Kinematic Redundancy

Definition

A manipulator is kinematically redundant when its number of actuated joints $n$ exceeds the dimension $m$ of the task it must accomplish, so that the inverse-kinematics map admits a continuum of joint-space solutions (the self-motion manifold) for a given task. The surplus $r=n-m>0$ is the degree of redundancy. In the space-robot setting the relevant task dimension is enlarged by the base: Sone et al. (2016) note that “the reactionless motion control task requires a manipulator arm with at least nine DoFs: six DoFs for the end effector task plus three-DoFs for base rotation.” Their study instead uses a seven-DOF arm against a reduced orientation-only end-effector task, so that the arm “comprises one degree of redundancy w.r.t. the end-effector and base rotation tasks, each of them requiring three-DoFs” ($7-3-3=1$). Redundancy is what makes kinematic_redundancy_resolution and null-space-based secondary objectives (e.g. reaction_null_space motion, task_priority_redundancy) possible.

Key Equations

Symbols per notation.md.

For an $m$-dimensional task with Jacobian $\mathbf{J}\in {\mathbb{R}}^{m\times n}$, the redundant joint solution is the particular (minimum-norm) solution plus an arbitrary self-motion in the Jacobian kernel:

$\dot{\mathbf{q}}={\mathbf{J}}^{+}\dot{\mathbf{x}}+(\mathbf{E}-{\mathbf{J}}^{+}\mathbf{J}){\dot{\mathbf{q}}}_a,r=n-m>0,$

where ${\mathbf{J}}^{+}$ is the pseudoinverse, $\mathbf{E}$ the identity, ${\dot{\mathbf{q}}}_a$ an arbitrary vector, and $\mathbf{N}=\mathbf{E}-{\mathbf{J}}^{+}\mathbf{J}$ the null-space projector (notation.md $\mathbf{N}$). Sone’s reaction-null-space motion is the special case in which the task Jacobian is the coupling inertia matrix ${\mathbf{M}}_{bm}\equiv {\tilde{\mathbf{M}}}_{\omega m}$ and the secondary motion lives entirely in its kernel:

$\dot{\mathbf{\theta }}={\mathbf{P}}_{\mathrm{RNS}}{\dot{\mathbf{\theta }}}_a,{\mathbf{P}}_{\mathrm{RNS}}=\mathbf{E}-{\tilde{\mathbf{M}}}_{\omega m}^{+}{\tilde{\mathbf{M}}}_{\omega m}.$

(Here $\mathbf{\theta }$ is Sone’s joint vector, the source-local name for $\mathbf{q}$.)

Source Support

• sone2016reactionless — exploits the one-DOF redundancy of a seven-DOF arm to perform a reactionless camera-inspection task; states the $\ge 9$-DOF requirement for reactionless end-effector tracking and layers prioritized subtasks (reactionless constraint, wrist orientation, wrist-position stabilization) through consecutive null-space projections.

Related Topics

• kinematic_redundancy_resolution — the procedure that uses this surplus to pick a unique joint solution; redundancy is its precondition.
• redundancy_resolution — the general (not necessarily kinematic) resolution problem; same surplus DOFs, broader objective set.
• reaction_null_space — the specific kernel (of the coupling inertia matrix) that redundant DOFs are projected into to obtain reactionless arm motion.
• task_priority_redundancy — the layered scheme by which the redundant DOFs realize several ordered subtasks without lower-priority tasks disturbing higher ones.
• kinematic_singularity — where the task Jacobian loses rank; redundancy can help avoid configuration (kinematic) singularities but introduces algorithmic_singularity when prioritized subtasks conflict.

Open Questions

• Sone treats the FFSR as free-floating ($\dot{\mathbf{\varphi }}=0$, reaction wheels off) for the RNS redundancy analysis, even though the platform physically carries wheels (a free-flying-adjacent system). For our free-flying robot the 6-DOF base is actuated, so base attitude need not be paid for out of arm redundancy — does the ”$\ge 9$ DOF for reactionless tracking” budget collapse, freeing the arm’s redundant DOF entirely for end-effector and secondary tasks?
• Reactionless motion is confined to an initial-state-dependent manifold in joint space; how much of the arm’s nominal redundancy survives that constraint, and does an actuated base remove the manifold restriction altogether?
• The prioritized null-space scheme makes the restricted Jacobian rank-deficient (an algorithmic singularity) when end-effector and reactionless subtasks conflict — for an actuated base is this conflict still present, or does it dissolve once base attitude is no longer a competing redundant-DOF claimant?