Singularities and the Jacobian

1.1 Speeds are simpler than positions

The position of the tool depends on the joint angles in a complicated way: sines, cosines, products. But the speed of the tool depends on the joint speeds in a very simple way: linearly. If you double all the joint speeds, the tool goes twice as fast. This linear link is a matrix, the Jacobian J.

1.2 One single joint

Take one link of length L turned by the angle θ: the end is at x = L cos θ, y = L sin θ. Differentiate with respect to time:

ẋ = −L sin θ · θ̇, ẏ = L cos θ · θ̇

The column (−L sin θ, L cos θ) is perpendicular to the link and has length L: the end moves on a circle, so its velocity is tangent to it. Longer link, faster tool: v = L · ω.

1.3 The two-joint arm

For the arm of the “Maths and motion” course, x = L₁ cos θ₁ + L₂ cos(θ₁+θ₂) and y = L₁ sin θ₁ + L₂ sin(θ₁+θ₂). Differentiating gives:

ẋẏ = −L₁ sin θ₁ − L₂ sin θ₁₂−L₂ sin θ₁₂L₁ cos θ₁ + L₂ cos θ₁₂L₂ cos θ₁₂ θ̇₁θ̇₂ (θ₁₂ = θ₁ + θ₂)

Read it geometrically: column 1 is the tool velocity when only joint 1 turns (1 rad/s). It is perpendicular to the line “joint 1 → tool” and its length is the distance joint 1 – tool. Column 2 is the same for joint 2: perpendicular to the line “joint 2 → tool”, length L₂.

Key terms

• Jacobian matrix
• joint speed
• tool velocity
• column of J
• linear / angular velocity

The Jacobian in the Master-level course →

2.1 The stretched arm

Put the two-joint arm with θ₂ = 0: the arm is perfectly stretched. Replace in the Jacobian of lesson 1 (cos θ₁₂ = cos θ₁, sin θ₁₂ = sin θ₁):

J = −(L₁+L₂) sin θ₁−L₂ sin θ₁(L₁+L₂) cos θ₁L₂ cos θ₁

Look at the two columns: both are proportional to the same vector (−sin θ₁, cos θ₁), which is perpendicular to the arm. Whatever the joint speeds, the tool can only move sideways. Along the arm, nothing is possible.

2.2 The test: the determinant

For the two-joint arm, a short calculation (lesson 5 of this course shows it again) gives:

det J = L₁ L₂ sin θ₂

2.3 Two families

• Boundary singularities sit at the edge of the workspace: arm stretched or folded. They are easy to avoid, simply do not ask the tool to go there.
• Internal singularities sit inside the workspace (shoulder and wrist of a 6-axis robot, lesson 3). They are the dangerous ones: the tool can pass through them while doing a perfectly normal task.

The first reflex of an engineer facing a new robot: find its singular poses and keep the useful workspace away from them.

Key terms

• singularity
• rank
• determinant
• boundary singularity
• internal singularity

Watch the stretched and folded poses on the SCARA page →

Most industrial robots have six axes and a spherical wrist (axes 4, 5 and 6 meet at one point, the wrist centre; see the 6-axis robot page). This shape splits the problem in two: joints 1-3 place the wrist centre, joints 4-6 orient the tool. The Jacobian then also splits, and its determinant becomes a product of simple factors. For the Stäubli RX-90 of the Master course:

det J = −C₃ D₃ RL₄ S₅ (S₂₃ RL₄ − C₂ D₃)

A product is zero when one factor is zero. Each zero factor is a family of singular poses:

3.1 Why the wrist singularity is the one you meet most

The three wrist axes give the tool angular speeds along z₄, z₅ and z₆. When q₅ = 0, z₆ = z₄: two joints now turn the tool about the same axis. The tool can no longer turn about the axis perpendicular to both z₄ and z₅. Worse, an infinity of pairs (q₄, q₆) give the same tool orientation.

3.2 Robots with an offset wrist (UR, cobots)

On a UR5 the wrist axes do not meet at one point, yet the same three families exist: wrist (q₅ = 0 or 180°), elbow (q₃ = 0 or 180°) and shoulder (the wrist centre above axis 1, inside the small cylinder made by the shoulder offset). Only the formulas change. Try the three of them on the 6-axis robot page and on the cobot page.

Key terms

• elbow singularity
• shoulder singularity
• wrist singularity
• spherical wrist
• wrist centre

Create a wrist singularity on the 6-axis robot page →

A singularity is not a cliff: it is the bottom of a valley. The robot already behaves badly on the slopes. We need a number that says how far we are.

4.1 The manipulability ellipse

Mathematically, if J = U Σ V^T (singular value decomposition), the semi-axes of the ellipse are the singular values σ₁ ≥ σ₂ ≥ … of J, in the directions of the columns of U. The tool is fast along the first, slow along the last.

4.2 Two numbers

• Manipulability (Yoshikawa): w = √det(J J^T) = σ₁ σ₂ … It is the “area” of the ellipse. It is zero at a singularity. For a square J, w = |det J|; for the two-joint arm, w = L₁ L₂ |sin θ₂|.
• Condition number: κ = σ_max / σ_min. It is 1 for a circle (the tool is equally good in all directions) and tends to infinity at a singularity. It is the number that says how much errors are amplified.

4.3 The force ellipse: the mirror image

The same matrix describes forces (see lesson 5): the force ellipse has the same axes but the inverse lengths 1/σ_i. Where the tool is slow it is strong, and where it is fast it is weak. A stretched arm is slow along its length but it can push very hard along it, because the bones carry the load. You choose the pose according to the task: speed (go to the fast direction) or force (go to the strong one). A hammer-like task wants a stretched arm, a fast pick and place wants a well-conditioned pose.

Key terms

• manipulability
• manipulability ellipse
• singular value
• SVD
• condition number
• isotropic

5.1 The joint speeds explode

To make the tool follow a Cartesian path at speed ẋ, the controller computes the joint speeds with the inverse Jacobian. For the two-joint arm:

J⁻¹ = (1 / (L₁L₂ sin θ₂)) L₂ cos θ₁₂L₂ sin θ₁₂−L₁ cos θ₁ − L₂ cos θ₁₂−L₁ sin θ₁ − L₂ sin θ₁₂

Every term is divided by det J = L₁L₂ sin θ₂. When it tends to zero, the joint speeds tend to infinity.

5.2 Forces: the same matrix, transposed

The power given by the motors, τ^T q̇, equals the power received by the tool, F^T ẋ = F^T J q̇. Hence the static model:

τ = J^T F

To push on the world with a force F at the tool, the motors must supply the torques τ. Now take the arm stretched along x (θ₁ = θ₂ = 0, L₁ = 0.5, L₂ = 0.3):

τ₁τ₂ = 00.800.3 F_xF_y = 0.8 F_y0.3 F_y

F_x does not appear: a load of 1000 N pulling on the tool along the arm needs zero motor torque. The structure carries it. But a force of 10 N sideways needs 8 N·m and 3 N·m.

5.3 Accuracy and noise

Going from the joints to the tool (forward) is benign: dx = J dq, the error of the encoders is passed on without amplification. Going from the tool to the joints is the problem: a tool error of dx asks for up to dx/σ_min of joint motion. With the arm of lesson 4, a tool step of 0.1 mm needs about 0.02° at θ₂ = 90° (σ_min = 0.25) but about 0.9° at θ₂ = 2° (σ_min = 0.0061): 40 times more, which can be below the resolution of the gearbox and the encoder. A small calibration error becomes a visible error.

5.4 What you see in a real program

Key terms

• inverse Jacobian
• static model
• joint torque
• duality
• speed limit

So far the singularity was in the robot. There is another one in the maths we use to write an orientation with three angles. It has the same bad effects on the controller, so it is worth knowing both.

6.1 Three angles, one lost

Cardan angles (roll-pitch-yaw): rotate by α about z, then β about y, then γ about x (all in the fixed frame): R = R_z(α) R_y(β) R_x(γ). Take the pitch β = 90° and compute the product:

R = 0sin(γ − α)cos(γ − α)0cos(γ − α)−sin(γ − α)−100

The matrix depends only on γ − α. Whatever α and γ are, if their difference is the same, the orientation is the same. We have three angles but only two numbers matter: a degree of freedom has disappeared. This is gimbal lock.

6.2 Representation singularity versus robot singularity

The angular velocity ω of the tool is related to the rates of the three angles by a matrix E:

ω_xω_yω_z = 0−sin αcos α cos β0cos αsin α cos β10−sin β α̇β̇γ̇ det E = −cos β

When β = ±90°, det E = 0: some angular velocities ω would need infinite rates α̇ and γ̇. The tool itself is perfectly fine; it is the way of writing its orientation that breaks. Engineers therefore distinguish two Jacobians: the geometric one (about ω, singular only when the robot is) and the analytical one (about the angle rates, which also inherits the singularity of E).

6.3 What to use instead

Key terms

• gimbal lock
• Cardan / Euler angles
• quaternion
• geometric Jacobian
• analytical Jacobian
• SLERP

Euler and Cardan angles in the Master-level course →

7.1 More joints than needed

A robot is redundant when it has more joints (n) than the task needs (m). Examples: a planar arm with 3 joints positioning a point in the plane (n = 3, m = 2); a 7-axis arm (KUKA iiwa, Franka Panda) for a 6-DOF task; your own arm (7 joints from the shoulder to the wrist) holding a glass: you can move the elbow freely while the glass stays still.

The Jacobian is no longer square: it has m rows and n > m columns. The system ẋ = J q̇ has infinitely many solutions.

7.2 The null space

7.3 Using the freedom

q̇ = J⁺ ẋ + (I − J⁺J) z

The first term does the main task (the tool). The matrix (I − J⁺J) projects any vector z into the null space, so the second term can only move the joints, never the tool. Choose z to optimise a secondary criterion:

• move away from the singularities (increase the manipulability w: z = k·∇w);
• stay away from the joint limits (keep each joint near the middle of its range);
• avoid obstacles with the elbow while the hand keeps working;
• minimise energy, or share the load between joints.

7.4 Redundancy and singularities

A redundant robot is singular when rank J < m, that is det(J J^T) = 0. This is a rarer event: the robot has many ways to reach the same tool pose, and if one of them is singular it can change the arm shape through the null space without disturbing the tool. A 7-axis arm can rotate its elbow to escape a shoulder or an elbow singularity. Only the singularities that lose a tool direction whatever the arm shape (typically a fully stretched arm) cannot be fixed.

Key terms

• redundant robot
• null space
• self-motion
• pseudo-inverse
• secondary task
• task redundancy

You cannot remove singularities of a given robot, but you can detect them, avoid them, or tame them. Professionals use all three, in this order.

8.1 Detect

Compute at each control step a distance measure (lesson 4): the manipulability w, the smallest singular value σ_min, or simply the key joint angles (q₅ for a wrist, q₃ for an elbow). Set a threshold, for example “warn when w falls under 10 % of its maximum” and “slow down under 3 %”. Simulation tools draw this as a colour map of the workspace: green far from singularities, red close.

8.2 Avoid: by design and by programming

8.3 Tame: the damped inverse

If the robot must go through a poorly conditioned zone, replace the exact inverse by the damped least squares (Wampler, Nakamura and Hanafusa):

q̇ = J^T (J J^T + λ² I)⁻¹ ẋ

It solves a compromise: make the tool error small and keep the joint speeds small, with λ setting the balance. Far from a singularity (σ ≫ λ) it gives the same answer as the exact inverse. Near it the joint speeds stay finite, at the price of a small tool error in the lost direction: the robot slows down instead of racing. A common refinement raises λ only when needed: λ² = λ₀²(1 − (w/w₀)²) for w < w₀, and 0 otherwise.

Two other taming tools: speed scaling (slow the whole path down so that no joint exceeds its limit; the path is kept, only the time changes) and task prioritisation (the position keeps the priority, the orientation is allowed to drift a little).

8.4 Parallel robots: three kinds of singularity

For a parallel robot (Delta, Stewart platform) the velocity relation is written A ẋ = B q̇, with x the platform and q the motors. Gosselin and Angeles named three cases:

This is a mirror of the serial case: for a serial robot the singular pose loses a tool direction (type 1); a parallel robot has this problem and the opposite one. That is why a parallel robot is always designed together with a map of its singularity-free workspace, and why the useful volume of a Delta robot is a small dome compared with the size of its frame.

See a Stewart platform in 3D: change its pose and watch the six leg lengths →

8.5 The cheat sheet

Key terms

• damped least squares
• speed scaling
• MoveJ / MoveL
• direct kinematic singularity
• inverse kinematic singularity
• singularity-free workspace

Singularities and workspace of the Delta robot →