M8-T6 — Jacobian Inverse and Pseudoinverse

Learning objectives

After this lesson, you should be able to:

• Solve ṗ = J q̇ by J⁻¹ for a square nonsingular Jacobian and verify the result.
• Use the pseudoinverse J⁺ = Jᵀ(JJᵀ)⁻¹ for a redundant arm and show that it gives the minimum joint-speed solution.
• Contrast the transpose, inverse and pseudoinverse methods and their costs and failure modes.
• Close the loop on position error to turn a velocity solver into an inverse-kinematics controller.

Introduction and motivation

A seven-joint arm must move its gripper along a path. At each control tick the planner supplies a desired hand velocity, and the controller must turn it into seven joint speeds. There are infinitely many solutions; which one should it take? The shortest one is smooth, uses the least motion and is exactly what the pseudoinverse delivers. For a six-joint arm the same step uses the plain inverse, until the arm approaches a singularity.

The intuition

The equation J q̇ = ṗ is a linear system, so Modules 3 and 6 tell us everything. Square and invertible: one answer. More joints than task dimensions: many answers, so choose the shortest. Too few effective directions: no exact answer, so choose the best fit. The pseudoinverse covers all three in one formula, which is why it is the workhorse of resolved-rate control.

Geometric and physical interpretation

Geometric view. The solutions of J q̇ = ṗ form an affine subspace: one particular solution plus the whole null space of J. The pseudoinverse solution J⁺ṗ is the point of that subspace closest to the origin, i.e. the one perpendicular to the null space, and it lies in the row space of J. Physical view. Minimum joint-speed means minimum kinetic-like effort in a weighted sense, so the motion is smooth and avoids needless self-motion; the leftover freedom is kept for secondary tasks (null-space lesson).

The mathematics

J

the m × n Jacobian at the current configuration.

J⁻¹

the ordinary inverse (square, nonsingular).

J⁺

the Moore-Penrose pseudoinverse (Module 6).

ṗ_des

the desired hand velocity, possibly plus a correction K(p_des − p) from position error.

Square, nonsingular

q̇ = J⁻¹ ṗ        (n = m, det J ≠ 0)

What it means. The unique solution. Cost is a linear solve; the condition number of J measures how much noise in ṗ is amplified in q̇.

Redundant: the pseudoinverse

q̇ = J⁺ ṗ = Jᵀ (J Jᵀ)⁻¹ ṗ        (n > m, rank J = m)

What it means. Among all q̇ with J q̇ = ṗ, this one has the smallest ‖q̇‖. Compute it with the SVD, J⁺ = V Σ⁺ Uᵀ, rather than the explicit formula.

Closing the loop

q̇ = J⁺ ( ṗdes + K (pdes − p) )

What it means. The correction term pulls the hand back toward the desired position, removing the drift that integrating velocities alone would cause.

Square and nonsingular: J⁻¹. Redundant: J⁺ gives the shortest joint-speed vector. Always add position feedback, and check the singular values before inverting.

Step-by-step derivation: the minimum-norm solution and the transpose method

1. Solution set. If q̇₀ is any solution of J q̇ = ṗ, then every solution is q̇₀ + z with z ∈ null(J) (Module 3). For a redundant arm the null space has dimension n − m ≥ 1.
2. Shortest solution. Minimise ½‖q̇‖² subject to J q̇ = ṗ. With a Lagrange multiplier λ: q̇ − Jᵀλ = 0, so q̇ = Jᵀλ lies in the row space.
3. Solve for λ. Substitute into the constraint: J Jᵀ λ = ṗ, so λ = (JJᵀ)⁻¹ṗ (JJᵀ is invertible when J has full row rank) and q̇ = Jᵀ(JJᵀ)⁻¹ ṗ = J⁺ṗ.
4. Check with the SVD. J = UΣVᵀ gives J⁺ = VΣ⁺Uᵀ and Jᵀ(JJᵀ)⁻¹ = VΣᵀUᵀ (UΣΣᵀUᵀ)⁻¹ = VΣᵀ(ΣΣᵀ)⁻¹Uᵀ = VΣ⁺Uᵀ ✓ (Module 6).
5. Jacobian transpose method. Instead of inverting, move along q̇ = α Jᵀ e where e = p_des − p. This decreases ½‖e‖² because d/dt ½‖e‖² = −eᵀ J q̇ = −α‖Jᵀe‖² ≤ 0. It needs no inverse and never blows up at a singularity, but converges slowly.
6. Cost. The inverse of a 6 × 6 matrix or an SVD of a 6 × 7 matrix is microseconds; the cost matters only for very large systems or kHz loops on small processors.
7. Feedback. An open-loop velocity command integrates errors (the linearisation and the numerical integration both drift). Adding K(p_des − p) makes the position error obey ė = −K e, an exponential decay with rate K when J⁺J behaves as identity on the task space.

Weighted pseudoinverse. To prefer some joints over others (a big shoulder over a light wrist) minimise q̇ᵀ W q̇ instead: q̇ = W⁻¹Jᵀ(J W⁻¹ Jᵀ)⁻¹ṗ. It is the weighted least squares of Module 5 applied to velocities.

When it fails. If J loses rank (singularity) JJᵀ becomes singular and J⁺ jumps or explodes; the next two lessons handle it. Even for full rank, a small σ gives enormous q̇ for a hand velocity along the weak direction.

A worked example: square inverse, pseudoinverse and the transpose

• Square case. J = [[−0.8, −0.8], [1.1, 0]] (the two-link arm at q = (0°, 90°)), det = 0.88, J⁻¹ = (1/0.88)[[0, 0.8], [−1.1, −0.8]]. For ṗ = (0.4, 0.55): q̇ = (0.8·0.55, −1.1·0.4 − 0.8·0.55)/0.88 = (0.5, −1.0) rad/s. Check: J q̇ = (−0.4 + 0.8, 0.55) = (0.4, 0.55) ✓.
• Redundant case. Three-link arm, links 1.1, 0.8, 0.7 at q = (0°, 90°, 0°): J = [[−1.5, −1.5, −0.7], [1.1, 0, 0]]. For the same ṗ = (0.4, 0.55), the second row gives q̇₁ = 0.5; the first row then needs −1.5 q̇₂ − 0.7 q̇₃ = 1.15.
• Pseudoinverse solution. q̇ = J⁺ṗ = (0.5, −0.630, −0.294), with ‖q̇‖ = 0.856. Check: −1.5·0.5 − 1.5(−0.630) − 0.7(−0.294) = −0.75 + 0.944 + 0.206 = 0.400 ✓.
• Another valid solution. q̇ = (0.5, −0.767, 0) (only the first two joints help the wrist) also satisfies the equation, but ‖q̇‖ = 0.915 > 0.856, so the pseudoinverse solution is indeed shorter.
• The null space. The difference of the two solutions lies in the null space, spanned by n = (0, −0.423, 0.906) (for example J n = 0 since the first row gives 0.634 − 0.634 = 0).
• Transpose method. For the square arm with position error e = (0.1, 0): Jᵀ e = (−0.08, −0.08). A step q̇ = α Jᵀ e with α = 1 moves the hand by J q̇ = (0.128, −0.088). Its dot product with e is positive (0.0128), so the error shrinks to first order, but the hand also drifts sideways: the direction is only roughly right, which is why the method converges slowly.

Robotics application

Manipulators. Resolved-rate controllers in industrial and collaborative arms run q̇ = J⁺(ṗ_des + K e) at 1 kHz; the library routines (KDL, MoveIt, Pinocchio) use the SVD or damped inverse underneath.

LiDAR scan matching. The Gauss-Newton step Δx = (JᵀJ)⁻¹Jᵀr is the pseudoinverse of the tall Jacobian applied to the residual, the left-inverse case of Module 5; see nonlinear least squares and ICP fundamentals.

UAVs. A drone with a manipulator uses the stacked Jacobian of body and arm; control allocation among redundant rotors uses a pseudoinverse of the allocation matrix; see fault detection, isolation and recovery for redistributing control after a motor failure.

Mobile robots and SLAM. Whole-body controllers for mobile manipulators use J⁺ to coordinate base and arm; pose-graph optimisers solve for state updates by pseudoinverse-like steps when the system is gauge-deficient; see pose graphs.

Engineering implications and limitations

Practical rules

• Compute J⁺ with the SVD (or a QR / damped solve), never with explicit inverses of JJᵀ. The SVD exposes the singular values and handles rank loss gracefully.
• Check σ_min before inverting. If it is below a threshold, switch to damping or truncation; do not discover the singularity from an exploding joint speed.
• Always include position feedback. Open-loop velocity integration drifts, and singular transits make the drift large.
• Saturate and scale. If any joint speed exceeds its limit, scale the whole vector q̇ down uniformly so the hand direction is preserved.
• Scale mixed units. Position (m) and orientation (rad) rows should be weighted, otherwise “shortest” q̇ favours the wrong rows.
• Keep the secondary freedom. For redundant arms the formula J⁺ṗ + (I − J⁺J)z keeps all solutions available (null-space lesson).

Limits

The pseudoinverse is a local, instantaneous solution. It does not avoid joint limits or obstacles, and a path computed by repeated J⁺ steps may end in different configurations for the same hand pose (it is not a function of the hand pose alone).

Common misconceptions

Engineering takeaways

1. Square nonsingular J: q̇ = J⁻¹ṗ is the unique solution.
2. Redundant J (wide): q̇ = J⁺ṗ = Jᵀ(JJᵀ)⁻¹ṗ is the minimum-norm solution among infinitely many; compute it via the SVD.
3. The Jacobian transpose method needs no inverse and is safe at singularities but slow.
4. Add position feedback q̇ = J⁺(ṗ_des + Ke) to cancel drift; saturate by uniform scaling.
5. Check the singular values first; near singularities switch to damping.

References and further reading

Primary papers and authoritative documentation.

• Modern Robotics (Lynch and Park), free preprint: chapter 6.2, numerical inverse kinematics: Newton-Raphson and the pseudoinverse
• Buss, Introduction to Inverse Kinematics with Jacobian Transpose, Pseudoinverse and Damped Least Squares: Jacobian transpose, pseudoinverse and damped least-squares methods for inverse kinematics; the site did not load when we last checked
• Boyd and Vandenberghe, Introduction to Applied Linear Algebra (VMLS): least-norm solutions and the pseudoinverse; the site did not load when we last checked
• MIT 18.06SC Linear Algebra (OCW): lecture on the pseudoinverse and least squares

Knowledge check

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What’s next

Both the inverse and the pseudoinverse have a weak spot: configurations where the Jacobian loses rank. The next lesson finds those singularities, explains what the arm cannot do there, and measures how close it is to them.

Related content

Nothing below is required to finish this lesson.

• Before
  M8-T5: Differential Kinematics

  Relate joint velocities to hand velocity with the Jacobian, and joint torques to hand forces with its transpose.

• Before
  M6-T7: The Moore-Penrose Pseudoinverse

  Build the pseudoinverse from the SVD and use it to solve rank-deficient and underdetermined systems.

• Lab
  M8-L1: Interactive Robotic Arm and Inverse Kinematics

  Drive a two-link arm to targets by inverse kinematics, watch the pseudoinverse blow up near a singularity, and fix it with damping.