2.1. Forward Kinematics

Forward Kinematics

Forward kinematics is the map from joint space to task space: given a vector of joint variables
$\Theta =({\theta }_1,\dots ,{\theta }_n),$
it computes the rigid body transformation of the end-effector relative to a fixed base frame,
$g_{ST}(\Theta )\in SE(3).$

For a serial manipulator, each joint defines a twist ${\xi }_i\in \mathfrak{se}(3)$. If $g_{ST}(0)$ denotes the home configuration (all joint variables equal to zero), then the configuration for arbitrary $\Theta$ is given by the Product of Exponentials (PoE) formula:
$g_{ST}(\Theta )=e^{{\hat{\xi }}_1{\theta }_1}{e}^{{\hat{\xi }}_2{\theta }_2}\dots e^{{\hat{\xi }}_n{\theta }_n}{g}_{ST}(0).$

Each exponential represents the rigid motion generated by one joint’s screw axis, expressed in the spatial frame. The order of multiplication matters: the rightmost exponential corresponds to the joint closest to the end-effector, while exponentials further left correspond to joints progressively closer to the base.

The rotational component of $g_{ST}(\Theta )$ is obtained by composing the individual joint rotations. The translational component is more subtle because upstream joints alter the coordinate frames in which downstream motions occur. This interaction is captured algebraically by the adjoint transformation:
${\xi }^{\prime }={\mathrm{Ad}}_g\xi ,$
which describes how a rigid transformation changes the coordinates of a twist.

The forward kinematic map $g:\mathcal{Q}\to SE(3)$ maps the configuration space $\mathcal{Q}\subset {\mathbb{R}}^n$ into the space of rigid body poses. It is deterministic and typically smooth, but not generally injective: multiple joint vectors may produce the same end-effector pose.

Intuition

The product of exponentials is a coordinate-free formulation of forward kinematics: you specify each joint’s screw axis at the home configuration and the home pose $g_{st}(0)$, and the formula handles everything else. This avoids the tedious, error-prone frame assignment of the Denavit-Hartenberg convention.

Reading the PoE formula right-to-left: first, the tool is at $g_{st}(0)$. Then joint $n$ moves, applying $e^{{\hat{\xi }}_n{\theta }_n}$. Then joint $n-1$ moves, applying $e^{{\hat{\xi }}_{n-1}{\theta }_{n-1}}$—but in the spatial frame, so it moves everything downstream. The leftmost exponential (joint 1) affects all other joints. This “upstream joints move downstream joints” structure is what makes the spatial Jacobian columns involve adjoint transformations by preceding joints.

Properties

Product of Exponentials

(Spatial PoE) All twists ${\xi }_i$ are expressed in the spatial (fixed) frame at the home configuration. The formula $g_{st}=e^{{\hat{\xi }}_1{\theta }_1}\cdots e^{{\hat{\xi }}_n{\theta }_n}{g}_{st}(0)$ computes the end-effector pose by applying each joint’s screw motion in the spatial frame.

(Body PoE) Alternatively, using twists ${\xi }_i^b$ expressed in the tool (body) frame at home configuration: $g_{st}=g_{st}(0)e^{{\hat{\xi }}_1^b{\theta }_1}\cdots e^{{\hat{\xi }}_n^b{\theta }_n}$. The body-frame twist of joint $i$ is ${\xi }_i^b={\mathrm{Ad}}_{g_{st}(0{)}^{-1}}{\xi }_i$.

(Number of Solutions) The PoE map is generally not injective: the inverse kinematics problem may have multiple solutions. For a 6-DoF manipulator, a generic reachable pose has finitely many solutions (up to 16 for certain geometries).

Home Configuration

(Choice of Home) The home configuration $g_{st}(0)$ is a design choice (often “arm stretched out”). Changing it rescales the twists accordingly. A common convention is to place the base at the spatial origin and extend all revolute joints to $\theta =0$.

(End-Effector Frame) The tool frame $\{T\}$ is rigidly attached to the last link. Its placement affects $g_{st}(0)$ but not the joint twists.

Examples

SCARA Forward Kinematics