5.2. Robot Jacobians

Robot Jacobians

The Jacobian of a serial manipulator is the matrix that maps joint velocities $\dot{\Theta }\in {\mathbb{R}}^n$ to the end-effector’s spatial or body velocity. It is the differential (linearization) of the forward kinematic map $g_{st}(\Theta )$ and plays a central role in velocity control, force analysis, and inverse kinematics.

For a serial chain with twists ${\xi }_1,\dots ,{\xi }_n$ and product-of-exponentials kinematics $g_{st}(\Theta )=e^{{\hat{\xi }}_1{\theta }_1}\cdots e^{{\hat{\xi }}_n{\theta }_n}{g}_{st}(0)$, the spatial Jacobian $J^s(\Theta )\in {\mathbb{R}}^{6\times n}$ satisfies
$V_{st}^s=J^s(\Theta )\dot{\Theta },$
where $V_{st}^s$ is the spatial velocity of the tool frame. Its columns are
$J^s(\Theta )=\left[\begin{array}{cccc}{\xi }_1& {\xi }_2^{\prime }& \cdots & {\xi }_n^{\prime }\end{array}\right],$
where the $i$-th column is the $i$-th joint twist transformed by all preceding joints:
${\xi }_i^{\prime }={\mathrm{Ad}}_{e^{{\hat{\xi }}_1{\theta }_1}\cdots e^{{\hat{\xi }}_{i-1}{\theta }_{i-1}}}{\xi }_i.$
In particular, ${\xi }_1^{\prime }={\xi }_1$ always (the first joint twist is already in the spatial frame), and subsequent columns depend on the current joint angles.

The body Jacobian $J^b(\Theta )\in {\mathbb{R}}^{6\times n}$ satisfies
$V_{st}^b=J^b(\Theta )\dot{\Theta },$
where $V_{st}^b$ is the body velocity of the tool frame. Its last column is always ${\xi }_n^b$ (the $n$-th joint twist in body-frame coordinates at the home configuration), and earlier columns are transformed by joints closer to the tool:
$J_i^b={\mathrm{Ad}}_{(e^{{\hat{\xi }}_i^b{\theta }_i}\cdots e^{{\hat{\xi }}_n^b{\theta }_n}{g}_{st}(0){)}^{-1}}{\xi }_i.$

The two Jacobians are related by the adjoint of the forward kinematics:
$J^s(\Theta )={\mathrm{Ad}}_{g_{st}(\Theta )}{J}^b(\Theta ).$

Configuration Dependence $\Theta$. A configuration where $\mathrm{rank}(J)<\min (6,n)$ is a singularity—the manipulator loses the ability to move in certain task-space directions. Singularity analysis is essential for trajectory planning and control.

Both Jacobians depend on the current joint configuration

Intuition

The Jacobian answers the question: “if I wiggle joint $i$ by $\delta {\theta }_i$, how does the end-effector move?” Each column of the Jacobian is the instantaneous twist that joint $i$ would produce if it alone were moving at unit speed. For the spatial Jacobian, this twist is expressed in the fixed spatial frame; for the body Jacobian, in the tool frame.

The key subtlety is that joint $i$‘s axis in the spatial frame is not its home-configuration axis ${\xi }_i$—it has been moved by joints $1$ through $i-1$. The adjoint transformation accounts for this: ${\xi }_i^{\prime }={\mathrm{Ad}}_{g_{1\cdots (i-1)}}{\xi }_i$ is the twist of joint $i$ after the preceding joints have repositioned it.

For the body Jacobian, the reasoning is reversed: joint $n$ (closest to the tool) always contributes its home-configuration body-frame twist, while joints further from the tool must be “un-transformed” by the joints between them and the tool.

The relationship $J^s={\mathrm{Ad}}_{g_{st}}{J}^b$ is exactly the change-of-frame formula for velocities: the same physical end-effector motion, reported in spatial vs. body coordinates.

Properties

Spatial Jacobian

(Column construction) Each spatial column is the joint’s home twist transformed by all preceding joints.

$J_i^s(\Theta )={\mathrm{Ad}}_{e^{{\hat{\xi }}_1{\theta }_1}\cdots e^{{\hat{\xi }}_{i-1}{\theta }_{i-1}}}{\xi }_i.$
Column 1 reduces to ${\xi }_1$ (no upstream joints); column 2 is ${\mathrm{Ad}}_{e^{{\hat{\xi }}_1{\theta }_1}}{\xi }_2$; the last column does not depend on ${\theta }_n$ because no joint is upstream of itself. Consequently, $J^s$ is a function of ${\theta }_1,\dots ,{\theta }_{n-1}$ only.

(Derivation) The column structure follows from differentiating the PoE formula and right-multiplying by $g_{st}^{-1}$.

Starting from $g_{st}=e^{{\hat{\xi }}_1{\theta }_1}\cdots e^{{\hat{\xi }}_n{\theta }_n}{g}_{st}(0)$:
${\dot{g}}_{st}={\sum }_{i=1}^n{e}^{{\hat{\xi }}_1{\theta }_1}\cdots e^{{\hat{\xi }}_{i-1}{\theta }_{i-1}}{\hat{\xi }}_i{e}^{{\hat{\xi }}_i{\theta }_i}\cdots e^{{\hat{\xi }}_n{\theta }_n}{g}_{st}(0){\dot{\theta }}_i.$
Right-multiplying by $g_{st}^{-1}$ and using $e^{\hat{\xi }\theta }\hat{\xi }{e}^{-\hat{\xi }\theta }=\hat{{\mathrm{Ad}}_{e^{\hat{\xi }\theta }}\xi }$, each term collapses to ${\hat{\xi }}_i^{\prime }{\dot{\theta }}_i$, yielding ${\hat{V}}^s={\sum }_i{\hat{\xi }}_i^{\prime }{\dot{\theta }}_i$.

(Revolute column) A revolute joint’s spatial column has the same twist structure after transformation, with axis and pivot carried along by the upstream chain.

At home, ${\xi }_i=\left[\begin{array}{c}-{\omega }_i\times q_i\\ {\omega }_i\end{array}\right]$. After the preceding joints rotate and translate the chain, the transformed column ${\xi }_i^{\prime }$ has the axis rotated to $R{\omega }_i$ and the effective pivot point shifted to $R{q}_i+p$, where $(R,p)$ is the upstream PoE product.

Body Jacobian

(Column construction) Each body column is the joint’s home twist transformed by the rigid motion from joint $i$ to the tool.

$J_i^b(\Theta )={\mathrm{Ad}}_{(e^{{\hat{\xi }}_i{\theta }_i}\cdots e^{{\hat{\xi }}_n{\theta }_n}{g}_{st}(0){)}^{-1}}{\xi }_i.$
The last column is $J_n^b={\mathrm{Ad}}_{g_{st}(0{)}^{-1}}{\xi }_n={\xi }_n^b$, which is configuration-independent. Symmetrically to the spatial case, $J^b$ depends on ${\theta }_2,\dots ,{\theta }_n$ but not ${\theta }_1$.

(Relation to spatial Jacobian) $J^s$ and $J^b$ are equivalent under the adjoint of the forward kinematics.

$J^s={\mathrm{Ad}}_{g_{st}}{J}^b$ and $J^b={\mathrm{Ad}}_{g_{st}^{-1}}{J}^s$. Because ${\mathrm{Ad}}_{g_{st}}$ is always invertible, $\mathrm{rank}(J^s)=\mathrm{rank}(J^b)$ — singularity is an intrinsic geometric property, independent of the frame in which the Jacobian is expressed.

Dimensions and Rank

(Dimensions) For $n$ joints, $J\in {\mathbb{R}}^{6\times n}$ with maximum possible rank $\min (6,n)$.

The $6$ comes from the dimension of $\mathfrak{se}(3)$; the $n$ from the number of joint-velocity inputs.

(Rank and mobility) Jacobian rank determines how much of the 6D task space the manipulator can access at a given configuration.

At a regular configuration:

• $n<6$: the image of $J$ is an $n$-dimensional subspace of task-space twists — the manipulator is kinematically incapable of moving in certain directions.
• $n=6$: $J$ is square and generically invertible, so arbitrary end-effector twists are reachable.
• $n>6$: $J$ has a non-trivial null space of dimension $n-\mathrm{rank}(J)$. Joint velocities in this null space produce zero end-effector motion — these are self-motions of a redundant manipulator, which null-space IK exploits.

Force Duality

(Transpose maps wrenches to torques) The dual of the velocity relation $V=J\dot{\Theta }$ is the static relation $\tau =J^{T}F$.

The principle of virtual work requires ${\tau }^T\dot{\Theta }=F^{T}V$ for all $\dot{\Theta }$; substituting $V=J\dot{\Theta }$ gives ${\tau }^T=F^{T}J$, i.e., $\tau =J^{T}F$. This duality lets the same matrix used for velocity control be reused for force control and gravity compensation, with no extra kinematic modeling.

Examples

Spatial Jacobian of a 2R Planar Manipulator

Consider a planar 2-revolute arm with link lengths ${\ell }_1,{\ell }_2$, both joints rotating about $z$. At home configuration, the joints are at the origin and at $({\ell }_1,0,0)$.

Column-by-column construction Joint twists at home: ${\xi }_1=\left[\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 1\end{array}\right]$ (rotation about $z$ through origin), ${\xi }_2=\left[\begin{array}{c}0\\ {\ell }_1\\ 0\\ 0\\ 0\\ 1\end{array}\right]$ (rotation about $z$ through $({\ell }_1,0,0)$, since $-\omega \times q=-(0,0,1)\times ({\ell }_1,0,0)=(0,{\ell }_1,0)$).

Column 1: $J_1^s={\xi }_1=\left[\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 1\end{array}\right]$.

Column 2: $J_2^s={\mathrm{Ad}}_{e^{{\hat{\xi }}_1{\theta }_1}}{\xi }_2$. Since joint 1 is a pure rotation about $z$ through the origin, $e^{{\hat{\xi }}_1{\theta }_1}=\left[\begin{array}{cc}{R}_z({\theta }_1)& 0\\ 0& 1\end{array}\right]$ and $\mathrm{Ad}=\left[\begin{array}{cc}{R}_z& 0\\ 0& R_z\end{array}\right]$. The angular part is unchanged ($\omega =(0,0,1)$ is invariant under $R_z$). The linear part becomes $R_z({\theta }_1)\left[\begin{array}{c}0\\ {\ell }_1\\ 0\end{array}\right]=\left[\begin{array}{c}-{\ell }_1{s}_1\\ {\ell }_1{c}_1\\ 0\end{array}\right]$.

$J^s({\theta }_1)=\left[\begin{array}{cc}0& -{\ell }_1{s}_1\\ 0& {\ell }_1{c}_1\\ 0& 0\\ 0& 0\\ 0& 0\\ 1& 1\end{array}\right]$

Singularity: $\det (J_{v_{xy}}^s)=0$ when ${\ell }_1{c}_1\cdot 0-(-{\ell }_1{s}_1)\cdot 0=0$—but more usefully, the planar $3\times 2$ sub-Jacobian (taking $v_x,v_y,{\omega }_z$ rows) loses rank when the two joint axes produce linearly dependent twists. For this 2R arm, this happens when ${\theta }_2=0$ or ${\theta }_2=\pi$ (arm fully extended or folded).

Spatial vs. Body Jacobian Relationship

For any configuration $\Theta$, verify that $J^s={\mathrm{Ad}}_{g_{st}}{J}^b$ by checking that both give the same end-effector velocity:

Verification $V^s=J^s\dot{\Theta }$ and $V^b=J^b\dot{\Theta }$, so $V^s={\mathrm{Ad}}_{g_{st}}{V}^b={\mathrm{Ad}}_{g_{st}}{J}^b\dot{\Theta }$. Since this holds for all $\dot{\Theta }$, we get $J^s={\mathrm{Ad}}_{g_{st}}{J}^b$. This confirms that singularities are frame-independent: $\mathrm{rank}(J^s)=\mathrm{rank}({\mathrm{Ad}}_{g_{st}}{J}^b)=\mathrm{rank}(J^b)$, since ${\mathrm{Ad}}_{g_{st}}$ is always invertible.