M8-T10 — Redundant Manipulators

Learning objectives

After this lesson, you should be able to:

• Compute the degree of redundancy n − m and relate it to the null space of J and to the self-motion manifold.
• Enumerate the one-parameter family of inverse-kinematics solutions of a planar three-link arm and compare members by manipulability.
• Choose among solutions using optimisation criteria: minimum joint motion, manipulability, joint limits, obstacle clearance.
• Describe how redundancy changes singularity behaviour and what extra control authority it gives.

Introduction and motivation

Industrial arms have six joints because six are needed for an arbitrary pose. Collaborative arms have seven because they work among people and clutter: the same gripper pose can be reached with the elbow high, low or sideways. Mobile manipulators add the base's three degrees of freedom on top of the arm's six. All of these are redundant, and every control, planning and calibration pipeline must decide what to do with the spare freedom.

The intuition

Fix the hand at a point. A two-link arm has two (mirror) shapes. Add a third link and a new freedom appears: the hand orientation, or equivalently how the wrist is held, can vary smoothly while the hand stays on the point. The set of all such arm shapes is a curve in joint space, the self-motion manifold; the null-space slider in the figure walks along it. Redundancy is choosing a point on that curve, now and at every moment of the motion.

Geometric and physical interpretation

Geometric view. The set of joint vectors reaching a given hand pose is a smooth manifold of dimension n − m (where J has full row rank): a curve for single redundancy, a surface for double. Its tangent space at q is the null space of J(q). Different components of the manifold are different “branches” (elbow up versus down). Physical view. Human arms are redundant: shoulder 3 + elbow 1 + wrist 3 = 7 joints for a 6-D hand pose, and the swivel of the elbow around the shoulder-to-wrist line is the one-parameter freedom we use to reach around obstacles.

The mathematics

n

number of joints; m: task dimension.

n − m

the degree of redundancy (at least n − rank J at any configuration).

M(p)

the self-motion manifold {q : f(q) = p}.

w(q)

manipulability √det(J Jᵀ), a common criterion to maximise.

Degrees of redundancy

dim null(J) = n − rank J ≥ n − m,    dim M(p) = n − m  (J of full row rank)

What it means. Rank-nullity (Module 3): the null space of J has dimension n minus the rank. At regular points this equals n − m, and the solutions of f(q) = p form a manifold of that dimension.

Complete velocity solution

q̇ = J⁺ ṗ + ( I − J⁺J ) z

What it means. The particular minimum-norm solution plus any null-space motion. Choosing z is choosing the secondary behaviour.

Optimisation criteria for z

z = k ∇H(q),   H = w(q)  (maximise)   or   H = −Σ((qᵢ − qᵢ,mid)/range)² (stay mid-range)   or   −1/dobstacle²

What it means. Gradient ascent on a quality measure H(q) inside the null space, so the hand is untouched. Combine several criteria with weights or priorities.

Redundancy n − m gives an (n − m)-dimensional family of arm shapes per hand pose. Pick the member by optimising a secondary criterion in the null space.

Step-by-step derivation: the solution family of a planar three-link arm

1. Free parameter. The hand position p = (x, y) of a three-link planar arm is two equations in three unknowns (q₁, q₂, q₃). Let φ = q₁ + q₂ + q₃ be the absolute orientation of the last link; treat it as the free parameter.
2. Reduce to a two-link problem. The wrist point is w = p − l₃(cos φ, sin φ), a function of the free parameter φ. The first two links must reach w: the closed-form two-link inverse kinematics applies, giving (q₁, q₂) (elbow up or down).
3. Recover the last joint. q₃ = φ − q₁ − q₂. Each φ for which w lies in the annulus of the first two links gives a valid configuration: the solution set is a curve in joint space parametrised by φ (two branches for the two elbow signs).
4. Tangent equals null space. Differentiating f(q(φ)) = p gives J (dq/dφ) = 0: the tangent of the curve is a null vector of J, consistent with the null-space lesson.
5. Compare members. Evaluate a criterion along the curve. Manipulability w = √det(JJᵀ) or the smallest singular value σ_min are computed from J at each member; joint-limit and obstacle costs are computed from q.
6. Local versus global. The velocity-level method (J⁺ṗ + Nz) climbs a criterion locally, step by step. Planning across the whole manifold (for example to switch elbow branch) is a global problem solved with sampling or optimisation over the family parameter.
7. Singularities of a redundant arm. Rank loss now requires J to lose rank as an m × n matrix, which needs all the columns to be parallel, not just two of them. It is a much smaller set: redundancy typically allows the arm to move around a singularity inside the manifold rather than through it.
8. Weighted preference. If some joints are costly (a heavy shoulder), minimise q̇ᵀWq̇ instead: J⁺_W = W⁻¹Jᵀ(JW⁻¹Jᵀ)⁻¹, which spends motion on cheap joints first.

Mobile manipulators. With a holonomic base the combined Jacobian is [J_base J_arm]; the null space then includes motions where the base moves and the arm compensates. A weighted pseudoinverse with larger weights on the base makes the arm do most of the work for small motions and the base take over for large ones. For a non-holonomic base the base velocity has only two independent inputs, so the effective Jacobian columns are those of (v, ω).

The elbow swivel. For a 7-DOF arm with a spherical shoulder and wrist, the hand pose fixes the shoulder-to-wrist line; the elbow can swivel around that line by an angle ψ, the single redundant parameter. Specifying ψ gives a closed-form solution, an alternative to the numerical null-space approach.

A worked example: one hand point, four arm shapes

Planar three-link arm, links 1.1, 0.8, 0.7. The hand must reach p = (1.5, 1.0). Choose the free orientation φ of the last link and solve the two-link inverse kinematics for the wrist point.

• φ = 0°. Wrist at (0.8, 1.0). Solution q = (13.0°, 96.9°, −109.9°); hand (1.500, 1.000) ✓; manipulability w = 1.12.
• φ = 30°. Wrist at (0.894, 0.65). q = (−6.5°, 110.9°, −74.4°); w = 1.39.
• φ = 60°. q = (−21.1°, 102.2°, −21.1°); w = 1.73.
• φ = 90°. q = (−18.8°, 73.8°, 35.0°); w = 1.92, the best of these four: the singular values are 2.25 and 0.853 (condition 2.6) against 2.05 and 0.548 (condition 3.7) at φ = 0°.
• Same hand, different shapes. All four place the hand at (1.5, 1.0) exactly; they differ in elbow height, in distance from the joint limits and in dexterity. A planner that wants dexterity picks φ near 90° here.
• Counting. n = 3, m = 2 gives one free parameter (φ), as the family above shows; the Jacobian has rank 2 at all four configurations, so the null space is one-dimensional everywhere.
• Mobile manipulator. A holonomic base (2 translations + yaw) plus the 3-link arm for a planar position task has n = 6, m = 2: four extra degrees of freedom, enough to optimise base placement, elbow posture and wrist orientation together.

Robotics application

Collaborative and humanoid arms. Seven-joint arms use elbow-swivel or null-space control to avoid obstacles and humans while keeping the gripper pose; humanoid whole-body controllers solve stacks of prioritised tasks in nested null spaces.

Mobile manipulation. Warehouse and household robots combine base and arm; base placement for a grasp is chosen by maximising manipulability over the base degrees of freedom; see frames, poses and footprints for the base geometry.

UAVs. Aerial manipulators have body (6) plus arm joints; over-actuated multirotors have more control inputs than body degrees of freedom; allocation is a null-space problem; see UAV coordinate frames and fault detection, isolation and recovery for reallocation after a failure.

Sensing and SLAM. A sensor on a redundant arm can be placed in many poses for the same task, so view planning and gauge-free optimisation (the null space of the information matrix) follow the same pattern; see pose graphs and LiDAR point clouds for SLAM.

Engineering implications and limitations

Practical rules

• Decide the secondary objective explicitly. Without one, the pseudoinverse alone makes the arm drift in the null space unpredictably (the self-motion is not conservative), and repeated cycles may not return it to its start.
• Prefer continuity. Warm-start from the previous configuration so the arm stays on one branch and does not jump between elbow-up and elbow-down.
• Keep margins to joint limits. Use the null space to centre joints; hard limits are enforced by clamping with the task re-solved in the remaining joints.
• Use priorities. Safety first (collision, limits), then the hand task, then comfort; solve nested null-space problems.
• Mind the weighting. Choose W so that “minimum motion” is physically meaningful (heavier joints cost more) and scale units.
• Test cyclic tasks. A closed hand path should return the joints to their start; if not, add a posture-restoring null-space term.

Limits

Local null-space methods get stuck in local optima of the secondary criterion and cannot switch branches. Global planning over the manifold is needed for large reconfigurations, and every method relies on an accurate Jacobian and kinematic model.

Common misconceptions

Engineering takeaways

1. Redundancy n − m is the dimension of the family of arm shapes for a hand pose; at regular points null(J) has dimension n − rank J ≥ n − m.
2. Solutions form a self-motion manifold: for the three-link arm a curve parametrised by the free orientation φ.
3. Velocity-level control is q̇ = J⁺ṗ + (I − J⁺J)z; choose z by gradient of a criterion (manipulability, joint-limit distance, obstacle clearance).
4. Compare solutions with manipulability or σ_min, use weights for unequal joint cost, and warm-start for continuity.
5. Redundancy buys avoidance and dexterity but demands explicit secondary objectives and global planning for large changes.

References and further reading

Primary papers and authoritative documentation.

• Modern Robotics (Lynch and Park), free preprint: chapter 6.3, kinematically redundant arms
• Buss, Introduction to Inverse Kinematics with Jacobian Transpose, Pseudoinverse and Damped Least Squares: redundant systems, null space and singularity handling in IK; the site did not load when we last checked
• Michigan ROB 101: Computational Linear Algebra: inverse kinematics for redundant systems
• UCSD ECE276A: Sensing and Estimation in Robotics: redundancy resolution and optimisation in kinematics

Knowledge check

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What’s next

Module 8 gave you the geometry and linear algebra of motion with certainty. Module 9 adds uncertainty: sensors are noisy, so every estimate comes with a covariance matrix, and the same matrices (symmetric, positive definite, with eigen-decompositions) describe how uncertain we are.

Related content

Nothing below is required to finish this lesson.

• Before
  M8-T9: Null-Space Control

  Use the null space of the Jacobian to pursue a secondary goal without disturbing the hand.