[Robotics] Velocity Kinematics and the Jacobian
In the previous posts we examined the forward kinematics of an open chain: given a joint configuration $\boldsymbol{\theta} \in \mathbb{R}^n$, we computed the end-effector frame $T(\boldsymbol{\theta}) \in SE(3)$. Velocity kinematics addresses the natural follow-up question. Given a joint-rate vector $\dot{\boldsymbol{\theta}} \in \mathbb{R}^n$, what is the instantaneous twist of the end-effector? Conversely, given a desired end-effector velocity, what joint rates produce it?
The map that answers the first question is linear at every configuration $\boldsymbol{\theta}$, and the matrix that realizes it is the manipulator Jacobian $J(\boldsymbol{\theta})$. The Jacobian is the central object of differential kinematics; it determines singular configurations, governs the duality between motion and force, and provides the foundation for inverse kinematics, manipulability analysis, motion planning, and force control. This post develops the Jacobian from first principles, builds both its geometric (space and body) and analytical forms, examines singularities and the manipulability ellipsoid, and derives the static-force relationship $\boldsymbol{\tau} = J^\top \mathcal{F}$.
From Joint Velocities to End-Effector Velocity
Suppose the end-effector pose is described by a minimal set of task-space coordinates $\mathbf{x} \in \mathbb{R}^m$ (for example, the $(x_1, x_2)$ tip position of a planar arm), and that the forward kinematics is expressed as
\[\mathbf{x}(t) = f(\boldsymbol{\theta}(t)).\]Differentiating with respect to time and applying the chain rule gives
\[\dot{\mathbf{x}} = \frac{\partial f(\boldsymbol{\theta})}{\partial \boldsymbol{\theta}} \, \dot{\boldsymbol{\theta}} = J(\boldsymbol{\theta}) \, \dot{\boldsymbol{\theta}},\]where $J(\boldsymbol{\theta}) \in \mathbb{R}^{m \times n}$ is, by definition, the Jacobian of $f$ at $\boldsymbol{\theta}$. This matrix is the linear sensitivity of the end-effector velocity to the joint-rate vector, and it depends on the current joint configuration.
As a concrete illustration, consider a planar 2R arm with link lengths $L_1$ and $L_2$ and joint angles $(\theta_1, \theta_2)$. Its forward kinematics is
\[\begin{aligned} x_1 &= L_1 \cos\theta_1 + L_2 \cos(\theta_1 + \theta_2), \\ x_2 &= L_1 \sin\theta_1 + L_2 \sin(\theta_1 + \theta_2). \end{aligned}\]Differentiating both equations with respect to time yields
\[\begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \end{bmatrix} = \underbrace{\begin{bmatrix} -L_1 \sin\theta_1 - L_2 \sin(\theta_1 + \theta_2) & -L_2 \sin(\theta_1 + \theta_2) \\ L_1 \cos\theta_1 + L_2 \cos(\theta_1 + \theta_2) & \;\;\;L_2 \cos(\theta_1 + \theta_2) \end{bmatrix}}_{J(\boldsymbol{\theta})} \begin{bmatrix} \dot{\theta}_1 \\ \dot{\theta}_2 \end{bmatrix}.\]Denoting the columns of $J(\boldsymbol{\theta})$ by $J_1(\boldsymbol{\theta})$ and $J_2(\boldsymbol{\theta})$, the tip velocity decomposes as
\[\mathbf{v}_{\text{tip}} \;=\; J_1(\boldsymbol{\theta}) \, \dot{\theta}_1 \;+\; J_2(\boldsymbol{\theta}) \, \dot{\theta}_2.\]Each column is the tip velocity produced by unit speed of the corresponding joint while every other joint is frozen. As long as $J_1$ and $J_2$ are linearly independent the tip can be driven in any direction in the plane by an appropriate choice of joint rates; whenever they become collinear, an entire direction of motion is lost. This basic algebraic-geometric picture generalizes to spatial chains, with the qualification that the end-effector “velocity” is then a six-dimensional twist rather than a planar vector.
The Geometric Jacobian
For a spatial open chain the natural notion of end-effector velocity is a twist $\mathcal{V} \in \mathbb{R}^6$, combining a linear and an angular velocity. The Jacobian that maps joint rates to a twist (with no reliance on a particular orientation parameterization) is called the geometric Jacobian. Throughout, let the chain have $n$ one-degree-of-freedom joints with joint variables $\boldsymbol{\theta} = (\theta_1, \ldots, \theta_n)$, fixed (space) frame $\{ s \}$ and end-effector (body) frame $\{ b \}$, and forward kinematics
\[T_{sb}(\boldsymbol{\theta}) \in SE(3).\]Two natural representations of the end-effector twist exist:
- the spatial twist $\mathcal{V}s = (\boldsymbol{\omega}_s, \mathbf{v}_s) \in \mathbb{R}^6$ defined by $[\mathcal{V}_s] = \dot{T}{sb} \, T_{sb}^{-1}$, expressed in $\{ s \}$;
- the body twist $\mathcal{V}b = (\boldsymbol{\omega}_b, \mathbf{v}_b) \in \mathbb{R}^6$ defined by $[\mathcal{V}_b] = T{sb}^{-1} \, \dot{T}_{sb}$, expressed in $\{ b \}$.
These two twists are related by the adjoint map $\mathcal{V}s = [\mathrm{Ad}{T_{sb}}] \, \mathcal{V}_b$. Correspondingly the chain admits two natural Jacobian matrices.
For an $n$-joint open chain with end-effector forward kinematics $T_{sb}(\boldsymbol{\theta}) \in SE(3)$, the **space Jacobian** $J_s(\boldsymbol{\theta}) \in \mathbb{R}^{6 \times n}$ satisfies
$$ \mathcal{V}_s \;=\; J_s(\boldsymbol{\theta}) \, \dot{\boldsymbol{\theta}}, $$ and the **body Jacobian** $J_b(\boldsymbol{\theta}) \in \mathbb{R}^{6 \times n}$ satisfies
$$ \mathcal{V}_b \;=\; J_b(\boldsymbol{\theta}) \, \dot{\boldsymbol{\theta}}. $$ The $i$-th column of $J_s$ (resp. $J_b$) is the spatial (resp. body) twist that results when $\dot{\theta}_i = 1$ and every other joint is held at rest, at the current configuration $\boldsymbol{\theta}$.
This columnwise interpretation is more than a mnemonic: it tells us exactly how to assemble each column from the geometry of the chain.
Space Jacobian: Column-by-Column Construction
Using the product of exponentials (PoE) formulation,
\[T(\boldsymbol{\theta}) \;=\; e^{[\mathcal{S}_1]\theta_1} \, e^{[\mathcal{S}_2]\theta_2} \cdots e^{[\mathcal{S}_n]\theta_n} \, M,\]where $\mathcal{S}i = (\boldsymbol{\omega}{si}, \mathbf{v}_{si}) \in \mathbb{R}^6$ is the screw axis of joint $i$ expressed in $\{ s \}$ at the zero configuration and $M \in SE(3)$ is the home configuration of $\{ b \}$, a direct calculation of $\dot{T} T^{-1}$ yields
\[\mathcal{V}_s \;=\; \mathcal{S}_1 \, \dot{\theta}_1 \;+\; \mathrm{Ad}_{e^{[\mathcal{S}_1]\theta_1}} (\mathcal{S}_2) \, \dot{\theta}_2 \;+\; \mathrm{Ad}_{e^{[\mathcal{S}_1]\theta_1} e^{[\mathcal{S}_2]\theta_2}} (\mathcal{S}_3) \, \dot{\theta}_3 \;+\; \cdots\]Reading off the coefficient of each $\dot{\theta}_i$ gives the columns of $J_s(\boldsymbol{\theta})$:
The $i$-th column of $J_s(\boldsymbol{\theta})$ is the screw axis of joint $i$ expressed in $\\{ s \\}$, after the first $i-1$ joints have been displaced from zero to their current values. Concretely,
$$ J_{s1}(\boldsymbol{\theta}) \;=\; \mathcal{S}_1, \qquad J_{si}(\boldsymbol{\theta}) \;=\; \mathrm{Ad}_{e^{[\mathcal{S}_1]\theta_1} \cdots e^{[\mathcal{S}_{i-1}]\theta_{i-1}}} (\mathcal{S}_i), \quad i = 2, \ldots, n. $$
This geometric reading is decisive: although the analytic derivation requires differentiating a matrix exponential, the columns of the Jacobian are simply the (configuration-dependent) screw axes themselves, viewed from $\{ s \}$.
Writing the screw of joint $i$ at the current configuration as $\mathcal{S}i^{\,\prime} = (\boldsymbol{\omega}_i, \mathbf{v}_i)$ in $\{ s \}$, the column $J{si} = (\boldsymbol{\omega}_i, \mathbf{v}_i)$ admits the following physical interpretation for the two standard joint types:
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Revolute joint. Let $\hat{\boldsymbol{\omega}}_i \in \mathbb{R}^3$ be the unit vector along the joint axis (in $\{ s \}$) and let $\mathbf{q}_i \in \mathbb{R}^3$ be any point lying on the joint axis. Then
\[J_{si} \;=\; \begin{bmatrix} \hat{\boldsymbol{\omega}}_i \\ -\hat{\boldsymbol{\omega}}_i \times \mathbf{q}_i \end{bmatrix} \;=\; \begin{bmatrix} \hat{\boldsymbol{\omega}}_i \\ \mathbf{q}_i \times \hat{\boldsymbol{\omega}}_i \end{bmatrix}.\]The lower (linear) part $-\hat{\boldsymbol{\omega}}_i \times \mathbf{q}_i$ is the linear velocity at the space-frame origin generated when the joint rotates at unit speed and the rest of the chain is rigid; this is the familiar relation $\mathbf{v} = \boldsymbol{\omega} \times \mathbf{r}$ from rigid-body kinematics, applied to the displacement from joint $i$ to the origin of $\{ s \}$.
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Prismatic joint. Let $\hat{\mathbf{v}}_i$ be the unit vector along the sliding direction in $\{ s \}$. Then
\[J_{si} \;=\; \begin{bmatrix} \mathbf{0} \\ \hat{\mathbf{v}}_i \end{bmatrix},\]reflecting the fact that prismatic motion contributes only linear, not angular, velocity.
This is the same construction that appears in the classical Denavit-Hartenberg-based derivation (Spong, Hutchinson, and Vidyasagar), in which one writes the upper three rows of the geometric Jacobian as
\[J_{\boldsymbol{\omega} i} = \begin{cases} \mathbf{z}_{i-1}, & \text{revolute joint } i, \\ \mathbf{0}, & \text{prismatic joint } i, \end{cases}\]and the lower three rows as
\[J_{\mathbf{v} i} = \begin{cases} \mathbf{z}_{i-1} \times (\mathbf{o}_n - \mathbf{o}_{i-1}), & \text{revolute joint } i, \\ \mathbf{z}_{i-1}, & \text{prismatic joint } i, \end{cases}\]where $\mathbf{z}{i-1}$ is the $z$-axis of frame $i-1$ in the base frame and $\mathbf{o}_n - \mathbf{o}{i-1}$ is the displacement from joint $i$ to the end-effector. The two formulations agree, but the PoE/screw-theoretic version is more compact, more symmetric, and avoids the need to introduce intermediate link frames.
The "stacking convention" $J_{si} = (\boldsymbol{\omega}_i, \mathbf{v}_i)$ used in *Modern Robotics* (Lynch and Park) places the angular part on top and the linear part on the bottom, and the corresponding twist is $\mathcal{V} = (\boldsymbol{\omega}, \mathbf{v})$. Spong's textbook uses the opposite convention $\mathcal{V} = (\mathbf{v}, \boldsymbol{\omega})$. The two formulations carry the same information but every formula carries an implicit row-order convention; one must be careful when transcribing equations between sources.
Body Jacobian and Its Relation to the Space Jacobian
The body Jacobian is derived analogously, using the alternative PoE form
\[T(\boldsymbol{\theta}) \;=\; M \, e^{[\mathcal{B}_1]\theta_1} \cdots e^{[\mathcal{B}_n]\theta_n},\]where $\mathcal{B}_i$ is the screw axis of joint $i$ expressed in the end-effector frame $\{ b \}$ at the zero configuration. Computing $[\mathcal{V}_b] = T^{-1} \dot{T}$ and reading off coefficients gives
\[J_{bn} \;=\; \mathcal{B}_n, \qquad J_{bi}(\boldsymbol{\theta}) \;=\; \mathrm{Ad}_{e^{-[\mathcal{B}_n]\theta_n} \cdots e^{-[\mathcal{B}_{i+1}]\theta_{i+1}}} (\mathcal{B}_i), \quad i = n-1, \ldots, 1.\]Each column of $J_b(\boldsymbol{\theta})$ is the screw axis of joint $i$ expressed in $\{ b \}$ at the current configuration. Note that the recursion runs in the opposite direction from the space case: in the space Jacobian one accumulates the displacements of joints before joint $i$, while in the body Jacobian one accumulates the displacements of joints after joint $i$ (because those joints displace $\{ b \}$ relative to the joint).
The space and body Jacobians are not independent; they describe the same physical map written in different frames. Starting from $\mathcal{V}s = [\mathrm{Ad}{T_{sb}}] \mathcal{V}_b$ and substituting $\mathcal{V}_s = J_s \dot{\boldsymbol{\theta}}$ and $\mathcal{V}_b = J_b \dot{\boldsymbol{\theta}}$, one obtains the change-of-frame law
$$ J_s(\boldsymbol{\theta}) \;=\; [\mathrm{Ad}_{T_{sb}}] \, J_b(\boldsymbol{\theta}), \qquad J_b(\boldsymbol{\theta}) \;=\; [\mathrm{Ad}_{T_{bs}}] \, J_s(\boldsymbol{\theta}). $$ Because $[\mathrm{Ad}_{T}]$ is always invertible, $J_s$ and $J_b$ have the same rank at every configuration. In particular they share the same singular configurations.
This adjoint relation is precisely what one would expect: the columns of $J_s$ and $J_b$ are twists (one per joint), and twists transform between frames by the adjoint map.
Worked Example: Space Jacobian of a Spatial RRRP Chain
To make the construction concrete, consider the spatial RRRP chain of Lynch and Park’s Example 5.2. The first three joints are revolute with axes parallel to $\hat{\mathbf{z}}s$ (the space-frame $z$-axis), and the fourth joint is prismatic along $\hat{\mathbf{z}}_s$. Let the link lengths be $L_1, L_2$, write $c_1 = \cos\theta_1$, $s_1 = \sin\theta_1$, $c{12} = \cos(\theta_1 + \theta_2)$, $s_{12} = \sin(\theta_1 + \theta_2)$.
Apply the recipe column by column:
- Column 1. $\boldsymbol{\omega}{s1} = (0, 0, 1)$, $\mathbf{q}_1 = (0, 0, 0)$ (the origin), so $\mathbf{v}{s1} = -\boldsymbol{\omega}_{s1} \times \mathbf{q}_1 = (0, 0, 0)$.
- Column 2. $\boldsymbol{\omega}{s2} = (0, 0, 1)$ (parallel to $\boldsymbol{\omega}{s1}$). Choose $\mathbf{q}2 = (L_1 c_1, L_1 s_1, 0)$, giving $\mathbf{v}{s2} = -\boldsymbol{\omega}_{s2} \times \mathbf{q}_2 = (L_1 s_1, -L_1 c_1, 0)$.
- Column 3. $\boldsymbol{\omega}{s3} = (0, 0, 1)$. Choose $\mathbf{q}_3 = (L_1 c_1 + L_2 c{12}, L_1 s_1 + L_2 s_{12}, 0)$, giving $\mathbf{v}{s3} = (L_1 s_1 + L_2 s{12}, -L_1 c_1 - L_2 c_{12}, 0)$.
- Column 4. Prismatic joint along $\hat{\mathbf{z}}s$: $\boldsymbol{\omega}{s4} = (0, 0, 0)$ and $\mathbf{v}_{s4} = (0, 0, 1)$.
Stacking the columns gives
\[J_s(\boldsymbol{\theta}) \;=\; \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 \\ 0 & L_1 s_1 & L_1 s_1 + L_2 s_{12} & 0 \\ 0 & -L_1 c_1 & -L_1 c_1 - L_2 c_{12} & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}.\]No explicit differentiation of the forward kinematic map was required; the entire matrix is assembled from joint axes and reference points. This is one of the key practical advantages of the PoE/screw formulation.
The Analytical Jacobian
The geometric Jacobian relates joint rates to a twist, which is intrinsically a six-dimensional object on $SE(3)$. In many tasks, however, it is more convenient to describe the end-effector pose with a minimal set of coordinates $\mathbf{q} \in \mathbb{R}^m$, for instance $(x, y, z, \phi, \theta, \psi)$ using Euler angles, $(x, y, z, r_x, r_y, r_z)$ using exponential coordinates for rotation, or a position-plus-quaternion description. The Jacobian that maps joint rates to the time derivative of these minimal coordinates is the analytical Jacobian.
Let the end-effector pose be parameterized by a minimal set of coordinates $\mathbf{q} = (\mathbf{x}, \boldsymbol{\alpha}) \in \mathbb{R}^m$, where $\mathbf{x}$ describes position and $\boldsymbol{\alpha}$ is a minimal orientation parameterization. The **analytical Jacobian** $J_a(\boldsymbol{\theta}) \in \mathbb{R}^{m \times n}$ is the matrix satisfying
$$ \dot{\mathbf{q}} \;=\; J_a(\boldsymbol{\theta}) \, \dot{\boldsymbol{\theta}}. $$
The analytical Jacobian differs from the geometric Jacobian because $\dot{\boldsymbol{\alpha}}$ is not the angular velocity $\boldsymbol{\omega}$. The two are related by a configuration-dependent matrix $T(\boldsymbol{\alpha})$ that depends on the chosen orientation parameterization.
For Euler angles $\boldsymbol{\alpha} = (\phi, \theta, \psi)$ in the ZYZ convention, one can show that
\[\boldsymbol{\omega} \;=\; T(\boldsymbol{\alpha}) \, \dot{\boldsymbol{\alpha}}, \qquad T(\boldsymbol{\alpha}) \;=\; \begin{bmatrix} c_\psi s_\theta & -s_\psi & 0 \\ s_\psi s_\theta & \;\;\; c_\psi & 0 \\ c_\theta & 0 & 1 \end{bmatrix}.\]Combining this with the body Jacobian relation $\mathcal{V}_b = (\boldsymbol{\omega}_b, \mathbf{v}_b) = J_b \dot{\boldsymbol{\theta}}$ gives
\[\begin{bmatrix} \dot{\mathbf{x}} \\ \dot{\boldsymbol{\alpha}} \end{bmatrix} \;=\; \underbrace{\begin{bmatrix} I & 0 \\ 0 & T(\boldsymbol{\alpha})^{-1} \end{bmatrix} J_b(\boldsymbol{\theta})}_{J_a(\boldsymbol{\theta})} \, \dot{\boldsymbol{\theta}},\]provided $T(\boldsymbol{\alpha})$ is invertible.
For exponential coordinates $\mathbf{r} = \hat{\boldsymbol{\omega}} \theta \in \mathbb{R}^3$ representing the rotation, Lynch and Park derive the relation $\boldsymbol{\omega}_b = A(\mathbf{r}) \dot{\mathbf{r}}$ with
\[A(\mathbf{r}) \;=\; I \;-\; \frac{1 - \cos \\|\mathbf{r}\\|}{\\|\mathbf{r}\\|^2} [\mathbf{r}] \;+\; \frac{\\|\mathbf{r}\\| - \sin \\|\mathbf{r}\\|}{\\|\mathbf{r}\\|^3} [\mathbf{r}]^2,\]so that the analytical Jacobian is related to the body Jacobian via
\[J_a(\boldsymbol{\theta}) \;=\; \begin{bmatrix} A(\mathbf{r})^{-1} & 0 \\ 0 & R_{sb}(\boldsymbol{\theta}) \end{bmatrix} J_b(\boldsymbol{\theta}).\]Whenever $T(\boldsymbol{\alpha})$ (or $A(\mathbf{r})$) is singular, the corresponding orientation parameterization fails to be a valid local chart of $SO(3)$. For ZYZ Euler angles this happens at $\theta = 0$ and $\theta = \pi$ ("gimbal lock"). Such configurations are **representational singularities** of the analytical Jacobian; they are artifacts of the chosen coordinates and not physical singularities of the manipulator. The singularities of the analytical Jacobian are therefore the union of the kinematic singularities and the representational singularities. The geometric Jacobian, working directly with twists, is free of representational singularities.
Singularities
For a six-DOF spatial manipulator, the Jacobian $J(\boldsymbol{\theta}) \in \mathbb{R}^{6 \times n}$ has maximum possible rank $\min(6, n)$. At most configurations this maximum is attained. Configurations where it is not attained are called kinematic singularities (or simply singularities), and they are the points where the end-effector loses the instantaneous ability to move in one or more directions.
A configuration $\boldsymbol{\theta} \in \mathbb{R}^n$ is a **kinematic singularity** of an open chain if
$$ \mathrm{rank} \, J(\boldsymbol{\theta}) \;<\; \min(6, n). $$ Equivalently, the columns of $J(\boldsymbol{\theta})$ fail to span the maximum-dimensional subspace of $\mathbb{R}^6$ that they could otherwise span. For a square Jacobian ($n = 6$) this is the condition $\det J(\boldsymbol{\theta}) = 0$.
Several elementary observations follow from the definition.
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Because $J_s = [\mathrm{Ad}{T{sb}}] J_b$ and $[\mathrm{Ad}_T]$ is always invertible, $J_s$ and $J_b$ have the same rank everywhere; the set of singular configurations is independent of whether one uses the space or body Jacobian.
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The set of singular configurations is also independent of the choice of fixed and end-effector frames, since relocating $\{ s \}$ amounts to left-multiplication of $T$ by a constant transformation, which does not change rank. Singularities are intrinsic to the mechanism, not to its mathematical description.
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At a singularity the loss of instantaneous mobility is mirrored by a gain of resistance: the chain can passively support arbitrary wrenches in the directions in which it has lost mobility, since no actuator can move the end-effector against them. This is the static dual of velocity singularity, made precise in the section on statics below.
Identifying singularities is important for several reasons (Spong, Sec. 5.9):
- Certain directions of end-effector motion are unattainable.
- Bounded end-effector velocities may require unbounded joint rates (the inverse mapping $\dot{\boldsymbol{\theta}} = J^{-1} \dot{\mathbf{x}}$ blows up).
- Bounded end-effector wrenches may require unbounded joint torques.
- Singularities often correspond to boundary points of the manipulator’s workspace.
- Inverse kinematics solutions may not be unique near a singularity (or fail to exist on one side).
Common Singularities of Open Chains
Several archetypal kinematic singularities occur in spatial six-DOF chains with revolute and prismatic joints. The analysis follows directly from the columnwise structure of $J_s$.
For a six-DOF open chain: 1. **Two collinear revolute joint axes.** If two revolute joints share the same axis, the corresponding two columns of $J_s$ are equal (or differ by a sign), so the rank drops. 2. **Three coplanar and parallel revolute joint axes.** With axes parallel and lying in a common plane, three columns of $J_s$ become linearly dependent. 3. **Four revolute joint axes intersecting at a common point.** Choosing the common point as the origin makes $\mathbf{v}_{si} = 0$ for those four columns, leaving only their (three-dimensional) angular parts; four columns cannot be linearly independent in $\mathbb{R}^3$. The classical "wrist singularity" of an elbow-type arm with a spherical wrist is an instance. 4. **Four coplanar revolute joints.** If four revolute axes are coplanar, both their angular parts and their linear parts lie in low-dimensional subspaces, forcing linear dependence. 5. **Six revolute joints intersecting a common line.** Equivalent to a Plücker-coordinate dependence; the Jacobian is then automatically singular.
Singularity of the Planar 2R Arm
Return to the planar 2R arm of the introduction. Its Jacobian was
\[J(\boldsymbol{\theta}) \;=\; \begin{bmatrix} -L_1 \sin\theta_1 - L_2 \sin(\theta_1 + \theta_2) & -L_2 \sin(\theta_1 + \theta_2) \\ \;\;\;L_1 \cos\theta_1 + L_2 \cos(\theta_1 + \theta_2) & \;\;\;L_2 \cos(\theta_1 + \theta_2) \end{bmatrix}.\]A direct calculation gives
\[\det J(\boldsymbol{\theta}) \;=\; L_1 L_2 \sin\theta_2.\]The arm is singular precisely when $\sin\theta_2 = 0$, i.e. $\theta_2 = 0$ (arm fully extended) or $\theta_2 = \pm \pi$ (arm fully folded back on itself). At these configurations the two Jacobian columns are collinear, and the tip cannot move radially — only tangentially with respect to the base. This is a textbook example of a singularity occurring on the boundary of the workspace: when fully extended, the tip lies on the circle of radius $L_1 + L_2$ and any radial motion would require leaving the workspace.
Decoupling of Spatial Singularities
A useful structural fact, valid for many industrial arms, is the decoupling of singularities for a six-DOF manipulator consisting of a three-DOF positioning arm followed by a three-DOF spherical wrist (three revolute axes intersecting at a common point). Partition the Jacobian as
\[J \;=\; \begin{bmatrix} J_{11} & J_{12} \\ J_{21} & J_{22} \end{bmatrix},\]where each block is $3 \times 3$. Choosing the wrist center as the reference point makes $J_{12} = 0$, so that
\[\det J \;=\; \det J_{11} \, \det J_{22}.\]Singular configurations are therefore the union of arm singularities ($\det J_{11} = 0$) and wrist singularities ($\det J_{22} = 0$). The latter occur whenever two of the three wrist axes become collinear (in particular, when the second wrist joint is at $\theta_5 = 0$); the former depend on the geometry of the arm and include configurations such as the wrist center lying on the base axis of rotation.
Manipulability
Singularity is a binary property — a configuration is either singular or it is not — but the proximity to singularity has a continuous and physically meaningful structure. The manipulability ellipsoid quantifies, at any nonsingular configuration, the directions in which the end-effector can move easily and those in which it can move only with great joint effort.
Consider an $n$-joint open chain with Jacobian $J(\boldsymbol{\theta}) \in \mathbb{R}^{m \times n}$ ($m \le n$). Let $\dot{\mathbf{q}} = J \dot{\boldsymbol{\theta}}$ denote the end-effector velocity. The set of joint velocities of unit norm,
\[\\{ \dot{\boldsymbol{\theta}} \in \mathbb{R}^n : \\| \dot{\boldsymbol{\theta}} \\| = 1 \\},\]is the unit sphere in joint-rate space. Its image under $J$ is, when $J$ has full row rank, an ellipsoid in $\mathbb{R}^m$. Explicitly, writing $\dot{\boldsymbol{\theta}} = J^{+} \dot{\mathbf{q}}$ (using the right pseudoinverse $J^{+} = J^\top (J J^\top)^{-1}$ to invert minimally),
\[1 \;=\; \dot{\boldsymbol{\theta}}^\top \dot{\boldsymbol{\theta}} \;=\; (J^{+} \dot{\mathbf{q}})^\top (J^{+} \dot{\mathbf{q}}) \;=\; \dot{\mathbf{q}}^\top (J J^\top)^{-1} \dot{\mathbf{q}}.\]This identifies the image as the ellipsoid
\[E_v \;=\; \\{ \dot{\mathbf{q}} \in \mathbb{R}^m : \dot{\mathbf{q}}^\top (J J^\top)^{-1} \dot{\mathbf{q}} \le 1 \\}.\]The set
$$ E_v(\boldsymbol{\theta}) \;=\; \\{ \dot{\mathbf{q}} \in \mathbb{R}^m : \dot{\mathbf{q}}^\top (J(\boldsymbol{\theta}) J(\boldsymbol{\theta})^\top)^{-1} \dot{\mathbf{q}} \le 1 \\} $$ is called the **manipulability ellipsoid** at $\boldsymbol{\theta}$. Its principal axes are the eigenvectors $\mathbf{u}_i$ of $A = J J^\top$, with semi-axis lengths $\sqrt{\lambda_i}$, where $\lambda_i$ are the corresponding eigenvalues.
The eigendecomposition reveals the geometric content: the longest semi-axis points in the direction in which the end-effector moves most freely per unit joint effort, while the shortest semi-axis identifies the “hardest” direction. As the configuration approaches a singularity, the smallest eigenvalue $\lambda_{\min}$ tends to zero and the corresponding semi-axis collapses; the ellipsoid degenerates to a lower-dimensional region (a disk, then a line segment, then a point), reflecting the loss of an instantaneous degree of motion.
For a $6 \times n$ Jacobian one typically splits
\[J(\boldsymbol{\theta}) \;=\; \begin{bmatrix} J_{\boldsymbol{\omega}}(\boldsymbol{\theta}) \\ J_{\mathbf{v}}(\boldsymbol{\theta}) \end{bmatrix},\]with $J_{\boldsymbol{\omega}}$ the top three rows (angular) and $J_{\mathbf{v}}$ the bottom three (linear), and analyzes the angular and linear manipulability ellipsoids separately, since the two have different physical units.
Yoshikawa’s Measure and the Condition Number
While the manipulability ellipsoid is a complete geometric picture, it is often desirable to summarize it with a single scalar. Three standard measures, all functions of $A = J J^\top$, are used:
Let $\lambda_1 \ge \lambda_2 \ge \cdots \ge \lambda_m > 0$ be the eigenvalues of $A = J J^\top$. Define: - **Yoshikawa's manipulability measure**: $\;\;\mu(\boldsymbol{\theta}) \;=\; \sqrt{\det\\!\big(J(\boldsymbol{\theta}) J(\boldsymbol{\theta})^\top\big)} \;=\; \sqrt{\lambda_1 \lambda_2 \cdots \lambda_m}$, proportional to the volume of the manipulability ellipsoid. - **Aspect ratio**: $\;\;\mu_1(\boldsymbol{\theta}) \;=\; \sqrt{\lambda_{\max}(A) / \lambda_{\min}(A)} \;\ge\; 1$, the ratio of the longest to the shortest principal semi-axis. - **Condition number**: $\;\;\mu_2(\boldsymbol{\theta}) \;=\; \lambda_{\max}(A) / \lambda_{\min}(A) \;\ge\; 1$, the square of the aspect ratio.
For a non-redundant manipulator ($n = m$), $J$ is square and
\[\mu(\boldsymbol{\theta}) \;=\; \sqrt{\det J \cdot \det J^\top} \;=\; |\det J(\boldsymbol{\theta})|.\]In particular, $\mu = 0$ exactly at singularities, which gives a clean way to detect singular configurations numerically. The aspect ratio and condition number diverge to $+\infty$ as $\boldsymbol{\theta}$ approaches a singularity, providing finer-grained measures of “how close” the configuration is to being singular.
A configuration at which $\mu_1 = 1$ (equivalently $\lambda_1 = \cdots = \lambda_m$) is called isotropic: the manipulability ellipsoid is a sphere, and the end-effector can move with equal facility in every direction. Isotropic configurations are generally the most “dexterous”; one can in fact pose the optimal design problem of choosing link lengths so as to maximize $\mu$ or to attain isotropy in a region of interest.
Manipulability of the Planar 2R Arm
For the planar 2R arm of the introduction, $\det J = L_1 L_2 \sin\theta_2$ and so (with $J$ square)
\[\mu(\boldsymbol{\theta}) \;=\; L_1 L_2 \, |\sin\theta_2|.\]The manipulability is maximized at $\theta_2 = \pm \pi/2$, i.e. with the elbow bent at a right angle. If one is free to choose the link lengths subject to a fixed total reach $L_1 + L_2 = L$, then maximizing $L_1 L_2$ at fixed sum gives $L_1 = L_2$. The most “dexterous” planar 2R arm therefore has equal link lengths and is most able to position the tip when the elbow is bent at $90^\circ$. This is one of the classical applications of manipulability analysis to robot design.
Statics: $\boldsymbol{\tau} = J^\top \mathcal{F}$
The Jacobian also governs the static relationship between joint torques and end-effector wrenches, and this is the second pillar of velocity kinematics. Suppose the manipulator is in static equilibrium and that the end-effector exerts a wrench $\mathcal{F} \in \mathbb{R}^6$ on its environment. What joint torques $\boldsymbol{\tau} \in \mathbb{R}^n$ are required?
The derivation is a one-line application of the principle of virtual work (equivalently, conservation of power in the limit of vanishing motion). Power at the joints equals power at the end-effector:
\[\boldsymbol{\tau}^\top \dot{\boldsymbol{\theta}} \;=\; \mathcal{F}^\top \mathcal{V}.\](Here $\mathcal{F}$ and $\mathcal{V}$ are expressed in the same frame; the relation holds frame by frame.) Substituting $\mathcal{V} = J(\boldsymbol{\theta}) \dot{\boldsymbol{\theta}}$,
\[\boldsymbol{\tau}^\top \dot{\boldsymbol{\theta}} \;=\; \mathcal{F}^\top J(\boldsymbol{\theta}) \dot{\boldsymbol{\theta}} \;=\; (J(\boldsymbol{\theta})^\top \mathcal{F})^\top \dot{\boldsymbol{\theta}}.\]Since the equality must hold for every $\dot{\boldsymbol{\theta}} \in \mathbb{R}^n$, we obtain the central static relation:
In static equilibrium, the joint-torque vector $\boldsymbol{\tau} \in \mathbb{R}^n$ required to exert an end-effector wrench $\mathcal{F} \in \mathbb{R}^6$ is
$$ \boxed{\;\boldsymbol{\tau} \;=\; J(\boldsymbol{\theta})^\top \, \mathcal{F}.\;} $$ Conventionally one writes $\boldsymbol{\tau} = J_s^\top \mathcal{F}_s = J_b^\top \mathcal{F}_b$ to emphasize that the equation holds in either frame, provided the Jacobian and the wrench are expressed in the same frame.
This is sometimes called the Jacobian transpose relation and is one of the most consequential identities in robotics. It says that the same matrix that maps velocities forward ($\dot{\boldsymbol{\theta}} \mapsto \mathcal{V}$) maps wrenches backward through its transpose ($\mathcal{F} \mapsto \boldsymbol{\tau}$). The forward map is the kinematic map; the backward map is its static dual. This duality between motion and force pervades robotics: it underlies hybrid motion/force control, impedance control, the analysis of grasping and form closure, and the design of force-controlled actuators.
Singularities and the Static Dual
The static identity makes the meaning of a kinematic singularity more vivid. At a singular configuration the Jacobian loses rank, so $J^\top$ does too. The null space of $J^\top$,
\[\mathrm{Null}(J^\top) \;=\; \\{ \mathcal{F} \in \mathbb{R}^6 : J(\boldsymbol{\theta})^\top \mathcal{F} = 0 \\},\]is the set of wrenches that produce zero joint torque. Equivalently, these are the wrenches the chain can resist passively: even with the joints fully unactuated, the linkage geometry alone supports loads in these directions. The dimension of this null space equals the rank deficiency of $J$.
Geometrically this matches our earlier statement that at a singularity the manipulator loses the ability to move in certain directions; precisely in those directions it gains the ability to resist forces. Think of an outstretched arm at full extension: it cannot move further radially, but it can resist arbitrary radial loads without any joint torque (the load is taken up directly by the link structure). The motion ellipsoid collapses along the radial axis; the dual force ellipsoid stretches to infinity along that axis.
The force ellipsoid is obtained from the analogous construction $\| \boldsymbol{\tau} \| = 1 \Rightarrow \mathcal{F}^\top (J J^\top) \mathcal{F} \le 1$ (under invertibility assumptions). Its eigenvectors coincide with those of the manipulability ellipsoid, but its semi-axis lengths are the reciprocals of those of the manipulability ellipsoid:
\[\ell_i^{\text{force}} \;=\; \frac{1}{\sqrt{\lambda_i(A)}} \;=\; \frac{1}{\ell_i^{\text{velocity}}}.\]In particular, the product of the volumes of the velocity and force ellipsoids is a constant independent of $\boldsymbol{\theta}$. Increasing the velocity manipulability decreases the force manipulability in the same direction, and vice versa. This is again the velocity-force duality, expressed quantitatively.
Redundancy and Force Control
When the manipulator is redundant ($n > 6$), the static map $\boldsymbol{\tau} = J^\top \mathcal{F}$ from a six-dimensional wrench to an $n$-dimensional torque is not surjective onto $\mathbb{R}^n$: there are joint-torque modes (the null space of $J$) that produce no end-effector force. These are precisely the torques that drive internal self-motions of the chain (motions that leave the end-effector fixed). Equivalently, the inverse map $\mathcal{F} \mapsto \boldsymbol{\tau}$ is no longer unique; we can add an arbitrary torque in the null space of $J$ without affecting the end-effector force, and this freedom can be used to optimize a secondary objective (joint-limit avoidance, energy minimization, posture preservation, etc.).
Conversely, when $n < 6$, $J^\top$ has nontrivial null space in the wrench domain. The chain cannot actively generate forces along any direction in $\mathrm{Null}(J^\top)$ — only resist them passively. The motorized door of Lynch and Park’s example ($n = 1$) is the simplest illustration: a single-axis door at the doorknob can actively push tangent to the swing arc, but it resists arbitrary perpendicular wrenches without consuming actuator effort.
A Worked Example: the Planar 2R Arm Revisited
To bring the pieces together, let us treat the planar 2R arm uniformly through Jacobian, singularity, and manipulability lenses. The arm lives in the plane and has Jacobian
\[J(\boldsymbol{\theta}) \;=\; \begin{bmatrix} -L_1 s_1 - L_2 s_{12} & -L_2 s_{12} \\ \;\;\;L_1 c_1 + L_2 c_{12} & \;\;\;L_2 c_{12} \end{bmatrix}\](here $c_i = \cos\theta_i$ etc.), with columns
\[J_1 \;=\; \begin{bmatrix} -L_1 s_1 - L_2 s_{12} \\ \;\;\;L_1 c_1 + L_2 c_{12} \end{bmatrix}, \quad J_2 \;=\; \begin{bmatrix} -L_2 s_{12} \\ \;\;\;L_2 c_{12} \end{bmatrix}.\]These are the tip velocities produced by rotating joint 1 (resp. joint 2) at unit rate while the other joint is locked, as the columnwise interpretation of the geometric Jacobian dictates.
The determinant evaluates to
\[\det J(\boldsymbol{\theta}) \;=\; L_1 L_2 (-s_1 c_{12} + c_1 s_{12}) \cdot (\text{after simplification}) \;=\; L_1 L_2 \sin\theta_2,\]so the singular configurations are $\theta_2 = 0$ (fully extended) and $\theta_2 = \pi$ (fully folded). At these values the two columns of $J$ are collinear; the tip can move only along a single direction (tangent to the reachable boundary circle), and any required radial motion would demand unbounded joint rates.
The manipulability measure in Yoshikawa’s sense is
\[\mu(\boldsymbol{\theta}) \;=\; |\det J(\boldsymbol{\theta})| \;=\; L_1 L_2 \, |\sin\theta_2|,\]which is zero at the singularities and maximized at $\theta_2 = \pm \pi/2$. The maximum value over all configurations is $L_1 L_2$. If we further constrain the design by fixing the total reach $L_1 + L_2$, the AM-GM inequality forces the maximum of $L_1 L_2$ to occur at $L_1 = L_2$.
The manipulability ellipsoid has principal semi-axes given by the square roots of the eigenvalues of $A = J J^\top$. At the configuration $L_1 = L_2 = 1$, $\boldsymbol{\theta} = (0, \pi/4)$,
\[J(\boldsymbol{\theta}) \;=\; \begin{bmatrix} -0.71 & -0.71 \\ \;\;\; 1.71 & \;\;\; 0.71 \end{bmatrix},\]and the ellipsoid is elongated along the direction in which $J_1$ and $J_2$ best span $\mathbb{R}^2$, narrow along the orthogonal direction; as $\theta_2 \to 0$ the narrow semi-axis collapses to zero and the ellipsoid degenerates to a line segment.
By the static duality, the force ellipsoid is the same shape with its semi-axis lengths inverted. Near $\theta_2 = 0$, the velocity ellipsoid collapses to a segment along the boundary tangent; the force ellipsoid stretches arbitrarily far in the perpendicular (radial) direction. This is why fully extending one’s arm to support a heavy weight is so much less tiring than holding it with the elbow bent: the radial load is borne by link geometry, not by muscle torque about the elbow joint. The kinematic singularity is, from the static viewpoint, a configuration of infinite mechanical advantage in the perpendicular direction.
Summary
We have built up the differential kinematics of an open chain around a single object — the manipulator Jacobian — and read it in two complementary directions.
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The geometric Jacobian $J(\boldsymbol{\theta}) \in \mathbb{R}^{6 \times n}$ maps joint rates to a twist of the end-effector. It comes in two flavors, the space Jacobian $J_s$ (twists in $\{ s \}$) and the body Jacobian $J_b$ (twists in $\{ b \}$), related by the adjoint of $T_{sb}$. Each column is the screw axis of one joint at the current configuration, expressed in the appropriate frame; for revolute joints the column is $(\hat{\boldsymbol{\omega}}_i, -\hat{\boldsymbol{\omega}}_i \times \mathbf{q}_i)$, for prismatic joints it is $(\mathbf{0}, \hat{\mathbf{v}}_i)$. No differentiation of the forward kinematic map is required.
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The analytical Jacobian $J_a(\boldsymbol{\theta})$ maps joint rates to the time derivative of a minimal pose parameterization. It differs from the geometric Jacobian by a representation-dependent matrix relating $\dot{\boldsymbol{\alpha}}$ to $\boldsymbol{\omega}$, and it suffers from representational singularities (e.g. gimbal lock) on top of the kinematic ones.
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A kinematic singularity is a configuration at which $J$ fails to attain maximum rank. Common cases include collinear revolute axes, coplanar parallel axes, and several axes meeting at a common point. Singularities are intrinsic to the mechanism — independent of frame choice, and shared by $J_s$ and $J_b$.
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The manipulability ellipsoid $\{ \dot{\mathbf{q}} : \dot{\mathbf{q}}^\top (J J^\top)^{-1} \dot{\mathbf{q}} \le 1 \}$ visualizes the ease of end-effector motion in different directions. Yoshikawa’s measure $\mu = \sqrt{\det(J J^\top)}$, the aspect ratio, and the condition number summarize it scalarly; $\mu = 0$ exactly at singularities, and $\mu = \det J $ when $J$ is square. - The static identity $\boldsymbol{\tau} = J(\boldsymbol{\theta})^\top \mathcal{F}$, derived from virtual work, expresses the duality between motion and force. At singularities the null space of $J^\top$ is nontrivial, identifying the directions in which the chain resists wrenches passively without joint torque. The corresponding force ellipsoid is the geometric inverse of the manipulability ellipsoid.
With the Jacobian in hand, the next natural questions concern its inverse — given a desired end-effector twist or pose, what joint rates or configuration realize it? These questions are taken up in the next post, on inverse kinematics and inverse velocity kinematics, where the rank structure and singularity analysis developed here play the central role.
Reference
[1] Kevin M. Lynch and Frank C. Park, Modern Robotics: Mechanics, Planning, and Control, Cambridge University Press, 2017. Chapter 5: Velocity Kinematics and Statics.
[2] Mark W. Spong, Seth Hutchinson, and M. Vidyasagar, Robot Modeling and Control, 2nd ed., Wiley, 2020. Chapter 5: Velocity Kinematics — The Manipulator Jacobian.
[3] Richard M. Murray, Zexiang Li, and S. Shankar Sastry, A Mathematical Introduction to Robotic Manipulation, CRC Press, 1994. Chapter 3: Manipulator Kinematics.
[4] T. Yoshikawa, “Manipulability of robotic mechanisms,” International Journal of Robotics Research, vol. 4, no. 2, pp. 3-9, 1985.
[5] Bruno Siciliano, Lorenzo Sciavicco, Luigi Villani, and Giuseppe Oriolo, Robotics: Modelling, Planning and Control, Springer, 2009.
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