Robot Jacobians and Singularities

Why Does This Exist?

The gripper is at the right point, but the next command asks it to slide farther outward along a fully extended arm. The joints can turn; none of their instantaneous motions produces that outward tip velocity. Reaching a position and being able to move through it in every direction are different questions.

A robot Jacobian is the local map from joint velocity to task velocity. A kinematic singularity is a configuration where that map loses rank relative to the maximum rank it can attain. The task matters: controlling planar position is different from controlling a full position and orientation.

Start with forward kinematics: it maps joint angles to tool position. A velocity is how quickly a position changes. The Jacobian differentiates that same mapping, so each column says what one moving joint contributes while the others are momentarily still. This page develops that local relationship, rather than solving a complete collision-free motion.

Think of It Like This

How It Actually Works

What is in a Jacobian column?

Let $q=(q_1,q_2)$ contain two joint angles in radians and $p=(x,y)$ the tip position in metres. A dot means differentiation with respect to time. For link lengths $l_1,l_2$, shoulder angle $q_1$ measured counterclockwise from base $x$, and elbow angle $q_2$ relative to the first link:

The matrix $J$ contains the partial derivatives of $p$ with respect to each angle:

Column one is the tip velocity per unit shoulder rate; column two is the contribution per unit elbow rate. This position Jacobian has units of metres per radian. A spatial Jacobian instead maps to a six-component twist, combining angular and linear velocity. Always declare the frame, reference point and component order before using it.

Work the bent arm, then straighten it

Set $l_1=l_2=1$ m and $q=(0,\pi/2)$. The tip is $(1,1)$ m and

With $\dot q=(0.2,-0.1)$ rad/s, shoulder motion contributes $(-0.2,0.2)$ m/s; elbow motion contributes $(0.1,0)$ m/s. Their sum is $(-0.1,0.2)$ m/s. The diagram places these two contributions head to tail, labelled 1 and 2. The mint arrow joins the original tip to their combined endpoint. It is a velocity vector, not the tip's position after a finite step. Both physical links have equal length within each panel; the panels use slightly different drawing scales for space.

At $q=(0,0)$ the tip is $(2,0)$ m and $J=\begin{bmatrix}0&0\\2&1\end{bmatrix}$. Every finite joint-rate choice gives $\dot x=0$. The matrix rank drops from two to one; the dashed horizontal request in the right panel is unavailable. Bending inward later can change $x$, but that does not supply a nonzero horizontal velocity at this exact instant. Modern Robotics on singularities formalizes rank loss and distinguishes it from simply having too few joints for a task.

What does damping buy?

The pseudoinverse $J^+$ gives the least-squares joint-rate solution with minimum Euclidean norm. Near a singularity, a small achievable task direction can require large joint rates. Damped least squares instead minimizes $\|J\dot q-v_d\|^2+\lambda^2\|\dot q\|^2$, where $v_d$ is requested tool velocity and $\lambda>0$ sets the trade-off. Its solution is

Here $I$ is the identity matrix, distinct from physical inertia. In our translational, revolute-only fixture, choose $\lambda=0.1$ in the corresponding Jacobian units. At the straight posture, requesting $v_d=(0.1,0.2)$ m/s gives $\dot q=(0.4,0.2)/5.01=(0.079840,0.039920)$ rad/s. The achieved velocity is $(0,0.199601)$ m/s. Damping softens inversion; it cannot create the missing horizontal direction or enforce explicit actuator limits. The formula follows by differentiating the stated quadratic objective. Joint bounds need a constrained solve or another explicit limit-handling policy.

The singular values of $J$ describe directional velocity amplification. Mapping a unit joint-rate sphere through $J$ gives a velocity ellipsoid; a collapsed axis reveals rank loss. Modern Robotics on manipulability derives this geometry. A raw norm combining metres per second and radians per second needs a declared task weighting; its condition number is otherwise unit-dependent.

Why does the transpose map forces?

Let $F$ be a tool force and $\tau$ its corresponding generalized joint load, using matching frames and sign conventions. Equality of instantaneous power gives $F^T\dot p=\tau^T\dot q$. Substituting $\dot p=J\dot q$ gives $\tau=J^TF$. For the bent fixture, $F=(0,2)$ N maps to $(2,0)$ N m. The tool power is $2(0.2)=0.4$ W; joint power is $2(0.2)+0(-0.1)=0.4$ W.

An external load contributes $J^TF$ to the robot's generalized force. Motors holding against that load supply the opposite contribution, alongside gravity compensation. Modern Robotics on statics derives the transpose through power. This is the bridge to impedance control, not a claim that transpose mapping alone controls contact.

Code

This dependency-free check verifies the analytic velocity against a small forward kinematics step, then checks the unachievable component of the damped request.

1from math import cos, sin, pi2
3def tip(q1: float, q2: float) -> tuple[float, float]:4    return cos(q1) + cos(q1 + q2), sin(q1) + sin(q1 + q2)5
6q = (0.0, pi / 2)7rates = (0.2, -0.1)8dt = 1e-69p0 = tip(*q)10p1 = tip(q[0] + dt * rates[0], q[1] + dt * rates[1])11velocity = tuple((b - a) / dt for a, b in zip(p0, p1))12assert max(abs(a-b) for a, b in zip(velocity, (-0.1, 0.2))) < 1e-613
14def damped_straight(vy: float, damping: float) -> tuple[float, float]:15    # J = [[0, 0], [2, 1]]: horizontal request cannot contribute.16    return 2 * vy / (5 + damping**2), vy / (5 + damping**2)17
18u = damped_straight(0.2, 0.1)19achieved = (0.0, 2 * u[0] + u[1])20assert abs(achieved[1] - 0.1996007984031936) < 1e-1221print(tuple(round(v, 6) for v in velocity))22print(tuple(round(v, 6) for v in achieved))

Output: (-0.1, 0.2) and (0.0, 0.199601). The finite difference checks a local derivative, not finite-step trajectory accuracy.

Watch Out For

The Quick Version

• A Jacobian maps joint rates to instantaneous task velocity at the current posture.
• Singularity means rank loss relative to the chosen task's attainable maximum.
• Damped inversion trades tracking accuracy for reduced sensitivity; missing directions stay missing.
• The transpose maps matching tool wrenches to generalized joint loads through equality of power.
• Kinematic feasibility leaves torque, collision and timing checks to the rest of the motion system.

What to Read Next

• Forward and Inverse Kinematics
• Degrees of Freedom and Robot Bodies
• Robot Impedance Control: Yielding Under Contact
• Robot Trajectory Generation and Time Scaling