i & a_i \sin\theta_i \\ 0 & \sin\alpha_i & \cos\alpha_i & d_i \\ 0 & 0 & 0 & 1 \end{bmatrix}

Approach 2 — Product of Exponentials (PoE)

Building on the Lie-group view from Part 1, screw theory writes each joint as a twist $\mathcal{S}_i \in \mathbb{R}^6$ (an axis of rotation/translation in space) and FK becomes a clean product of matrix exponentials (Brockett's formula):

T^{tip}_{base}(\mathbf q) = e^{[\mathcal{S}_1]q_1}\, e^{[\mathcal{S}_2]q_2}\cdots e^{[\mathcal{S}_n]q_n}\; M

where $M$ is the tip pose at the home configuration. No per-link frame assignment is required — every twist is expressed in a single base frame — which is why modern libraries lean toward PoE.

Which parameterization? Competing conventions and their costs

Method	Params/joint	Strengths	Shortcomings
Denavit–Hartenberg	4	Compact; ubiquitous in textbooks/industry	Ambiguous frame assignment; ill-defined for parallel/intersecting axes; two incompatible variants (standard vs. modified)
Product of Exponentials	6 (twist)	Single base frame; geometrically transparent; no singular conventions	More numbers per joint; requires Lie-group machinery
URDF/kinematic tree	6 (pose) per link	Direct, tool-friendly (ROS, simulators)	Verbose; no analytical insight; easy to mis-specify inertials

The takeaway: D-H is the lingua franca, but its compactness hides edge cases (it literally cannot represent two parallel adjacent joint axes without an arbitrary choice). PoE pays a small storage cost to make those edge cases disappear.

Python Implementation: Building a D-H Chain

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Continuum Kinematics: When the Chain Never Ends

For a rigid robot the product above has $n$ terms and we are done. A continuum arm has no discrete joints — it bends continuously. Its shape is a space curve $\mathbf r(s)$ parameterized by arc length $s \in [0, L]$ , and FK becomes an integral rather than a product:

T(s) = T(0)\,\exp\!\left(\int_0^{s} [\boldsymbol{\xi}(\sigma)]\, d\sigma\right)

where $\boldsymbol\xi(s)$ is the local curvature/twist along the backbone. In practice we adopt one of two modeling regimes:

Piecewise Constant Curvature (PCC). Approximate the arm as a small number of constant-curvature arcs, each described by curvature $\kappa$ , bending plane angle $\phi$ , and length $\ell$ . This is the workhorse for cable-driven arms and reduces an infinite-dimensional problem to a handful of parameters — at the cost of accuracy when external loads break the constant-curvature assumption.
Cosserat rod theory. Model the backbone as a continuum that can bend, twist, shear, and stretch, governed by partial differential equations. This is far more accurate (and is the basis of modern soft-robot dynamics) but requires numerically integrating a boundary-value problem.

Continuum model	DOF captured	Accuracy	Cost
Constant curvature (PCC)	bending only, per segment	Good for light, cable-driven arms	Cheap, closed-form per arc
Pseudo-rigid-body (many small links)	bending, discretized	Tunable via segment count	Grows with #segments
Cosserat rod (PDE)	bend, twist, shear, extend	Highest; handles loads/gravity	Solve a BVP/PDE per step

This is the high-dim vs. low-dim tension made concrete: to model a tentacle as a serial chain you must pick a number of segments $N$ . Small $N$ is fast but inaccurate; large $N$ approaches the true continuum but inflates the configuration dimension toward infinity. There is no free lunch — only a dial between speed and fidelity.

Focus on the Octopus: Finding the Tip

Traditional industrial arms have rigid links and discrete joints, so D-H parameters are highly effective. Our metallic continuum octopus tentacle instead bends along its entire length.

To approximate its forward kinematics we treat the tentacle as a series of many short constant-curvature segments, each bending slightly. The "joint variables" $\mathbf q$ become the curvature parameters of each segment, which we map from the underlying actuation (cable pull lengths or pneumatic pressures) through a constant-curvature kinematic map. Chaining dozens of these small transforms — exactly like the D-H loop above, but with curvature-driven blocks — gives the 3D coordinate of the tip.

The subtlety unique to our hardware: the map from cable lengths to curvature is itself nonlinear and shifts as cables stretch and rub against the segment walls (we return to this "actuation-space vs. configuration-space" gap in Part 5). For now, the key idea is that FK for a continuum robot is the limit of FK for a rigid one — and that limit is where the degrees of freedom explode.

In Part 3 we flip the problem: given a desired tip position, what configuration achieves it? Welcome to Inverse Kinematics — and the curse of redundancy.

Further reading: Brockett (1984) on the product-of-exponentials formula; Webster & Jones (2010), "Design and Kinematic Modeling of Constant Curvature Continuum Robots: A Review"; "Rod models in continuum and soft robot control: a review" (arXiv:2407.05886).